Bhagya

Punjab State Board PSEB 8th Class Maths Book Solutions Chapter 9 Algebraic Expressions and Identities InText Questions and Answers.

PSEB 8th Class Maths Solutions Chapter 9 Algebraic Expressions and Identities InText Questions

Try These: [Textbook Page No. 139]

1. Write two terms which are like:

Question (i)
7xy
Solution:
14xy, 21xy

Question (ii)
4mn²
Solution:
8mn² , – 11mn²

Question (iii)
21
Solution:
– 5l, 9l

Try These : [Textbook Page No. 142]

1. Can you think of two more such situations, where we may need to multiply algebraic expressions?
Solution:
1. Aarush purchased x notebooks and y pens. If cost of a notebook and a pen is same ₹ z, what amount has he to pay? → ₹ z (x + y)
2. Shailja wants to spread a carpet in her room having length (l + 5) m and breadth (b – 2) m. Find the area of the carpet. → (l + 5) (b – 2) m²

Try These: [Textbook Page No. 143]

1. Find 4x × 5y × 7z.
Solution:
4x × 5y × 7z = 4 × 5 × 7 × x × y × z
= 140 xyz

2. Find 4x × 5y × 7z. First find (4x × 5y) and multiply it by 7z; or first find (5y × 7z) and multiply it by 4x. Is the result the same? What do you observe? Does the order in which you carry out the multiplication matter?
Solution:
(4x × 5 y) = 4 × 5 × x × y
= 20xy
Now, 20xy × 7z = 20 × 7 × xy × z
= 140xyz … (i)
Also, (5y × 7z) = 5 × 7 × y × z = 35 yz
Now, 35yz × 4x = 35 × 4 × yz × x
= 140xyz … (ii)
Yes, the result is same.
We can conclude that product remains same if we change order of the terms.

3. Complete the table for area of a rectangle with given length and breadth.
Solution:

length	breadth	area
3x	5y	3x × 5y = 15xy
9y	4y2	9y × 4y2 = 36y3
4ab	5bc	4ab × 5be = 20ab2c
2l2m	3lm2	2l2m × 3lm2 = 6l3m3

Try These : [Textbook Page No. 144]

1. Find the product:

Question (i)
2x (3x + 5xy)
Solution:
= (2x × 3x) + (2x × 5xy)
= 6x² + 10x²y

Question (ii)
a² (2ab – 5c)
Solution:
= (a² × 2ab) – (a² × 5c)
= 2a³b – 5a²c

Try These: [Textbook Page No. 145]

1. Find the product: (4p² + 5p + 7) × 3p
Solution:
(4p² + 5p + 7) × 3p
= (4p² × 3p) + (5p × 3p) + (7 × 3p)
= 12p³ + 15p² + 21p

Try These : [Textbook Page No. 149]

1. Put -b in place of b in Identity (I). Do you get Identity (II)?
Solution:
Identity (I): (a + b)² = a² + 2ab + b²
Let us put (- b) instead of b [a + (- b)]²
= a² + 2a (- b) + (- b)²
∴ (a – b)²
= a² – 2ab + b²
Identity (II): (a – b)² = a² – 2ab + b²
Yes, we get Identity (II).

Try These : [Textbook Page No. 149]

1. Verify Identity (IV), for a = 2, b = 3, x = 5.
Solution:
Identity (IV):
(x + a) (x + b) = x² + (a + b) x + ab
Substitute a = 2, b = 3 and x = 5
LHS
= (x + a) (x + b)
= (5 + 2) (5 + 3)
= (7)(8)
= 56

RHS
= x² + (a + b) x + ab
= (5)² + (2 + 3) × 5 + (2 × 3)
= 25 + (5) × 5 + (6)
= 25 + 25 + 6 = 56
∴ LHS = RHS
∴ The given identity is true for the given values.

2. Consider, the special case of Identity (IV) with a = b, what do you get ? Is it related to Identity (I)?
Solution:
When a = b (∴ Take y for both)
(x + a) (x + b) = x² + (a + b) x + ab
Substitute a = y and b = y
(x + y)(x + y) = x² + (y + y)x + (y × y)
= x² + (2y) x + (y × y) = x² + 2xy + y²
∴ Yes, it is the same as Identity ( I).

3. Consider, the special case of Identity (IV) with a = -c and b = -c. What do you get ? Is it related to Identity (II) ?
Solution:
Identity (IV):
(x + a)(x + b) = x² + (a + b) x + ab
Substitute (- c) instead of a and (- c) instead of b,
(x – c) (x – c)
= x² + [(-c) + (-c)]x + [(-c) × (-c)]
= x² + [- 2c] x + (c²)
= x² – 2cx + c²
∴ Yes, it is the same as Identity (II).

4. Consider the special case of Identity (IV) with b = – a. What do you get ? Is it related to Identity (III)?
Solution :
Identity (IV):
(x + a) (x + b) = x² + (a + b) x + ab
Substitute (-a) instead of b,
(x + a) (x – a)
= x² + [a + (- a)] x + [a × (- a)]
= x² + (a – a) x + [- a²]
= x² + (0) x – a²
= x² – a²
∴ Yes, it is the same as Identity (III).

Punjab State Board PSEB 8th Class Maths Book Solutions Chapter 7 Cubes and Cube Roots Ex 7.1 Textbook Exercise Questions and Answers.

PSEB Solutions for Class 8 Maths Chapter 7 Cubes and Cube Roots Ex 7.1

1. Which of the following numbers are not perfect cubes ?

Question (i).
216
Solution:
216
\(\begin{array}{l|l}
2 & 216 \\
\hline 2 & 108 \\
\hline 2 & 54 \\
\hline 3 & 27 \\
\hline 3 & 9 \\
\hline 3 & 3 \\
\hline & 1
\end{array}\)
216 = 2 × 2× 2 × 3 × 3 × 3
Here, the prime factors 2 and 3 appear in a group of three (triples).
∴ 216 is a perfect cube.
216 = 2³ × 3³

Question (ii).
128
Solution:
\(\begin{array}{l|l}
2 & 128 \\
\hline 2 & 64 \\
\hline 2 & 32 \\
\hline 2 & 16 \\
\hline 2 & 8 \\
\hline 2 & 4 \\
\hline 2 & 2 \\
\hline & 1
\end{array}\)
128 = 2 × 2 × 2 × 2 × 2 × 2 × 2
Here, the prime factor 2 does not appear in a group of three.
∴ 128 is not a perfect cube.

Question (iii).
1000
Solution:
\(\begin{array}{l|l}
2 & 1000 \\
\hline 2 & 500 \\
\hline 2 & 250 \\
\hline 5 & 125 \\
\hline 5 & 25 \\
\hline 5 & 5 \\
\hline & 1
\end{array}\)
1000 = 2 × 2 × 2 × 5 × 5 × 5
Here, the prime factors 2 and 5 appear in a group of three.
∴ 1000 is a perfect cube.
1000 = 2³ × 5³

Question (iv).
100
Solution:
\(\begin{array}{l|l}
2 & 100 \\
\hline 2 & 50 \\
\hline 5 & 25 \\
\hline 5 & 5 \\
\hline & 1
\end{array}\)
100 = 2 × 2 × 5 × 5
Here, the prime factors 2 and 5 do not appear in a group of three.
∴ 100 is not a perfect cube.

Question (v).
46656
Solution:
\(\begin{array}{l|l}
2 & 46656 \\
\hline 2 & 23328 \\
\hline 2 & 11664 \\
\hline 2 & 5832 \\
\hline 2 & 2916 \\
\hline 2 & 1458 \\
\hline 3 & 729 \\
\hline 3 & 243 \\
\hline 3 & 81 \\
\hline 3 & 27 \\
\hline 3 & 9 \\
\hline 3 & 3 \\
\hline & 1
\end{array}\)
46656 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3 × 3 × 3 × 3
Here, the prime factors 2 and 3 appear in a group of three.
∴ 46656 is a perfect cube.
46656 = 2³ × 2³ × 3³ × 3³

2. Find the smallest number by which each of the following numbers must be multiplied to obtain a perfect cube:

Question (i).
243
Solution:
\(\begin{array}{l|l}
3 & 243 \\
\hline 3 & 81 \\
\hline 3 & 27 \\
\hline 3 & 9 \\
\hline 3 & 3 \\
\hline & 1
\end{array}\)
243 = 3 × 3 × 3 × 3 × 3
Here, the prime factor 3 does not appear in a group of three.
∴ 243 is not a perfect cube.
(243) × 3 = (3 × 3 × 3 × 3 × 3) × 3
∴ 729 = 3³ × 3³
Thus, 3 is the smallest number by which 243 must be multiplied to get a perfect cube.

Question (ii).
256
Solution:
\(\begin{array}{l|l}
2 & 256 \\
\hline 2 & 128 \\
\hline 2 & 64 \\
\hline 2 & 32 \\
\hline 2 & 16 \\
\hline 2 & 8 \\
\hline 2 & 4 \\
\hline 2 & 2 \\
\hline & 1
\end{array}\)
256 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2
Here, the prime factor 2 does not appear in a group of three.
∴ 256 is not a perfect cube.
(256) × 2
= (2 × 2 × 2 × 2 × 2 × 2 × 2 × 2) × 2
∴ 512 = 2³ × 2³ × 2³
Thus, 2 is the smallest number by which 256 must be multiplied to get a perfect cube.

Question (iii).
72
Solution:
\(\begin{array}{l|l}
2 & 72 \\
\hline 2 & 36 \\
\hline 2 & 18 \\
\hline 3 & 9 \\
\hline 3 & 3 \\
\hline & 1
\end{array}\)
72 = 2 × 2 × 2 × 3 × 3
Here, the prime factor 3 does not appear in a group of three.
∴ 72 is not a perfect cube.
(72) × 3
= (2 × 2 × 2 × 3 × 3) × 3
∴ 216 = 2³ × 3³
Thus, 3 is the smallest number by which 72 must be multiplied to get a perfect cube.

Question (iv).
675
Solution:
\(\begin{array}{l|l}
3 & 675 \\
\hline 3 & 225 \\
\hline 3 & 75 \\
\hline 5 & 25 \\
\hline 5 & 5 \\
\hline & 1
\end{array}\)
∴ 675 = 3 × 3 × 3 × 5 × 5
Here, the prime factor 5 does not appear in a group of three.
∴ 675 is not a perfect cube.
(675) × 5
= (3 × 3 × 3 × 5 × 5) × 5
∴ 3375 = 3³ × 5³
Thus, 5 is the smallest number by which 675 must be multiplied to get a perfect cube.

Question (v).
100
Solution:
\(\begin{array}{l|l}
2 & 100 \\
\hline 2 & 50 \\
\hline 5 & 25 \\
\hline 5 & 5 \\
\hline & 1
\end{array}\)
100 = 2 × 2 × 5 × 5
Here, the prime factors 2 and 5 do not appear in a group of three.
∴ 100 is not a perfect cube.
(100) × 2 × 5 = (2 × 2 × 5 × 5) × 2 × 5
∴ 1000 = 2³ × 5³
Thus, 10 is the smallest number by which 100 must be multiplied to get a perfect cube.

3. Find the smallest number by which each of the following numbers must be divided to obtain a perfect cube:

Question (i).
81
Solution:
\(\begin{array}{l|l}
3 & 81 \\
\hline 3 & 27 \\
\hline 3 & 9 \\
\hline 3 & 3 \\
\hline & 1
\end{array}\)
81 = 3 × 3 × 3 × 3
Here, the prime factor 3 is left over after making triples of 3.
∴ 81 is not a perfect cube.
(81) ÷ 3 = (3 × 3 × 3 × 3) ÷ 3
∴ 27 = 3 × 3 × 3 = 33
Thus, 3 is the smallest number by which 81 must be divided to get a perfect cube.

Question (ii).
128
Solution:
\(\begin{array}{l|l}
2 & 128 \\
\hline 2 & 64 \\
\hline 2 & 32 \\
\hline 2 & 16 \\
\hline 2 & 8 \\
\hline 2 & 4 \\
\hline 2 & 2 \\
\hline & 1
\end{array}\)
128 = 2 × 2 × 2 × 2 × 2 × 2 × 2
Here, the prime factor 2 is left over after making triples of 2.
∴ 128 is not a perfect cube.
(128) ÷ 2
= (2 × 2 × 2 × 2 × 2 × 2 × 2) ÷ 2
∴ 64 = 2 × 2 × 2 × 2 × 2 × 2
= 2³ × 2³
Thus, 2 is the smallest number by which 128 must be divided to get a perfect cube.

Question (iii).
135
Solution:
\(\begin{array}{l|l}
3 & 135 \\
\hline 3 & 45 \\
\hline 3 & 15 \\
\hline 5 & 5 \\
\hline & 1
\end{array}\)
135 = 3 × 3 × 3 × 5
Here, the prime factor 5 is left over after making triples of 3.
∴ 135 is not a perfect cube.
(135) ÷ 5 = (3 × 3 × 3 × 5) ÷ 5
∴ 27 = 3 × 3 × 3
= 3³
Thus, 5 is the smallest number by which 135 must be divided to get a perfect cube.

Question (iv).
192
Solution:
\(\begin{array}{l|l}
2 & 192 \\
\hline 2 & 96 \\
\hline 2 & 48 \\
\hline 2 & 24 \\
\hline 2 & 12 \\
\hline 2 & 6 \\
\hline 3 & 3 \\
\hline & 1
\end{array}\)
192 = 2 × 2 × 2 × 2 × 2 × 2 × 3
Here, the prime factor 3 is left over after making triples of 2.
∴ 192 is not a perfect cube.
(192) ÷ 3
= (2 × 2 × 2 × 2 × 2 × 2 × 3) ÷ 3
∴ 64 = 2 × 2 × 2 × 2 × 2 × 2
= 2³ × 2³
Thus, 3 is the smallest number by which 192 must be divided to get a perfect cube.

Question (v).
704
Solution:
\(\begin{array}{l|l}
2 & 704 \\
\hline 2 & 352 \\
\hline 2 & 176 \\
\hline 2 & 88 \\
\hline 2 & 44 \\
\hline 2 & 22 \\
\hline 11 & 11 \\
\hline & 1
\end{array}\)
704 = 2 × 2 × 2 × 2 × 2 × 2 × 11
Here, the prime factor 11 is left over after making triples of 2.
∴ 704 is not a perfect cube.
(704) ÷ 11
= (2 × 2 × 2 × 2 × 2 × 2 × 11) ÷ 11
∴ 64 = 2 × 2 × 2 × 2 × 2 × 2
= 2³ × 2³
Thus, 11 is the smallest number by which 704 must be divided to get a perfect cube.

4. Parikshit makes a cuboid of plasticine of sides 5 cm, 2 cm, 5 cm. How many such cuboids will he need to form a cube ?
Solution:
Sides of the cuboid are 5 cm, 2 cm and 5 cm (Given).
∴ Volume of the cuboid = l × b × h
= 5 cm × 2 cm × 5 cm
To form it as a cube, its dimensions should be in the group of triplets.
Here, the prime factors 2 and 5 are not in group of three.
∴ Volume of the required cube
= (5 cm × 5 cm × 2 cm) × 5 cm × 2 cm × 2 cm
= (5³ × 2³) cm³
Thus, Parikshit needed 5 × 2 × 2 = 20 cuboids to form a cube.

Punjab State Board PSEB 8th Class Maths Book Solutions Chapter 9 Algebraic Expressions and Identities Ex 9.5 Textbook Exercise Questions and Answers.

PSEB Solutions for Class 8 Maths Chapter 9 Algebraic Expressions and Identities Ex 9.5

1. Use a suitable identity to get each of the following products:

Question (i)
(x + 3) (x + 3)
Solution:
= (x + 3)²
= (x)² + 2(x)(3) + (3)²
[∵ (a + b)² = a² + 2ab + b²]
= x² + 6x + 9

Question (ii)
(2y + 5) (2y + 5)
Solution:
= (2y + 5)²
= (2y)² + 2 (2y)(5) + (5)²
[∵ (a – b)² = a² – 2ab + b²]
= 4y² + 20y + 25

Question (iii)
(2a – 7) (2a – 7)
Solution:
= (2a – 7)²
= (2a)² – 2(2a)(7) + (7)²
[∵ (a – b)² = a² – 2ab + b²]
= 4a² – 28a + 49

Question (iv)
(3a – \(\frac {1}{2}\))(3a – \(\frac {1}{2}\))
Solution:
= (3a – \(\frac {1}{2}\))²
= (3a)² – 2(3a)(\(\frac {1}{2}\)) + (\(\frac {1}{2}\))²
[∵ (a – b)² = a² – 2ab + b²]
= 9a² – 3a + \(\frac {1}{4}\)

Question (v)
(1.1m – 0.4) (1.1m + 0.4)
Solution:
= (1.1m)² – (0.4)²
[∵ (a + b) (a – b) = a² – b²]
= 1.21m² – 0.16

Question (vi)
(a² + b²) (-a² + b²)
Solution:
= (b² + a²) (b² – a²)
= (b²)² – (a²)²
[∵ (a + b) (a – b) = a² – b²]
= b⁴ – a⁴

Question (vii)
(6x – 7) (6x + 7)
Solution:
= (6x)² – (7)²
[∵ (a + b) (a – b) = a² – b²]
= 36x² – 49

Question (viii)
(-a + c) (-a + c)
Solution:
= (-a + c)²
= (-a)² + 2 (-a) (c) + (c)²
[∵ (a + b)² = a² + 2ab + b²]
= a² – 2ac + c²

Question (ix)
(\(\frac{x}{2}+\frac{3 y}{4}\))
Solution:
= (\(\frac{x}{2}+\frac{3 y}{4}\))²
= (\(\frac {x}{2}\))² + 2(\(\frac {x}{2}\))(\(\frac {3y}{4}\)) + (\(\frac {3y}{4}\))²
[∵ (a + b)² = a² + 2ab + b²]
= \(\frac{x^{2}}{4}+\frac{3 x y}{4}+\frac{9 y^{2}}{16}\)

Question (x)
(7a – 9b) (7a – 9b)
Solution:
= (7a – 9b)²
= (7a)² – 2(7a)(9b) + (9b)²
[∵ (a – b)² = a² – 2ab + bsup>2]
= 49a² – 126ab + 81b²

2. Use the identity (x + a) (x + b) = x² + (a + b) x + ab to find the following products:

Question (i)
(x + 3) (x + 7)
Solution:
Identity : (x + a) (x + b) = x² + (a + b) x + ab
= (x)² + (3 + 7)x + (3) (7)
= x² + (10) x + 21
= x² + 10x + 21

Question (ii)
(4x + 5) (4x + 1)
Solution:
= (4x)² + (5 + 1) 4x + (5)(1)
= 16x² + (6) 4x + 5
= 16x² + 24x + 5

Question (iii)
(4x – 5) (4x – 1)
Solution:
= (4x)² + (- 5 – 1) 4x + (- 5) (- 1)
= 16x² + (- 6) 4x + 5
= 16x² – 24x + 5

Question (iv)
(4x + 5) (4x- 1)
Solution:
= (4x)² + (5 – 1) 4x + (5) (- 1)
= 16x² + (4) 4x – 5
= 16x² + 16x – 5

Question (v)
(2x + 5y) (2x + 3y)
Solution:
= (2x)² + (5y + 3y) 2x + (5y) (3y)
= 4x² + (8y) 2x + 15y²
= 4x² + 16xy + 15y²

Question (vi)
(2a² + 9) (2a² + 5)
Solution:
= (2a²)² + (9 + 5) 2a² + (9)(5)
= 4a⁴ + (14)2a² + 45
= 4a⁴ + 28a² + 45

Question (vii)
(xyz – 4) (xyz – 2)
Solution:
= (xyz)² + (- 4 – 2) xyz + (- 4)(- 2)
= x²y²z² + (- 6) xyz + 8
= x²y²z² – 6xyz + 8

3. Find the following squares by using the identities:

Question (i)
(b – 7)²
Solution:
= (b)² – 2 (b)(7) + (7)²
[∵ (a – b)² = a² – 2ab + b²]
= b² – 14 b + 49

Question (ii)
(xy + 3z)²
Solution:
= (xy)² + 2 (xy)(3z) + (3z)²
[∵ (a + b)² = a² + 2ab + b²]
= x²y² + 6xyz + 9z²

Question (iii)
(6x² – 5y)²
Solution:
= (6x²)² – 2 (6x²) (5y) + (5y)²
[∵ (a – b)² = a² – 2ab + b²]
= 36x⁴ – 60x²y + 25y²

Question (iv)
(\(\frac {2}{3}\)m + \(\frac {3}{2}\)n)²
Solution:
= (\(\frac {2}{3}\)m)² + 2(\(\frac {2}{3}\)m)(\(\frac {3}{2}\)n) + (\(\frac {3}{2}\)n)²
[∵ (a + b)² = a² + 2ab + b²]
= \(\frac {4}{9}\)m² + 2mn + \(\frac {9}{4}\)n²

Question (v)
(0.4p – 0.5q)²
Solution:
= (0.4p)² – 2 (0.4p)(0.5q) + (0.5q)²
[∵ (a – b)² = a² – 2ab + b²]
= 0.16p² – 0.4pq + 0.25q²

Question (vi)
(2xy + 5y)²
Solution:
= (2xy)² + 2 (2xy)(5y) + (5y)²
[∵ (a + b)² = a² + 2ab + b²]
= 4x²y² + 20xy² + 25 y²

4. Simplify:

Question (i)
(a² – b²)²
Solution:
= (a²)² – 2(a²)(b²) + (b²)²
= a⁴ – 2a²b² + b⁴

Question (ii)
(2x + 5)² – (2x – 5)²
Solution:
= [(2x)² + 2(2x)(5) + (5)²] – [(2x)² – 2 (2x)(5) + (5)²]
= [4x² + 20x + 25] – [4x² – 20x + 25]
= 4x² + 20x + 25 – 4x² + 20x – 25
= 4x² – 4x² + 20x + 20x + 25 – 25
= 40x

Question (iii)
(7m – 8n)² + (7m + 8n)²
Solution:
= [(7m)² – 2(7m)(8n) + (8n)²] + [(7m)² + 2 (7m)(8n) + (8n)²]
= [49m² – 112mn + 64n²] + [49m² + 112mn + 64n²]
= 49m² – 112mn + 64n² + 49m² + 112mn + 64n²
= 49m² + 49m² – 112mn + 112mn + 64n² + 64n²
= 98m² + 128n²

Question (iv)
(4m + 5n)² + (5m + 4n)²
Solution:
= [(4m)² + 2 (4m)(5n) + (5n)²] + [(5m)² + 2 (5m)(4n) + (4n)²]
= 16m² + 40mn + 25n² + 25m² + 40mn + 16n²
= 16m² + 25m² + 40mn + 40 mn + 25n² + 16n²
= 41m² + 80mn + 41n²

Question (v)
(2.5p – 1.5q)² – (1.5p – 2.5q)²
Solution:
= [(2.5p)² – 2 (2.5p)(1.5q) + (1.5q)²] – [(1.5p)² – 2 (1.5p)(2.5q) + (2.5q)²]
= [6.25p² – 7.5pq + 2.25q²] – [2.25p² – 7.5pq + 6.25q²]
= 6.25p² – 7.5pq + 2.25q² – 2.25p² + 7.5pq – 6.25q²
= 6.25p² – 2.25p² – 7.5pq + 7.5pq + 2.25q² – 6.25q²
= 4p² – 4q²

Question (vi)
(ab + bc)² – 2ab²c
Solution:
= [(ab)² + 2 (ab)(bc) + (bc)²] – 2ab²c
= a²b² + 2ab²c + b²c² – 2ab²c
= a²b² + 2ab²c – 2ab²c + b²c²
= a²b² + b²c²

Question (vii)
(m² – n²m)² + 2m³n²
Solution:
= [(m²)² – 2 (m²)(n²m)² + (n²m)²] + 2m³n²
= m⁴ – 2 m³n² + n⁴m² + 2 m³ n²
= m⁴ – 2 m³n² + 2 m³n² + n⁴m²
= m⁴ + m²n⁴

5. Show that:

Question (i)
(3x + 7)² – 84x = (3x – 7)²
Solution:
LHS = (3x + 7)² – 84x
= (3x)² + 2(3x)(7) + (7)² – 84x
= 9x² + 42x + 49 – 84x
= 9x² + 42x – 84x + 49
= 9x² – 42x + 49

RHS = (3x – 7)²
= (3x)² – 2(3x)(7) + (7)²
= 9x² – 42x + 49
Thus, LHS = RHS
∴ (3x + 7)² – 84x = (3x – 7)²

Question (ii)
(9p – 5q)² + 180pq = (9p + 5q)²
Solution:
LHS = (9p – 5q)² + 180pq
= (9p)² – 2(9p)(5q) + (5q)² + 180pq
= 81p² – 90pq + 25q² + 180pq
= 81p² – 90pq + 180pq + 25q²
= 81p² + 90pq + 25q²

RHS = (9p + 5q)²
= (9p)² + 2(9p)(5q) + (5q)²
= 81p² + 90pq + 25q²
Thus, LHS = RHS
∴ (9p – 5q)² + 180pq = (9p + 5q)²

Question (iii)
(\(\frac {4}{3}\)m – \(\frac {3}{4}\)n)² + 2mn = \(\frac {16}{9}\)m² + \(\frac {9}{16}\)n²
Solution:
LHS = [\(\frac {4}{3}\)m – \(\frac {3}{4}\)n]² + 2mn
= [(\(\frac {4}{3}\)m)² – 2(\(\frac {4}{3}\)m)(\(\frac {3}{4}\)n) + (\(\frac {3}{4}\)n)²] + 2mn
= \(\frac {16}{9}\)m² – 2mn + \(\frac {9}{16}\)n² + 2mn
= \(\frac {16}{9}\)m² – 2mn + 2mn + \(\frac {9}{16}\)n²
= \(\frac {16}{9}\)m² + \(\frac {9}{16}\)n² = RHS
Thus, LHS = RHS
∴ (\(\frac {4}{3}\)m – \(\frac {3}{4}\)n)² + 2mn = \(\frac {16}{9}\)m² + \(\frac {9}{16}\)n²

Question (iv)
(4pq + 3q)² – (4pq – 3q)² = 48pq²
Solution:
LHS = (4pq + 3q)² – (4pq – 3q)²
= [(4pq)² + 2 (4pq)(3q) + (3q)²] – [(4pq)² – 2 (4pq)(3q) + (3q)²]
= [16p²q² + 24pq² + 9q²] – [16p²q² – 24pq² + 9q²]
= 16p²q² + 24pq² + 9q² – 16p²q² + 24pq² – 9q²
= 16p²q² – 16p²q² + 24pq² + 24pq² + 9q² – 9q²
= 48pq² = RHS
Thus, LHS = RHS
∴ (4pq + 3q)² – (4pq – 3q)² = 48pq²

Question (v)
(a – b) (a + b) + (b – c) (b + c) + (c – a) (c + a) = 0
Solution:
LHS = (a – b)(a + b) + (b – c)(b + c) + (c – a) (c + a)
= (a² – b²) + (b² – c²) + (c² – a²)
= a² – b² + b² – c² + c² – a²
= a² – a² + b² – b² + c² – c²
= 0 = RHS
Thus, LHS = RHS
∴ (a -b)(a + b) + (b- c) (b + c) + (c – a) (c + a) = 0

6. Using identities, evaluate:

Question (i)
71²
Solution:
= (70 + 1)²
= (70)² + 2 (70)(1) + (1)²
[∵ (a + b)² = a² + 2ab + b²]
= 4900 + 140 + 1
= 5041

Question (ii)
99²
Solution:
= (100 – 1)2
= (100)² – 2(100)(1) + (1)²
[∵ (a – b)² – a² – 2ab + b²]
= 10000 – 200 + 1
= 9801

Question (iii)
102²
Solution:
= (100 + 2)²
= (100)² + 2 (100)(2) + (2)²
[∵ (a + b)² = a² + 2ab + b²]
= 10000 + 400 + 4
= 10404

Question (iv)
998²
Solution:
= (1000 – 2)²
= (1000)² – 2 (1000)(2) + (2)²
[∵ (a – b)² = a² – 2ab + b²]
= 1000000 – 4000 + 4
= 996004

Question (v)
5.2²
Solution:
= (5 + 0.2)²
= (5)² + 2 (5)(0.2) + (0.2)²
[∵ (a + b)² = a² + 2ab + b²]
= 25 + 2 + 0.04
= 27 + 0.04
= 27.04

Question (vi)
297 × 303
Solution:
= (300 – 3) × (300 + 3)
= (300)² – (3)²
[∵ (a – b)(a + b) = a² – b²]
= 90000 – 9
= 89991

Question (vii)
78 × 82
Solution:
= (80 – 2) × (80 + 2)
= (80)² – (2)²
[∵ (a – b)(a + b) = a² – b²]
= 6400 – 4
= 6396

Question (viii)
8.9²
Solution:
= (9 – 0.1)²
= (9)² – 2(9)(0.1) + (0.1)²
[∵ (a – b)² = a² – 2ab + b²]
= 81 – 1.8 + 0.01
= 81.01 – 1.8
= 79.21

Question (ix)
10.5 × 9.5
Solution:
= (10 + 0.5) × (10 – 0.5)
= (10)² – (0.5)²
[∵ (a + b)(a – b) = a² – b²]
= 100 – 0.25
= 99.75

7. Using a² – b² = (a + b) (a – b), find:

Question (i)
51² – 49²
Solution:
= (51 + 49) (51 – 49)
= (100) × (2)
= 200

Question (ii)
(1.02)² – (0.98)²
Solution:
= (1.02 + 0.98) (1.02 – 0.98)
= (2.0) × (0.04)
= 0.08

Question (iii)
153² – 147²
Solution:
= (153 + 147) (153 – 147)
= (300) × (6)
= 1800

Question (iv)
12.1² – 7.9²
Solution:
= (12.1 + 7.9) (12.1 – 7.9)
= 20 × 4.2
= 84

8. Using (x + a)(x + b) = x² + (a + b) x + ab, find:

Question (i)
103 × 104
Solution:
= (100 + 3) × (100 + 4)
= (100)² + (3 + 4) × 100 + (3)(4)
= 10000 + 700 + 12
=10712

Question (ii)
5.1 × 5.2
Solution:
= (5 + 0.1) (5 + 0.2)
= (5)² + (0.1 + 0.2) × 5 + (0.1)(0.2)
= 25 + (0.3) × 5 + 0.02
= 25 + 1.5 + 0.02
= 26.52

Question (iii)
103 × 98
Solution:
= (100 + 3) (100-2)
= (100)² + (3 – 2) 100 + (3)(-2)
= 10000 + 100 – 6
= 10094

Question (iv)
9.7 × 9.8
Solution:
= (10 – 0.3) (10 – 0.2)
= (10)² + [(-0.3) + (-0.2)] 10 + (-0.3) (-0.2)
= 100 + [-0.5] × 10 + 0.06
= 100 – 5 + 0.06
= 95.06

Punjab State Board PSEB 8th Class Maths Book Solutions Chapter 8 Comparing Quantities Ex 8.1 Textbook Exercise Questions and Answers.

PSEB Solutions for Class 8 Maths Chapter 8 Comparing Quantities Ex 8.1

1. Find the ratio of the following.

Question (a).
Speed of a cycle 15 km per hour to the speed of scooter 30 km per hour.
Solution:
Speed of a cycle = 15 km/h
Speed of a scooter = 30 km / h
∴ Ratio of the speed of a cycle to the speed of a scooter
= \(\frac{15 \mathrm{~km} / \mathrm{h}}{30 \mathrm{~km} / \mathrm{h}}\)
= \(\frac {1}{2}\)
= 1 : 2

Question (b).
5 m to 10 km
Solution:
[Note : Unit of both quantities should be same.]
1 km = 1000 m
∴ 10 km = 10 × 1000 m
= 10,000 m
∴ Ratio of 5 m to 10 km = \(\frac{5 \mathrm{~m}}{10 \mathrm{~km}}\)
= \(\frac{5 \mathrm{~m}}{10000 \mathrm{~m}}\)
= \(\frac{1}{2000}\)
= 1 : 2000

Question (c).
50 paise to ₹ 5
Solution:
[Note : Unit of both quantities should be same.]
₹ 1 = 100 paise
∴ ₹ 5 = 500 paise
∴ Ratio of 50 paise to ₹ 5 = \(\frac{50 \text { paise }}{₹ 5}\)
= \(\frac{50 \text { paise }}{500 \text { paise }}\)
= \(\frac{1}{10}\)
= 1 : 10

2. Convert the following ratios to percentages.

Question (a).
3 : 4
Solution:
Given ratio = 3 : 4
∴ Percentage = (\(\frac{3}{4}\) × 100) %
= (3 × 25) %
= 75 %

Question (b).
2 : 3
Solution:
Given ratio = 2 : 3
∴ Percentage = (\(\frac{2}{3}\) × 100) %
= (\(\frac{200}{3}\)) %
= 66 \(\frac{2}{3}\)%

3. 72% of 25 students are interested in Mathematics. How many are not interested in Mathematics?
Solution:
Total number of students = 25
Students interested in Mathematics = 72%
∴ Students who are not interested in Mathematics = (100 – 72) %
= 28 %
Number of students who are not interested in Mathematics = 28% of 25
= \(\frac{28}{100}\) × 25
= \(\frac{28}{4}\)
= 7
Thus, 7 students are not interested in Mathematics.

4. A football team won 10 matches out of the total number of matches they played. If their win percentage was 40, then how many matches did they play in all ?
Solution:
Number of matches won by the football team = 10
Let x matches be played by the team.
∴ 40% of x = 10
∴ \(\frac{40}{100}\) × x = 10
∴ x = \(\frac{10 \times 100}{40}\)
= 25
Thus, the football team played 25 matches in all.

5. If Chameli had ₹ 600 left after spending 75% of her money, how much did she have in the beginning?
Solution:
Let Chameli had in the beginning ₹ x
Percentage of money spent by Chameli = 75 %
Percentage of money left with Chameli = (100 – 75)%
= 25%
But money left = ₹ 600 (Given)
∴ 25% of x = 600
∴ \(\frac{25}{100}\) × x = 600
∴ x = \(\frac{600 \times 100}{25}\)
∴ x = 2400
Thus, Chameli had ₹ 2400 in the beginning.

6. If 60% people in a city like cricket, 30% like football and the remaining like other games, then what per cent of the people like other games? If the total number of people is 50 lakh, find the exact number who like each type of game.
Solution:
Percentage of people who like cricket = 60 %
Percentage of people who like football = 30 %
∴ Percentage of people who like other games = [ 100 – (60 + 30)]%
= (100 – 90)%
= 10 %
Total number of people = 50,00,000 (Given)
Now,
People who like cricket
= 60% of 50,00,000
= \(\frac {1}{2}\) × 50,00,000
= 60 × 50000
= 3000000
= 30 lakh

People who like football
= 30% of 5000000
= \(\frac {30}{100}\) × 5000000
= 30 × 50000
= 1500000
= 15 lakh

People who like other games
= 10% of 5000000
= \(\frac {10}{100}\) × 5000000
= 500000
= 5 lakh
Thus, number of people who like
cricket = 30 lakh,
football = 15 lakh
and other games = 5 lakh

Punjab State Board PSEB 8th Class Maths Book Solutions Chapter 9 Algebraic Expressions and Identities Ex 9.2 Textbook Exercise Questions and Answers.

PSEB Solutions for Class 8 Maths Chapter 9 Algebraic Expressions and Identities Ex 9.2

1. Find the product of the following pairs of monomials:

Question (i)
4, 7p
Solution:
= 4 × 7p
= 4 × 7 × p
= 28p

Question (ii)
– 4p, 7p
Solution:
= – 4p × 7p
= (-4 × 7) × p × p
= – 28p²

Question (iii)
– 4p, 7pq
Solution:
= (- 4p) × 7pq
= – 4 × 7 × p × pq
= – 28p²q

Question (iv)
4p³, – 3p
Solution:
= 4p³ × (- 3p)
= 4 × (- 3) × p³ × p
= – 12p⁴

Question (v)
4p, 0
Solution:
= 4p × 0
= 0

2. Find the areas of rectangles with the following pairs of monomials as their s lengths and breadths respectively:
(p, q); (10m, 5n); (20x², 5y²); (4x, 3x²); (3mn, 4np)
Solution:
Area of a rectangle = length × breadth
(i) length = p, breadth = q
∴ Area = length × breadth
= p × q = pq unit²

(ii) length = 10m, breadth = 5n
∴ Area = length × breadth
= 10m × 5n
= 50mn unit²

(iii) length = 20x², breadth = 5y²
∴ Area = length × breadth
= 20x² × 5y²
= 100x²y² unit²

(iv) length = 4x, breadth 3x²
∴ Area = length × breadth
= 4x × 3x²
= 12x³ unit²

(v) length = 3mn, breadth = 4np
∴ Area = length × breadth
= 3mn × 4np
= 12 mn²p unit²

3. Complete the table of products:
Solution:

First monomial → Second monomial ↓	2x	-5 y	3x2	-4xy	7x2y	– 9x2y2
2x	4x2	-10xy	6x3	-8x2y	14x3y	– 18x3y2
– 5 y	– 10xy	25y2	– 15x2y	20xy2	– 35x2y2	45x2y3
3x2	6x3	– 15x2y	9x4	– 12x3y	21x4y	– 27x4y2
– 4xy	– 8x2y	20xy2	-12x3y	16x2y2	– 28x3y2	36x3y3
7x2y	14x3y	– 35x2y2	21x4y	– 28x3y2	49x4y2	– 63x4y3
– 9x2y2	– 18x3y2	45x2y3	– 27x4y2	36x3y3	– 63x4y3	81x4y4

4. Obtain the volume of rectangular boxes with the following length, breadth and height respectively:

Question (i)
5a, 3a², 7a⁴
Solution:
Volume of rectangular box = length × breadth × height
length = 5a, breadth = 3a², height = 7a⁴
∴ Volume = length × breadth × height
= 5a × 3a² × 7a⁴
= (5 × 3 × 7) × a × a² × a⁴
= 105a⁷ cubic unit

Question (ii)
2p, 4q, 8r
Solution:
length = 2p, breadth = 4q, height = 8r
∴ Volume = length × breadth × height
= 2p × 4q × 8r
= (2 × 4 × 8) × p × q × r
= 64pqr cubic unit

Question (iii)
xy, 2x²y, 2xy²
Solution:
length = xy, breadth = 2x²y, height = 2 xy²
∴ Volume = length × breadth × height
= xy × 2x²y × 2xy²
= (1 × 2 × 2) × xy × x²y × xy²
= 4x⁴y⁴ cubic unit

Question (iv)
a, 2b, 3c
Solution:
length = a, breadth = 2b, height = 3c
∴ Volume = length × breadth × height
= a × 2b × 3c
= (1 × 2 × 3) × a × b × c
= 6abc cubic unit

5. Obtain the product of:

Question (i)
xy, yz, zx
Solution:
= xy × yz × zx
= [x × y) × (y × z) × (z × x)
= x × x × y × y × z × z
= x²y²z²

Question (ii)
a, -a², a³
Solution:
= (a) × (- a²) × (a³)
= – a × a² × a³
= – a⁶

Question (iii)
2, 4y, 8y², 16y³
Solution:
= 2 × 4y × 8y² × 16y³
= 2 × 4 × 8 × 16 × y × y² × y³
= 1024y⁶

Question (iv)
a, 2b, 3c, 6abc
Solution:
= a × 2b × 3c × 6abc
= 1 × 2 × 3 × 6 × a × b × c × abc
= 36 a²b²c²

Question (v)
m, – mn, mnp
Solution:
= (m) × (- mn) × (mnp)
= – (m × mn × mnp)
= – m³n²p

Punjab State Board PSEB 8th Class Maths Book Solutions Chapter 7 Cubes and Cube Roots InText Questions and Answers.

PSEB 8th Class Maths Solutions Chapter 7 Cubes and Cube Roots InText Questions

Try These (Textbook Page No. 111)

Find the one’s digit of the cube of each of the following numbers:
(i) 3331
(ii) 8888
(iii) 149
(iv) 1005
(v) 1024
(vi) 77
(vii) 5022
(viii) 53
Solution:

Sl. No.	Number	Number ending in	Units place digit of the cube
(i)	3331	1	1
(ii)	8888	8	2
(iii)	149	9	9
(iv)	1005	5	5
(v)	1024	4	4
(vi)	77	7	3
(vii)	5022	2	8
(viii)	53	3	7

Some interesting patterns: (Textbook Page No. 111)

Observe the following pattern of sums of odd numbers.

Try These (Textbook Page No. 111)

1. Express the following numbers as the sum of odd numbers using the above pattern ?
(a) 6³
(b) 8³
(c) 7³
Solution:
From above pattern, we can conclude n³ = [n(n – 1) + 1] + [n(n – 1) + 3] + [n(n – 1) + 5]… + n terms
(a) 6³
Here, n = 6, n- 1=5
6 (6 – 1) + 1 → 6 × 5 + 1 → 31

OR
= [6(6 – 1) + 1] + [6(6 – 1) + 3] + [6(6 – 1) + 5] + [6(6 – 1) + 7] + [6(6 – 1) + 9] + [6(6 – 1) + 11]
= (6 × 5 + 1) + (6 × 5 + 3) + (6 × 5 + 5) + (6 × 5 + 7) + (6 × 5 + 9) + (6 × 5 + 11)
= (30 + 1) + (30 + 3) + (30 + 5) + (30 + 7) + (30 + 9) + (30 + 11)
= 31 +33 + 35 + 37 + 39 + 41
= 216

(b) 8³
Here, n = 8, n – 1 = 7
8 (8 – 1) + 1 → 8 × 7 + 1 → 57

OR
= [8(8 – 1) + 1] + (8(8 – 1) + 3] + [8(8 – 1) + 5] + [8(8 – 1) + 7] + [8(8 – 1) + 9] + [8(8 – 1) + 11] + [8(8 – 1) + 13] + [8(8 – 1) + 15]
= (8 × 7 + 1) + (8 × 7 + 3) + (8 × 7 + 5) + (8 × 7 + 7) + (8 × 7 + 9) + (8 × 7 + 11) + (8 × 7 + 13) + (8 × 7 + 15)
= (56 + 1) + (56 + 3) + (56 + 5) + (56 + 7) + (56 + 9) + (56 + 11) + (56 + 13) + (56 + 15)
= 57 + 59 + 61 + 63 + 65 + 67 + 69 + 71
= 512

(c) 7³
Here, n = 7, n – 1 = 6
7 × 6 + 1 → 42 + 1 → 43

OR
= [7(7 – 1) + 1] + [7(7 – 1) + 3] + [7(7 – 1) + 5] + [7(7 – 1) + 7] + [7(7 – 1) + 9] + [7(7 – 1) + 11] + [7(7 – 1) + 13]
= (7 × 6 + 1) + (7 × 6 + 3) + (7 × 6 + 5) + (7 × 6 + 7) + (7 × 6 + 9) + (7 × 6 + 11) + (7 × 6 + 13)
= (42 + 1) + (42 + 3) + (42 + 5) + (42 + 7) + (42 + 9) + (42 + 11) + (42 + 13)
= 43 + 45 + 47 + 49 + 51 + 53 + 55
= 343

Consider the following pattern:
2³ – 1³ = 1 + 2 × 1 × 3
3³ – 2³ = 1 + 3 × 2 × 3
4³ – 3³ = 1 + 4 × 3 × 3
Using the above pattern, find the value of the following:
(i) 7³ – 6³
(ii) 12³– 11³
(iii) 20³ – 19³
(iv) 51³ – 50³
Solution:
From above pattern, we can conclude
n³ – (n – 1)³ = 1 + n × (n – 1) × 3
(i) 7³ – 6³ = 1 + 7 × 6 × 3
= 1 + 126
= 127

(ii) 12³ – 11³ = 1 + 12 × 11 × 3
= 1 + 396
= 397

(iii) 20³ – 19³ = 1 + 20 × 19 × 3
= 1 + 1140
= 1141

(iv) 51³ – 50³ = 1 + 51 × 50 × 3
= 1 + 7650
= 7651

Try These (Textbook Page No. 112)

1. Which of the following are perfect cubes?

Question (1).
400
Solution:
\(\begin{array}{l|l}
2 & 400 \\
\hline 2 & 200 \\
\hline 2 & 100 \\
\hline 2 & 50 \\
\hline 5 & 25 \\
\hline 5 & 5 \\
\hline & 1
\end{array}\)
400 = 2 × 2 × 2 × 2 × 5 × 5
Here, the prime factors 2 and 5 do not appear in triples.
∴ 2 × 5 × 5 is left over.
∴ 400 is not a perfect cube.

Question (2).
3375
Solution:
\(\begin{array}{l|l}
3 & 3375 \\
\hline 3 & 1125 \\
\hline 3 & 375 \\
\hline 5 & 125 \\
\hline 5 & 25 \\
\hline 5 & 5 \\
\hline & 1
\end{array}\)
3375 = 3 × 3 × 3 × 5 × 5 × 5
Here, the prime factors 3 and 5 appear in triples.
No factor is left over.
∴ 3375 is a perfect cube.
3375 = 3³ × 5³

Question (3).
8000
Solution:
\(\begin{array}{l|l}
2 & 8000 \\
\hline 2 & 4000 \\
\hline 2 & 2000 \\
\hline 2 & 1000 \\
\hline 2 & 500 \\
\hline 2 & 250 \\
\hline 5 & 125 \\
\hline 5 & 25 \\
\hline 5 & 5 \\
\hline & 1
\end{array}\)
8000 = 2 × 2 × 2 × 2 × 2 × 2 × 5 × 5 × 5
Here, the prime factors 2 and 5 appear in triples.
No factor is left over.
∴ 8000 is a perfect cube.
8000 = 2³ × 2³ × 5³

Question (4).
15625
Solution:
\(\begin{array}{l|l}
5 & 15625 \\
\hline 5 & 3125 \\
\hline 5 & 625 \\
\hline 5 & 125 \\
\hline 5 & 25 \\
\hline 5 & 5 \\
\hline & 1
\end{array}\)
15625 = 5 × 5 × 5 × 5 × 5 × 5
Here, the prime factor 5 appear in triples.
No factor is left over.
∴ 15625 is a perfect cube.
15625 = 5³ × 5³

Question (5).
9000
Solution:
\(\begin{array}{l|l}
2 & 9000 \\
\hline 2 & 4500 \\
\hline 2 & 2250 \\
\hline 3 & 1125 \\
\hline 3 & 375 \\
\hline 5 & 125 \\
\hline 5 & 25 \\
\hline 5 & 5 \\
\hline & 1
\end{array}\)
9000 = 2 × 2 × 2 × 3 × 3 × 5 × 5 × 5
Here, among the prime factors 2 and 5 appear in triples but 3 does not appear in triple.
3 × 3 is left over.
∴ 9000 is not a perfect cube.

Question (6).
6859
Solution:
\(\begin{array}{l|l}
19 & 6859 \\
\hline 19 & 361 \\
\hline 19 & 19 \\
\hline & 1
\end{array}\)
6859 = 19 × 19 × 19
Here, the prime factor 19 appears in triple.
No factor is left over.
∴ 6859 is a perfect cube.
6859 = 193

Question (7).
2025
Solution:
\(\begin{array}{l|l}
3 & 2025 \\
\hline 3 & 675 \\
\hline 3 & 225 \\
\hline 3 & 75 \\
\hline 5 & 25 \\
\hline 5 & 5 \\
\hline & 1
\end{array}\)
2025 = 3 × 3 × 3 × 3 × 5 × 5
Here, the prime factor 3 appears in triple, but 3 × 5 × 5 is left over.
∴ 2025 is not a perfect cube.

Question (8).
10648
Solution:
\(\begin{array}{r|l}
2 & 10648 \\
\hline 2 & 5324 \\
\hline 2 & 2662 \\
\hline 11 & 1331 \\
\hline 11 & 121 \\
\hline 11 & 11 \\
\hline & 1
\end{array}\)
10648 = 2 × 2 × 2 × 11 × 11 × 11
Here, the prime factors 2 and 11 appear in triples.
No factor is left over.
∴ 10648 is a perfect cube.
10648 = 2³ × 11³

Think, Discuss and Write (Textbook Page No. 113)

1. Check which of the following are perfect cubes:
(i) 2700
(ii) 16000
(iii) 64000
(iv) 900
(v) 125000
(vi) 36000
(vii) 21600
(viii) 10000
(ix) 27000000
(x) 1000
What pattern do you observe in these perfect cubes ?
Solution:
(i) 2700
The number is ending with two zeros. If a number ends with three zeros or a multiple of 3 zeros, it may be a perfect cube.
∴ 2700 is not a perfect cube.

(ii) 16000
The number is ending with three zeros.
So it may be a perfect cube.
But, 16 is not a perfect cube.
∴ 16000 is not a perfect cube.

(iii) 64000
The number is ending with three zeros.
So it may be a perfect cube.
64 is a perfect cube. (∵ 4³ = 64)
∴ 64000 is a perfect cube.

(iv) 900
The number is ending with two zeros.
So it is not a perfect cube.
∴ 900 is not a perfect cube.

(v) 125000
The number is ending with three zeros.
So it may be a perfect cube.
125 is a perfect cube. (∵ 5³ = 125)
∴ 125000 is a perfect cube.

(vi) 36000
The number is ending with three zeros.
So it may be a perfect cube.
But, 36 is not a perfect cube.
∴ 36000 is not a perfect cube.

(vii) 21600
The number is ending with two zeros.
So it is not a perfect cube.
∴ 21600 is not a perfect cube.

(viii) 10000
The number is ending with four zeros.
So it is not a perfect cube.
∴ 10000 is not a perfect cube.

(ix) 27000000
The number is ending with six zeros.
So it may be a perfect cube.
27 is a perfect cube. (∵ 3³ = 27)
∴ 27000000 is a perfect cube.

(x) 1000
The number is ending with three zeros.
So it may be a perfect cube.
1 is a perfect cube, (∵ 1³ = 1)
∴ 1000 is a perfect cube.

Think, Discuss and Write (Textbook Page No. 115)

1. State true or false for any integer m, m² < m³. Why ?
Solution:
It seems true, but not always true.
m × m = m² and m × m × m = m³
∴ m² < m³
e.g. if m = 1
∴ m² = 1² = 1 and m³ = 1³ = 1
∴ m² ≮  m³, but m² = m³
If m = (- 1)
∴ m² = (- 1)² = 1 and m³ = (- 1)³ = (- 1)
∴ m² ≮  m³, but m2 > m³
So the above statement is not always true.

Punjab State Board PSEB 12th Class English Book Solutions English Grammar Narration / Direct and Indirect Speech Exercise Questions and Answers, Notes.

PSEB 12th Class English Grammar Narration / Direct and Indirect Speech

We may report the words of a speaker in two ways.

1. We may quote his actual words. This is called the Direct Speech.
2. We may report what he said without using his exact words. This is called the Indirect Speech.

Example : He said, “My brother is on tour.” (Direct)
He said that his brother was on tour. (Indirect)
Note : In the Direct Speech we use inverted commas to mark off the exact words of the speaker. In the Indirect Speech, we use neither exact words of the speaker nor the inverted commas.

The verb that reports the exact words of a speaker in Direct Speech is called the Reporting Verb.
The speech which the Reporting Verb actually reports is called the Reported Speech.
Rules for Changing Direct Speech into Indirect Speech.
(A) Change of Tense
1. The tense of the Reporting Verb is never changed. ,
2. The tense of the Reported Speech is changed as under :
(a) If the tense of the Reporting Verb is Present or Future, the tense of the verb in the Reported Speech is not changed.
(b) If the tense of the Reporting verb is past, the tense of the verb in the Reported Speech is changed into the corresponding past tense; as,

1. Present Indefinite	is changed into	Past Indefinite
2. Present Continuous	is changed into	Past Continuous
3. Present Perfect	is changed into	Past Perfect
4. Present Perfect Continuous	is changed into	Past Perfect Continuous
5. Past Indefinite	is changed into	Past Perfect
6. Past Continuous	is changed into	Past Perfect Continuous

(B) Change of Personal Pronouns

1. Pronouns of the first person in the Reported Speech are changed into the person of the subject of the Reporting Verb.
2. Pronouns of the second person in the Reported Speech are changed into the person of the object of the Reporting Verb.
3. Pronouns of the third person in the Reported Speech are generally not changed in the Indirect Speech.

(C) Change of Adjectives and Adverbs:
Words expressing nearness are generally changed into words expressing distance. Thus

(D) Use of Conjunction:
When a speech is changed from Direct into Indirect Speech, the conjunction ‘that’
generally inserted after the reporting verb. An exception in this respect is, however, made in the case of Interrogative and Imperative sentences.

1. Change of Assertive Sentences

In changing an Assertive sentence into Indirect Narration, change in the person of pronouns and tense is done as explained in the rules.
Conjunction ‘that’ is used.

Direct	Indirect
1. Ram says to me, “I am going home.”	Ram tells me that he is going home.
2. Ram said, “I go home.”	Ram said that he went home.
3. Ram said, “I have gone home.”	Ram said that he had gone home.
4. Ram said, “I have been going home.”	Ram said that he had been going home.
5. Ram said, “I went home.”	Ram said that he had gone home.
6. Ram said, “I was going home.”	Ram said that he had been going home.
7. Ram said, “I shall go home.”	Ram said that he would go home.
8. Ram said, “I shall be going home.”	Ram said that he would be going home.
9. Ram said, “I shall have gone home.”	Ram said that he would have gone home.
10. Ram said, “I shall have been going home.”	Ram said that he would have been going home.
11. Mohan said to me, “I am a poor student.”	Mohan told me that he was a poor student.
12. He said to me, “I have an urgent piece of work at home.”	He told me that he had an urgent piece of work at home.
13. I said to the teacher, “I shall never be late for school.”	I told the teacher that I would never be late for school.
14. Mohan said, “I have no time to talk to you now.”	Mohan said that he had no time to talk to me then.

NOTE : If the reported speech expresses a universal truth or habitual fact, the tense in it is not changed; as,

Direct	Indirect
1. The teacher said, “The earth moves round	The teacher said that the earth moves round the sun.
2. He said, “Health is wealth.”	He said that health is wealth.
3. He said, “God is everywhere.”	He said that God is everywhere.
4. Ram said, “Man is mortal.”	Ram said that man is mortal.
5. The teacher said to the boys, “Slow and and steady wins the race.”	The teacher told the boys that slow steady wins the race.
6. He said, “Two and two make four.”	He said that two and two make four.

2. Change of Interrogative Sentences

In converting an interrogative sentence into Indirect Speech, the following rules are observed:
1. The Reporting Verb is changed into ‘ask’, ‘enquire’ etc.

2. If the question begins with ‘who’, ‘when’, ‘where’, ‘what’, ‘from’, ‘why’, etc. no conjunction is used. These words follow in Reporting Verb unchanged.

3. When the question begins with a helping verb such as : ‘is’, ‘am’, ‘was’, ‘were’, ‘has’, ‘have’, ‘had’, ‘do’, ‘does’, ‘did’, ‘can’, ‘may’ etc. ‘if’ or ‘whether’ is used to introduce Indirect Speech.

The rules for the change of tense and pronouns as already given are observed.
The sentence changes its form from interrogative to assertive, helping verb goes with the verb and the question mark is removed.

Direct	Indirect
1. He said to me, “What is your name ?”	He asked me what my name was.
2. The teacher said to the student, “Why were you absent from school ?”	The teacher enquired of the student why he had been absent from the school.
3. He said to me, “Where do you live ?”	He asked me where I lived.
4. The teacher said to the boy, “How much money do you want ?”	The teacher asked the boy how much money he wanted.
5. Ram said to me, “Whom do you want to see ?	Ram asked me whom I wanted to see.
6. The teacher said to the boy, “Why are you	The teacher asked the boy why he was late again.

3. Change of Imperative Sentences

In converting an imperative sentence into Indirect Speech, following rules are observed:
1. The Reporting Verb is changed into order, command, advise, forbid, request, beg, ask etc. in accordance with the sense that the sentence carries.

2. The Imperative mood of the verb of the Reported Speech is changed into infinitive i.e. ‘to’ is followed by the first form of the verb.

3. In the negative sentences ‘to’ is used after ‘not’.

4. Sometimes an object is to be supplied to the Reporting Verb from the context.

Direct	Indirect
1. The headmaster said to the peon, “Ring the bell.”	The headmaster ordered the peon to ring the bell.
2. He said to me,“Please wait here till I return.”	He requested me to wait there till he returned.
3. The teacher said to the students, “Sit	The teacher ordered the students to sit down and not to make a noise.
4. I said to him, “Work hard if you want to get a prize.”	I advised him to work hard, if he wanted to get a prize.
5. The thief said to the magistrate, “Pardon	The thief begged the magistrate

4. Change of Exclamatory Sentences

Exclamatory sentences express the feelings of pain, pleasure, a wish, a desire etc. on the part of the speaker.
Rules for the change of such sentences into Indirect form of speech are as under:

1. The Reporting verb say is changed into exclaim, regret, admit, confess, cry out etc.
2. The sentence becomes an assertive one and sign of exclamation is replaced by a full stop.
3. Do away with words such as ‘Hurrah !’ ‘Alas’, ‘Bravo’ and express their sense by use of joy, sorrow, applaud.
4. Changes in Tenses and Pronouns are the same as discussed already.

Direct	Indirect
1. The players said, “Hurrah ! we have won the match.”	The players exclaimed with joy that they had won the match.
2. The old man said, “Alas ! my only son is	The old man exclaimed with sorrow that his only son was dead.
3. He said, “How clever I am !”	He exclaimed that he was very clever.
4. She said, “What a fine building it is !”	She exclaimed that it was a very fine building.
5. The teacher said, “Ah ! what a foolish	The teacher exclaimed with regret that he had to deal with a very foolish man.

5. Optative Sentences

Rules for change into Indirect Speech:
1. The reporting verb is changed into ‘wish’, ‘pray’ etc.
2. The sentence becomes assertive and sign of exclamation is replaced by a full stop.
3. In most cases, conjunction “that” is used after the reporting verb.

Direct	Indirect
1. The beggar said, “May you live long !”	The beggar prayed that I might live long.
2. He said to her, “May you succeed !”	He wished that she might succeed.
3. “May you go to hell !” said the hermit	The hermit prayed that the wretch might go to hell.
4. He said, “Would that I had all the pleasures of the world.”	He wished that he had all the pleasures of the world.
5. Prem said to me, “Good morning, sir.”	Prem respectfully wished me good morning.

Exercise 1

Change the form of Narration:

1. Tom said, “I don’t want to stay home from school.”	1. Tom said that he did not want to stay home from school.
2. ‘Oh, Sid, don’t juggle me !’ said Tom.	2. Tom exclaimed and told Sid not to juggle him.
3. She said, “Tom, you should not have frightened me like this.”	3. She told Tom that he should not have frightened her like that.
4. “Oh auntie, my sore toe is mortified.”	4. Tom told his auntie surprisingly that his sore toe was mortified.
5. “Now, that your tooth is out, Tom, you should go to school,” she said.	5. She advised Tom to go to school then because his tooth was out.
6. “What is the matter with you child ?” the aunt repeatedly said.	6. The aunt repeatedly asked the child lovingly what the matter with him was.
7. “I forgive you everything, Sid,” said Tom.	7. Tom told Sid that he forgave him everything.
8. “How long have you been this way ?” Sid said.	8. Sid asked Tom how long he had been like that way.
9. “Tom, why didn’t you wake me sooner ?” said Sid.	9. Sid asked Tom why he had not woken him sooner.
10. Aunt Polly said, “What is the matter with your tooth ?”	10. Aunt Polly asked him what the matter with his tooth was.
11. She said to Tom, “Open your mouth.”	11. She ordered Tom to open his mouth.
12. Tom said, “Oh, please Auntie, don’t pull it out.”	12. Tom requested his auntie not to pull it out.
13. Sid said to Tom, “Don’t wait, come quickly.”	13. Sid told Tom not to wait and asked

Exercise 2

Change the form of Narration:

1. “How many days is it since you touched your books ?”Father asked Swaminathan.	1. Father asked Swaminathan how many days it was since he had touched his books.
2. “Do you think you have passed the B.A. ?” Father asked Swami.	2. Father asked Swami if he thought he had passed the B.A.
3. “Do you want the whole day to throw out the spider ?” Father asked.	3. Father asked Swami if he wanted the whole day to throw out the spider.
4. “Get a piece of cloth, boy. That is not the way to clean things,” said Father.	4. Father told Swami to get a piece of cloth as that was not the way to clean things.
5. Swami announced, “Krishna must pay six annas.”	5. Swami announced that Krishna must pay six annas.
6. Swami said, “It seems I have got to tidy up these books of mine.”	6. Swami said that it seemed he had got to tidy up those books of his.
7. Father said to Swami, “Look at the way you have kept your English text.”	7. Father told Swami to look at the way he had kept his English text.
8. Swami said to his father, “Where am I to go for a piece of cloth ?”	8. Swami asked his father where he was to go for a piece of cloth.
9. “Can you give me a piece of cloth ?” said Swami to his mother.	9. Swami asked his mother if she could give him a piece of cloth.
10. Mother said to Swamis father, “There is no limit to his loafing in the sun.”	10. Mother told Swami’s father that there was no limit to his loafing in the sun.
11. Father said to Swami, “Have you done the sum ?”	11. Father asked Swami if he had done the sum.
12. Father said to Swami, “Where are you going ?”	12. Father asked Swami where he was going.
13. “How much does Ram want for his mangoes ?”asked the father.	13. The father asked him how much Ram wanted for his mangoes.
14. Father shouted and asked him, “Did you find the cloth ?”	14. Father shouted to ask him if he had found the cloth.
15. “You were not here yesterday,” said Swamis father.	15. Swamis father reminded (him) that he had not been there the previous day.

Exercise 3

Change the form of Narration:

1. “You must take the dog out, the woman bitterly.	1. He told the woman bitterly that she must take the dog out.
2. “I won’t go on the top in such weather,” said the woman.	2. The woman said that she would not go on the top in such weather.
3. “I will go on the top,” said the lady.	3. The lady said that she would go on the top.
4. “It is nonsense,” said her male companion.	4. Her male companion said angrily that it was nonsense.
5. “Let’s get our fares back,” said one of the passengers.	5. One of the passengers suggested that they should get their fares back.
6. “No fares back,” said the conductor.	6. The conductor told the passengers that he would give no fares back.
7. “Well, he has got his rules,” he said.	7. He remarked that he had got his rules.
8. “Shameful.” “Call the police, let’s make him give us our fares back.”	8. The passengers deplored that the behaviour of the conductor was shameful. It was suggested to call the police and to make him (the conductor) give them their fares back.
9. “You can take my name and address,” said the woman.	9. The woman told him (courageously) that he could take her name and address.
10. “Call the police,” said someone.	10. Someone suggested that the police should be called.
11. “I’ve got my rules,” said the conductor.	11. The conductor claimed or remarked that he had got his rules.

Exercise 4

Change the form of Narration:

1. “What is the treasure?” I enquired with interest.	1. I enquired with interest what the treasure was.
2. “Pearls,” he answered, “marvellous, flawless pearls.”	2. He answered that the pearls were really marvellous and flawless.
3. “And where are they to be found ?” I asked.	3. I asked him where the pearls were to be found.
4. “On the pearl banks of the Persian Gulf,” he replied.	4. He replied that they were to be found on the pearl banks of the Persian Gulf.
5. “The thin men are the divers,” he explained.	5. He explained that the thin men were the divers.
6. “Who is the man sitting cross-legged on a bench ?” I asked.	6. I asked him who the man sitting cross-legged on a bench was.
7. “That is the captain,” said my host.	7. My host said that that was the captain.
8. “There are not many men killed by sharks,’’said Mr. Hudson.	8. Mr. Hudson said that there were not many men killed by sharks.
9. “What is in the pots ?” I asked my host.	9. I asked my host what in the pots
10. “I could show you one,” said Mr. Hudson.	10. Mr. Hudson told me that he could show me one.
11. Mr. Hudson said, “This is a tiny copy of the Koran, the Arabs Bible.”	11. Mr. Hudson told me that it was a tiny copy of the Koran, the Arab’s Bible.
12. He asked me, “Do you notice a strong smell of herbs ?”	12. He asked me if I noticed a strong smell of herbs.
13. He replied, “The pearl fleet is at work quite near here.”	13. He replied that the pearl fleet was at work quite near there.
14. The writer said to Hudson, “Is it not dan-gerous to work here ?”	14. The writer asked Hudson if it was not dangerous to work there.
15. The host said to me, “Will you like to hear a song ?”	15. The host asked me if I would like to hear a song.
16. Mr. Hudson warned me, “Look out for the jelly fish.”	16. Mr. Hudson warned me to look out for the jelly fish.

Exercise 5

Change the form of Narration:

1. My hostess said, “I will show you over the church after service.”	1. My hostess told me that she would show me over the church after service.
2. “I hope you slept well,” said my host at breakfast next morning.	2. My host at breakfast next morning told me that she hoped that I had slept well.
3. “Oh ! that is the washing pool,” said my hostess.	3. My hostess pointed out with surprise that that was the washing pool.
4. “She would call that a kit,” said my hostess.	4. My hostess told me that she would call that a kit.
5. “Would you like to come with us to a service in the Maori Church ?” asked she.	5. She asked me if I would like to come with them to a service in the Maori Church.
6. “I’ll make your tea for you,” said the Maori woman.	6. The Maori woman told me that she would make my tea for me.
7. “What a tongue-twister !” I could not help saying.	7. I could not help remarking that it was a very bad tongue-twister.
8. “But isn’t it very dangerous to live there at all ?” I wanted to know.	8. I asked to know if it was not dangerous to live there at all.
9. My hostess said, “That’s not really wood, you know-it is flax.”	9. My hostess told me that that was not really wood. She reminded me that it was flax.
10. I said, “How amazing !”	10. I exclaimed that it was very amazing.

Our readers should note that all exercises on the grammatical items prescribed in the syllabus have been solved at the appropriate places in MBD Guide.

Exercise 1 (Textual)

Change the Narration:
1. She said to me, “I was present in the class yesterday.”
2. He says to me, “You may ask your brother to help you.”
3. She will say to me, “They are very brave.”
4. He told me that hard work pays in life.
5. Rohtas said, “I prefer death to dishonour.”
6. The stranger said, “I want to stay here for the night.”
7. The man told Ravi that he knew his father.
8. My uncle said that he had bought that house in 2005.
9. Ashok will say to Sumit, “I will try to stand first.”
10. Rishi said to me, “He needs help.”
Answer:
1. She told me that she had been present in the class the previous day.
2. He tells me that I may ask my brother to help me.
3. She will tell me that they are very brave.
4. He told me that hard work pays in life.
5. Rohtas said that he prefers death to dishonour.
6. The stranger said that he wanted to stay there for the night.
7. The man said to Ravi, “I know your father.”
8. My uncle said, “I bought this house in 2005.”
9. Ashok will tell Sumit that he will try to stand first.
10. Rishi told me that he needed help.

Exercise 2 (Textual)

Change the Narration:
1. The leader said, “Trust in God.”
2. My friend said to me, “Let me go home now as it is already twelve.”
3. She said, “What a lovely scene !”
4. My mother said to me, “Learn your lesson everyday.”
5. Columbus said to the courier, “Oh, leave me alone.”
6. The policeman advised me to obey the traffic rules.
7. Isha asked him when the postman came.
8. He asked, “Will you serve me faithfully ?”
9. You said, “You must work hard.”
10. He says to me, “I will leave you now.”
Answer:
1. The leader told them to trust in God.
2. My friend told me to let him go home then as it was already twelve.
3. She exclaimed with delight that it was a very lovely scene.
4. My mother advised me to learn my lesson everyday.
5. Feeling disgusted, Columbus told the courier to leave him alone.
6. The policeman said to me, “Obey the traffic rules.”
7. Isha said to him, “When did the postman come ?”
8. He asked me if I would serve him faithfully.
9. You told him that he must work hard.
10. He tells me that he will leave then.

Exercise 3 (Textual)

Change the Narration:
1. She said, “We are all sinners.”
2. “Run away, children,” said the mother.
3. “Do not you know the way home ?”asked I.
4. He said to him, “You should respect your elders.”
5. The principal said, “The authorities are trying their best to improve things.”
6. The teacher said, “Honesty is the best policy.”
7. Alice said to his wife, “Tom is coming for lunch today.”
8. I said, “I have my own likes and dislikes.”
9. You said to him, “I have seen this man somewhere.”
10. Mrs. Sonia says, “I am working on this computer.”
Answer:
1. She said that they are all sinners.
2. The mother ordered the children to run away.
3. I asked him if he did not know the way home.
4. He advised him that he should respect his elders.
5. The principal announced that the authorities were trying their best to improve things.
6. The teacher said that honesty is the best policy.
7. Alice told his wife that Tom was coming for lunch that day.
8. I said that I have my own likes and dislikes.
9. You told him that you had seen that man somewhere.
10. Mrs. Sonia says that she is working on that computer.

Exercise 4 (Textual)

Change the Narration:
1. He said to me, “I do not believe you.”
2. Rama said to Arjun, “Go away.”
3. He said to me, “What are you doing ?”
4. The Prince said, “It gives me great pleasuerc to be here this evening.”
5. He said, ‘I shall go as soon as it is possible.”
6. He said to him, “Is not your name Ahmed ?”
7. “Sit down, boys,” said the teacher.
8. Alice said, “How clever I am!”
9. The teacher said to him, “Do not read so fast.”
10. Abdul said that he had seen that picture.
Answer:
1. He told me that he did not believe me.
2. Rama told Arjun to go away.
3. He asked me what I was doing.
4. The Prince remarked that it gave him great pleasure to be there that evening.
5. He said that he would go as soon as it was possible.
6. He asked him if his name was not Ahmed.
7. The teacher asked the boys to sit down.
8. Alice exclaimed proudly that she was very clever.
9. The teacher told him not to read so fast.
10. Abdul said, “I have seen that picture.”

Exercise 5 (Textual)

Change the Narration:
1. You said, “My father returned from Cuttack last night.”
2. The saint said, “We should not hate the sinners.”
3. Jogi said that he had stood first.
4. Geeta said to Rohini, “Examination is drawing near.”
5. He says, “I go to the temple everyday.”
6. He said to me, “Hard work is the key to success.”
7. She said, “I was ill yesterday.”
8. “I want to take that journalist out on patrol,” he said to Priya.
9. The science teacher said to the students, “The earth moves round the sun.”
10. She said to me, “God will help you.”
Answer:
1. You said that your father had returned from Cuttack the previous night.
2. The saint said that they should not hate the sinners.
3. Jogi said, “I have stood first.”
4. Geeta informed Rohini that examination was drawing near.
5. He says that he goes to the temple everyday.
6. He told me that hard work is the key to success.
7. She said that she had been ill the previous day.
8. He told Priya that he wanted to take that journalist on patrol.
9. The science teacher told the students that the earth moves round the sun.
10. She assured me that God will help me.

Exercise 6 (Textual)

Change the Narration:
1. Kausalya said to Rama, “Do not desire to possess the moon.”
2. An old mouse said, “Who will bell that cat ?”
3. “Go down to the bazaar. Bring me some oil and a lump of ice,” ordered Latif.
4. Isha said to him, “What is it that makes you stronger and braver than any other man.”
5. “Have you anything to tell me ?”asked his master.
6. He said to me, “Wait until I come.”
7. He said, “Oh ! That’s a nuisance.”
8. “Which way did she go ?” asked the young man.
9. My father said to me, “I have often told you not to play with fire.”
10. I wrote that I would visit you the next day.
Answer:
1. Kausalya forbade Rama to desire to possess the moon.
2. An old mouse asked who would be willing to bell the cat.
3. Latif ordered him to go down to the bazaar and bring him some oil and a lump of ice.
4. Isha asked him as to what it was that made him stronger and brave than any other man.
5. His master asked him if he had anything to tell him.
6. He told me to wait until he came.
7. He exclaimed angrily that it was a great nuisance.
8. The young man asked him which way she had gone.
9. My father warned me that he had often told me not to play with fire.
10. I wrote, “I shall visit you tomorrow.”

Punjab State Board PSEB 8th Class Maths Book Solutions Chapter 8 Comparing Quantities Ex 8.3 Textbook Exercise Questions and Answers.

PSEB Solutions for Class 8 Maths Chapter 8 Comparing Quantities Ex 8.3

1. Calculate the amount and compound interest on:

Question (a)
₹ 10,800 for 3 years at 12\(\frac {1}{2}\) per annum compounded annually.
Solution:
Here, P = ₹ 10,800;
R = 12\(\frac {1}{2}\) % = \(\frac {25}{2}\) %;
T = 3 years; ∴ n = 3

Amount = ₹ 15,377.34
Compound’interest = Amount – Principal
= ₹ (15377.34 – 10800)
= ₹ 4577.34

Question (b)
₹ 18,000 for 2\(\frac {1}{2}\) years at 10% per annum compounded annually.
Solution:
Here, P = ₹ 18,000; R = 10 %;
T = 2\(\frac {1}{2}\) years; ∴ n = 2 + \(\frac {1}{2}\)

Amount = ₹ 22,869
Compoimd interest = Amount – Principal
= ₹ (22869 – 18000)
= ₹ 4869

Question (c)
₹ 62,500 for 1\(\frac {1}{2}\) years at 8% per annum compounded half yearly.
Solution:
Here, the interest is compounded half-yearly.
Here, P = ₹ 62,500; R = \(\frac {8}{2}\) = 4 %
T = 1\(\frac {1}{2}\) years ∴ n = \(\frac {3}{2}\) × 2 = 3

Amount = ₹ 70,304
Compound interest = Amount – Principal
= ₹ (70304 – 62500)
= ₹ 7804

Question (d)
₹ 8000 for 1 year at 9 % per annum compounded half-yearly.
(You could use the year-by-year calculation using SI formula to verify.)
Solution:
Here, the interest is compounded half-yearly.
Here, P = ₹ 8000; R = \(\frac {9}{2}\) %;
T = 1 year ∴ n = 2

Amount = ₹ 8736.20
Compound interest = Amount – Principal
= ₹ (8736.20 – 8000)
= ₹ 736.20
[Note : By finding simple interest also we can calculate.)
SI = \(\frac {PRT}{100}\)
= \(\frac{8000 \times 9 \times 1}{2 \times 100}\)
= ₹ 376.20
Thus, total interest of 1 year
= ₹ (360 + 376.20)
= ₹ 736.20

Question (e)
₹ 10,000 for 1 year 8% per annum compounded half yearly.
Solution:
Here, the interest is compounded half-yearly.
Here, P = ₹ 10,000; R = \(\frac {8}{2}\) = 4 %;
T = 1 year ∴ n = 1 × 2 = 2

Amount = ₹ 10,816
Compound interest = Amount – Principal
= ₹ (10816 – 10000)
= ₹ 816

2. Kamala borrowed ₹ 26,400 from a Bank to buy a scooter at a rate of 15% p.a. compounded yearly. What amount will she pay at the end of 2 years and 4 months to clear the loan?
(Hint: Find A for 2 years with interest is compounded yearly and then find SI on the 2nd year amount for \(\frac {4}{12}\) years)
Solution:
[Note: Here, find amount after 2 years by compound interest. This amount is principal for \(\frac {4}{12}\) year. For this \(\frac {4}{12}\) year, find simple interest.)
Here, P = ₹ 26,400; R = 15%;
T = 2 years ∴ n = 2

Now, ₹ 34,914 will be principal to find interest of 4 months.
SI = \(\frac {PRT}{100}\)
= \(\frac{34914 \times 15 \times 4}{100 \times 12}\)
= \(\frac{174570}{100}\)
= ₹ 174570
= ₹ 1745.70
Amount = ₹ (34914 + 1745.70)
= ₹ 36,659.70
Thus, Kamala Mil have to pay ₹ 36,659.70 to clear the loan.

3. Fabina borrows ₹ 12,500 at 12% per annum for 3 years at simple interest and Radha borrows the same amount for the same time period at 10% per annum, compounded annually. Who pays more interest and by how much?
Solution:
For Fabina:
Here, P = ₹ 12,500; R = 12%; T = 3 years
SI = \(\frac{\mathrm{P} \times \mathrm{R} \times \mathrm{T}}{100}\)
= \(\frac{12500 \times 12 \times 3}{100}\)
= 125 × 12 × 3
= ₹ 4500
Simple interest = ₹ 4500

For Radha:
Here, P = ₹ 12,500; R = 10%;
T = 3 years ∴ n = 3


Amount = ₹ 16,637.50
Compound interest = Amount – Principal
= ₹ (16637.50 – 12500)
= ₹ 4137.50
Fabina has to pay ₹ 4500 as interest and Radha has to pay ₹ 4137.50 as interest.
∴ Fabina has to pay more interest.
Difference in interest = ₹ (4500 – 4137.50)
= ₹ 362.50
Thus, Fabina has to pay ₹ 362.50 more than Radha as interest.

4. I borrowed ₹ 12,000 from Jamshed at 6 % per annum simple interest for 2 years. Had I borrowed this sum at 6% per annum compound interest, what extra amount would I have to pay?
Solution:
For Simple Interest:
Here, P = ₹ 12,000; R = 6 %; T = 2 years
SI = \(\frac{\mathrm{P} \times \mathrm{R} \times \mathrm{T}}{100}\)
= \(\frac{12000 \times 6 \times 2}{100}\)
= 120 × 6 × 2
= ₹ 1440
SI = ₹ 1440

For Compound Interest:
Here, P = ₹ 12,000, R = 6 %
T = 2 years ∴ n = 2

= ₹ 13,483.20
Amount = ₹ 13,483.20
CI = A – P
= ₹ (13483.20 – 12000)
= ₹ 1483.20
∴ Extra amount to be paid
= ₹ (1483.20 – 1440)
= ₹ 43.20
Thus, I have to pay ₹ 43.20 as extra amount.

5. Vasudevan invested ₹ 60,000 at an interest rate of 12% per annum compounded half-yearly. What amount would he get.

Question (i)
after 6 months?
Solution:
Interest after 6 months:
Here, P = ₹ 60,000; R = \(\frac {12}{2}\) = 6%;
T = 6 months ∴ n = 1

∴ Amount = ₹ 63,600

Question (ii)
after 1 year?
Solution:
After 1 year:
∴ Amount = ₹ 67,416
Here, P = ₹ 60,000; R = \(\frac {12}{2}\) = 6 %;
T = 1 year ∴ n = 2

∴ Amount = ₹ 67, 146
Thus, Vasudevan will get ₹ 63,600 after 6 months and ₹ 67,416 after 1 year.

6. Arif took a loan of ₹ 80,000 from a bank. If the rate of interest is 10% per annum, find the difference in amounts he would be paying after 1\(\frac {1}{2}\) years if the interest is

Question (i)
compounded annually.
Solution:
Calculation of CI annually:
Here, P = ₹ 80,000; R = 10 %;
T = 1\(\frac {1}{2}\) year

For 1st year:
R = 10% and n = 1

∴ Amount of CI after 1 year = ₹ 88,000
Now, calculate simple interest of ₹ 88,000 for 6 months.
P = ₹ 88,000; R = 10 %;
T = 6 months = \(\frac {1}{2}\) year
∴ Interest = \(\frac{P \times R \times T}{100}\)
= \(\frac{88000 \times 10 \times 1}{100 \times 2}\)
= 4400
∴ Interest of 6 months = ₹ 4400
Thus, A = P + I
= ₹ (88000 + 4400)
= ₹ 92,400
Thus, according to CI, Arif has to pay ₹ 92,400

Question (ii)
compounded half yearly.
Solution:
If interest is compounded half yearly
Here, P = ₹ 88000, R = \(\frac {10}{2}\) = 5%
T = 1\(\frac {1}{2}\) years ∴ n = 3

∴ Amount = ₹ 92,610
Thus, according to half-yearly CI, Arif has to pay ₹ 92,610
∴ Difference = ₹ (92,610 – 92,400)
= ₹ 210

7. Maria invested ₹ 8,000 in a business. She would be paid interest at 5% per annum compounded annually. Find

Question (i)
The amount credited against her name at the end of the second year,
Solution:
(i) Here, P = ₹ 8000, R = 5 %,
T = 2 years ∴ n = 2


Thus, amount credited against Marlas name at the end of second year is ₹ 8820.

Question (ii)
The interest for the 3rd year.
Solution:
To find the interest for the 3rd year:
P = ₹ 8820, R = 5%, T = 1 years
SI = \(\frac{\mathrm{PRT}}{100}=\frac{8820 \times 5 \times 1}{100}\) = 441
The interest for 3rd year is ₹ 441
OR
The interest for the 3rd year:
Here, P = ₹ 8000, R = 5 %, T = 3 years ∴ n = 3

∴ At the end of 3rd year ₹ 9261 will be credited against Maria’s name.
∴ Interest of the 3rd year = Amount of 3 years – Amount of 2 years
= ₹ (9261 – 8820)
= ₹ 441

8. Find the amount and the compound interest on ₹ 10,000 for 1\(\frac {1}{2}\) years at 10% per annum, compounded half-yearly. Would this interest be more than the interest he would get if it was compounded annually?
Solution:
(i) Here, interest is compounded half-yearly.
Here, P = ₹ 10,000; R = \(\frac {10}{2}\) = 5 %;
T = 1\(\frac {1}{2}\) years ∴ n = 3

Amount = ₹ 11,576.25
CI = A – P
= ₹ (11576.25 – 10000)
= ₹ 1576.25

(ii) Here, interest is compounded yearly. CI for 1 year:
Here, P = ₹ 10,000; R = 10%;
T = 1 year ∴ n = 1

Amount at the end of 1 year = ₹ 11,000
∴ CI = A – P
= ₹ (11000 – 10000)
= ₹ 1000
Now, calculate SI for 6 months.
Here, P = ₹ 11,000; R = 10%; T = \(\frac {1}{2}\) year
SI = \(\frac{\mathrm{P} \times \mathrm{R} \times \mathrm{T}}{100}=\frac{11000 \times 10 \times 1}{100 \times 2}\)
= 550
∴ Total interest of 1\(\frac {1}{2}\) years = ₹ (1000 + 550)
= ₹ 1550
After comparing (i) and (ii), we can conclude ₹ 1576.25 > ₹ 1550
∴ Yes, the interest is more if compounded half-yearly.

9. Find the amount which Ram will get on ₹ 4096, if he gave it for 18 months at 12\(\frac {1}{2}\) % per annum, interest being compounded half-yearly.
Solution:
Here, interest is compounded half-yearly.
Here, P = ₹ 4096, R = 12\(\frac{1}{2} \times \frac{1}{2}=\frac{25}{4}\)
T = 18 months = 1\(\frac {1}{2}\) years ∴ n = 3

Amount = ₹ 4913
Thus, Ram will get ₹ 4913 at the end of period.

10. The population of a place increased to 54,000 in 2003 at a rate of 5% per annum

Question (i)
find the population in 2001.
Solution:
[Note: In 1st case, we have to find P as population of 2003 is given. From that we have to find population of 2001.]
Population in 2003 = 54,000
Here, A = 54,000; R = 5%; T = 2 years ∴ n = 2

∴ P = 48979.59 (approx)
P = 48980 (approx)
Thus, the population in 2001 is 48,980.

Question (ii)
What would be its population in 2005?
Solution:
Here, P = 54,000; R = 5 %;
T = 2 years ∴ n = 2

Thus, the population in 2005 is 59,535.

11. In a Laboratory, the count of bacteria in a certain experiment was increasing at the rate of 2.5% per hour. Find the bacteria at the end of 2 hours if the count was initially 5,06,000.
Solution:
Initial count of bacteria = 5,06,000
Rate of increasing = 2.5% per hour
Bacteria count after 2 hours
Here, P = 5,06,000, R = 2.5 % = \(\frac {5}{2}\)%;
T = 2 hours ∴ n = 2

A = 531616 (approx)
Thus, the number of bacteria count after 2 hours will be 5,31,616 (approx).

12. A scooter was bought at ₹ 42,000. Its value depreciated at the rate of 8 % per annum. Find its value after one year.
Solution:
CP of a scooter = ₹ 42,000
Here, P = ₹ 42,000; R = 8 %; T = 1 ∴ n = 1
R = – 8 % (as depreciation)

Thus, the value of a scooter after 1 year will be ₹ 38,640.

Punjab State Board PSEB 12th Class English Book Solutions English Grammar Active and Passive Voice Exercise Questions and Answers, Notes.

PSEB 12th Class English Grammar Active and Passive Voice

There are two Voices:
1. Active Voice
2. Passive Voice.

1. Active Voice. The verb is said to be in the active voice when the subject acts; as,

• He sings a song.
• She wrote a story.
• We are telling a story.
• The farmer waters the fields.

2. Passive Voice. The verb is said to be in the passive voice when the object acts; as,

• A song is sung by him.
• A story was written by her.
• A story is being told by us.
• The fields are watered by the farmer.

Rules for changing a verb from the ACTIVE VOICE to the PASSIVE VOICE are:
1. The object of the verb in the Active Voice is made the subject in the Passive Voice.

2. If a transitive verb has two objects, either of them becomes the subject of the Passive Voice and the other remains unchanged.

3. The subject of the verb becomes the object of some preposition; as,
He gave me a book.
A book was given to me by him.

4. The verb is changed according to the following rules:

Tense	Changed into
Present Indefinite	(sings) = is, am, are + third form (sung).
Past Indefinite	(sang) = was, were + third form (sung).
Future Indefinite	(sing) = shall be, will be + third form (sung).
Present Continuous	(singing) = is, am, are + being + third form (sung).
Past Continuous	(singing) was, were + being + third form (sung)
Present Perfect	(sung) = has been, have been + third form.
Past Perfect	(sung) = had been + third form.
Future Perfect	(sung) = shall have been, will have been + third form.

Please Note. Use be + third form after can, could, shall, should, may, might, must, ought to.

5. When the agent is not expressed in the Passive Voice, we must supply it in the Active Voice:
Books were sold.
(Reeta) sold the books.

6. When a sentence is an imperative one, then the word “let” is used in the Passive voice:
Read your books.
Let your books be read.

7. Prepositions following the verbs in the ACTIVE VOICE must follow them again in the PASSIVE VOICE:
He laughed at me.
I was laughed at by him.

8. In interrogative sentences, the helping verb should always precede the Doer of the action in the Active Voice and the Receiver of the action in the Passive Voice:
Did he post the letter ?
Was the letter posted by him ?

9. THE PERSONAL PRONOUNS are changed as under:

(A) Present Indefinite Tense

1. Active : I hate Physics like poison.
Passive : Physics is hated like poison by me.
2. Active : She sings a song.
Passive : A song is sung by her.
3. Active : Who teaches you history ?
Passive : By whom are you taught history ?
4. Active : Does he help you ?
Passive : Are you helped by him ?

(B) Past Indefinite Tense

1. Active : I hated Physics like poison.
Passive : Physics was hated like poison by me.
2. Active : She sang a song.
Passive : A song was sung by her.
3. Active : Who taught you history ?
Passive : By whom were you taught history ?
4. Active : Did he help you ?
Passive : Were you helped by him ?

(C) Future Indefinite Tense

1. Active : I will hate Physics like poison.
Passive : Physics shall be hared like posion by me.
2. Active : She will sing a song.
Passive : A song will be sung by her.
3. Active : Who will teach you history ?
Passive : By whom will you be taught history ?
4. Active : Will he help you ?
Passive : Will you be helped by him ?

(D) Present Continuous Tense

1. Active : The peon is ringing the bell.
Passive : The bell is being rung by the peon.
2. Active : Why are you making a noise ?
Passive : Why is a noise being made by you ?
3. Active : Who is ringing the bell ?
Passive : By whom is the bell being rung ?
4. Active : Is he singing a song ?
Passive : Is a song being sung by him ?

(E) Past Continuous Tense

1. Active : They were singing a song.
Passive : A song was being sung by them.
2. Active : What were they doing ?
Passive : What was being done by them ?
3. Active : Was he teaching the class ?
Passive : Was the class being taught by him ?
4. Active : The peon was ringing the bell.
Passive : The bell was being rung by the peon.

(F) Present Perfect Tense

1. Active : She has insulted her mother.
Passive : Her mother has been insulted by her.
Active : Have you stolen the book ?
Passive : Has the book been stolen by you ?
3. Active : Who has rung the bell ?
Passive : By whom has the bell been rung ?
4. Active : She has spoken the truth.
Passive : The truth has been spoken by her.

(G) Past Perfect Tense

1. Active : She had insulted her mother.
Passive : Her mother had been insulted by her.
2. Active : Had you stolen the book ?
Passive : Had the book been stolen by you ?
3. Active : Who had rung the bell ?
Passive : By whom had the bell been rung ?
4. Active : She had spoken the truth.
Passive : The truth had been spoken by her.

(H) Future Perfect Tense

1. Active : She shall have insulted her father.
Passive : Her father will have been insulted by her.
2. Active : Will you have seen the picture by then ?
Passive .: Will the picture have been seen by you by then ?
3. Active : She will have caught the bird.
Passive : The bird will have been caught by her.
4. Active He will have stolen my pen.
Passive : My pen will have been stolen by him.

(I) Use of “Should”, “Must”, “Ought”, “Can”, “Could”, “May”, “Might” Etc. (Auxiliary Verbs)

1. Active : We should not laugh at the poor.
Passive : The poor should not be laughed at by us.
2. Active : You must teach him.
Passive : He must be taught by you.
3. Active : We ought to obey our elders.
Passive : Our elders ought to be obeyed by us.
4. Active : You can do it.
Passive : It can be done by you.
5. Active : I could teach him a lesson.
Passive : He could be taught a lesson by me.
6. Active : May I see your pen ?
Passive : May your pen be seen by me ?
7. Active : He might follow this path.
Passive : This path might be followed by him.
8. Active : May God bless you with a son !
Passive : May you be blessed with a son by God !

(J) Use of “Who”, “What”, “Why”, “When” Etc.

1. Active : Who beat you ?
Passive : By whom were you beaten ?
2. Active : Who broke your pen ?
Passive : By whom was your pen broken ?
3. Active : What have you done ?
Passive : WTiat has been done by you ?
4. Active : WTiy did you abuse him ?
Passive : Why was he abused by you ?
5. Active : When do you visit the temple ?
Passive : WTien is the temple visited by you ?

(K) Use Of “Know”, “Contain”, “Please”, “Astonish” “Surprise”, “Disgust”, “Alarm” Etc.

1. Active : Do you know him ?
Passive : Is he known to you ?
2. Active : This cup contains milk.
Passive : Milk is contained in this cup.
3. Active : His behaviour displeased me.
Passive : I was displeased at his behaviour.
4. Active : I cannot please everybody.
Passive : Everybody cannot be pleased with me.
5. Active : Her conduct astonished me.
Passive : I was astonished at her conduct.
6. Active : This news alarmed me.
Passive : I was alarmed at this news.

(L) Imperative Sentences

1. Active : Post this letter.
Passive : Let this letter be posted.
2. Active : Obey your elders.
Passive : Let your elders be obeyed.
3. Active : Do not pluck flowers.
Passive : You are prohibited to pluck flowers.
4. Active : Sit down.
Passive : You are ordered to sit down.
5. Active : Speak the truth.
Passive : You are advised to speak the truth.
6. Active : Please help me.
Passive : You are requested to help me.

NOTE : There are some Transitive verbs that give the sense of the Passive Voice without being changed into the Passive Voice.
1. Active : Quinine tastes bitter.
Passive : Quinine is bitter when tasted.
2. Active : This milk tastes sweet.
Passive : This milk is sweet when tasted.

Exercise 1

Change the Voice:

1. I like my teacher.
2. Do manners reveal character ?
3. Columbus discovered America.
4. This man has cut down the trees.
5. Hari did not open the door.
6. I will win him over.
7. The old man takes the snuff.
8. A cruel boy killed the bird.
9. I had never seen a zoo before.
10. Will she have written a letter ?
Answer:
1. My teacher is liked by me.
2. Is character revealed by manners ?
3. America was discovered by Columbus.
4. The trees have been cut down by this man.
5. The door was not opened by Hari.
6. He will be won over by me.
7. The snuff is taken by the old man.
8. The bird was killed by a cruel boy.
9. A zoo had never been seen by me before.
10. Will a letter have been written by her ?

Exercise 2 (textual)

Change the Voice:
1. Does she know you ?
2. Will you post the letter ?
3. She has done her duty.
4. Avoid bad company.
5. Are you expecting him today ?
6. I cannot lift this heavy box.
7. May I see your book ?
8. You are wasting your time.
9. When will she have finished her work ?
10. Why do you not call in the doctor ?
Answer:
1. Are you known to her ?
2. Will the letter be posted by you ?
3. Her duty has been done by her.
4. Bad company should be avoided.
5. Is he being expected by you today ?
6. This heavy box cannot be lifted by me.
7. May your book be seen by me ?
8. Your time is being wasted by you.
9. When will her work have been finished by her ?
10. Why is the doctor not called in by you ?

Exercise 3 (Textual)

Change the Voice:
1. Her uncle looks after her.
2. A favour will be done to him by me.
3. How is Sharda known to you ?
4. He was shocked at his sisters stupidity.
5. Why are you laughing at me ?
6. Who abused you ?
7. Had the picture been painted by her ?
8. This pot contains milk.
9. Were the villagers beating the terrorists ?
10. Who does not love his motherland ?
Answer:
1. She is looked after by her uncle.
2. I will do him a favour.
3. How do you know Sharda ?
4. His sister’s stupidity shocked him.
5. Why am I being laughed at by you ?
6. By whom were you abused ?
7. Had she painted the picture ?
8. Milk is contained in this pot.
9. Were the terrorists being beaten by the villagers ?
10. By whom is his motherland not loved ?

Exercise 4 (Textual)

Change the Voice:
1. His father praised him.
2. Kalidas wrote Sakuntalam.
3. The teacher was pleased by the boy’s work.
4. He kept me waiting.
5. Lata was singing a song.
6. They had already consulted the lawyer.
7. Many toys had been purchased by Rani.
8. The paper published the news.
9. Did you wring the clothes ?
10. Which book do you want ?
Answer:
1. He was praised by his father.
2. Sakuntalam was written by Kalidas.
3. The boy’s work pleased the teacher.
4. I was kept waiting by him.
5. A song was being sung by Lata.
6. The lawyer had already been consulted by them.
7. Rani had purchased many toys.
8. The news was published by the paper.
9. Were the clothes wrung by you ?
10. Which book is wanted by you ?

Exercise 5

Change the Voice:
1. Do not keep bad company.
2. They will have missed the train.
3. I shall not show you my book.
4. You are to help him.
5. Who invented the gramophone ?
6. I was given your message by him.
7. Open the door.
8. What do you want ?
9. The fire will have destroyed the house.
10. We ought to love our country.
Answer:
1. Let bad company not be kept.
2. The train will have been missed by them.
3. My book will not be shown to you.
4. He is to be helped by you.
5. By whom was the gramophone invented ?
6. He gave me your message.
7. Let the door be opened.
8. What is wanted by you ?
9. The house will have been destroyed by the fire.
10. Our country ought to be loved.

Exercise 6 (Textual)

Change the Voice:
1. I was pleased with his conduct.
2. The sudden noise frightened the horse.
3. The man cut down the tree.
4. The people will soon forget it.
5. We elected Thomas captain.
6. Somebody has put out the light.
7. We prohibit smoking.
8. He was refused admission.
9. Those cars were built by robots.
10. Why did he defraud you of your earnings ?
Answer:
1. His conduct pleased me.
2. The horse was frightened by the sudden noise.
3. The tree was cut down by the man.
4. It will soon be forgotten by the people.
5. Thomas was elected captain by us.
6. The light has been put out.
7. Smoking is prohibited.
8. The Principal refused him admission.
9. Robots built those cars. .
10. Why were you defrauded of your earnings by him ?

Additional Exercise Voice Based on Grammar Items

Exercise 1

Change the voice of the following sentences:
1. Crowd thronged the stadium.
2. Arjun is seeing a deer.
3. I have not liked kites.
4. Who is waiting for me ?
5. Learn your lessons.
6. Mend your business.
7. I am known to him.
8. You displeased her.
9. This pot contains coffee.
10. You had to do it.
11. Indians adore Prime Minister Narendra Modi.
12. Work must be finished.
13. Did he sing a song ?
14. Can you lend me your book.
15. Do you like apples ?
Answer:
1. The stadium was thronged with crowd.
2. A deer is being seen by Arjun.
3. Kites have not been liked by me.
4. By whom am I being waited for ?
5. Let your lessons be learnt ?
6. Let your business be mended ?
7. He knows me.
8. She was displeased at you.
9. Coffee is contained in this pot.
10. It had to be done by you.
11. Prime Minister is adored by Indians.
12. Workers must finish work.
13. Was a song sung by him ?
14. Can your book be lent to me (by you) ?
15. Apples are liked by me.

Punjab State Board PSEB 12th Class English Book Solutions English Grammar Transformation of Sentences (Simple, Compound & Complex) Exercise Questions and Answers, Notes.

PSEB 12th Class English Grammar Transformation of Sentences (Simple, Compound & Complex)

(Simple, Compound Complex)

Definition of Simple Sentences. A simple sentence is one that contains one finite verb, either expressed or understood. It simply means that it has one finite verb and is consequently a single independent clause.
Sentence : (S = Subject, V = Verb, O = Object)
Examples:

1. Boys are running.
2. Boys and girls play.
3. The handsome boy gave his mother a red rose.
4. He gave his brother a book.

Compound Sentences. Let us have a look at the sentence. He rose and went to the door of his room. There are two finite verbs-rose and went. It is made up of two clauses, each containing finite verb and joined by the conjunction ‘and’. Neither of the two clauses in the sentence is dependant on the other for its grammatical function. Such a sentence is called a compound sentence.

Complex Sentence. It is a sentence made up of a main clause and one or more subordinate clauses.
I met a boy who has plenty of push and drive, (one subordinate clause).
You may leave when I tell you to.
I liked the Shimla climate because it was healthy, but Suresh preferred the summer of Delhi, (two subordinate clauses).

Table showing conjunctions used in compound and complex sentences.

In Compound Sentences	In Complex Sentences
and	who, whom, which
but	whose
or	that
nor	if, unless
neither	when, while, till, until
either	before, after
neither ……… nor	where
either ……….. or	why
as well as	as, because, since
	so, that, lest
	though

(a) Conversion of simple sentences into complex:
1. By adding a Noun Clause to the Principal Clause.

Simple	Complex
1. You seem to be a look	It seems that you are a fool.
2. She confessed her guilt.	She contessed that she was guilty.
3. The child appears to be angry.	It appears that the child is angry.
4. It looks like rain.	It looks that it will rain.

2. By adding an Adverb Clause to the Principal Clause.

Simple	Complex
(a) 1. He is too fat to run.	He is so fat that he cannot run.
2. This load is too heavy to carry.	This load is so heavy that one cannot carry it.
(b) 1. He is the tallest boy in the class.	No other boy in the class is as tall as he.
2. I shall write to you on reaching Delhi.	I shall write to you when I reach Delhi.
3. India is the land of my birth.	India is the land where I was born.

3. By adding an Adjective Clause to the Principal Clause.

Simple	Complex
1. Columbus was the first to discover America.	He is a man on whom you can depend.
2. He is a man to be depended upon.	Columbus was the first who discovered America.
3. This is the way to earn huge	This is the way in which huge profits can be earned.

Conversion of Simple Sentences into Compound Sentences.
A Simple Sentence can be changed into a Compound Sentence by expanding a word or a phrase into co-ordinate clause by using a co-ordinate conjunction.

Compound Sentences are converted into Simple ones by using a participle or a prepositional phrase or a Gerund or Infinitive in place of one of the co-ordinate clauses.

Interchange of Simple and Compound Sentences.

1. Simple : He was honoured for his honesty.
He was hottest and so he was honoured.
2. Simple : Taking his gun he shot at the bird.
He took his gun and shot at the bird.
3. Simple : You must work hard to pass the examination.
You must work hard or you will not pass the examination.

Interchange of compound and complex Sentences:

1. Compound : Spare the rod and spoil the child.
Complex : If you spare the rod, you will spoil the child.
2. Compound : He confessed his guilt or he would have been punished.
Complex : If he had not confessed his guilt, he would have been punished.
3. Compound : Go wherever you like, only do not stay here.
Complex : You may go wherever you like but you do not stay here.
4. Compound : Leave this room or I shall compel you to do so.
Complex : Unless you leave this room, I shall compel you to do so.
5. Compound : He was a poor man, but he was always honest.
Complex : He was always honest although he was poor.
6. Compound : He stood up to speak and everyone was at once silent.
Complex : Everyone was at once silent when he stood up to speak.
7. Compound : He was very tired and therefore he fell sound asleep.
Complex : He fell sound asleep, as he was very tired.
8. Compound : Confess your fault and I will pardon you.
Complex : If you confess your fault, I will pardon you.
9. Compound : He lost a watch, but he has found it again.
Complex : He has found the watch that he lost.
10. Compound : He worked very hard therefore he succeeded.
Complex : He succeeded because he worked very hard.

Exercise 1

Change the following simple/compound sentences into complex sentences:

1. This was a vague possibility.	1. This was a possibility which was vague.
2. The old lady made one end of the silk thread fast to Tom’s tooth.	2. It was the old lady who made one end of the silk thread fast to Tom’s tooth.
3. It makes my flesh crawl to hear you.	3. It makes my flesh crawl when I hear you.
4. The groans ceased and the pain vanished from the toe.	4. When the groans ceased, the pain vanished from the toe.
5. Monday morning found Tom Sawyer miserable.	5. When Monday morning came, Tom Sawyer found himself miserable.
6. He generally began that day by wishing he had no intervening holiday.	6. He generally began that day by wishing that he had no intervening holiday.
7. When he found no ailment, he investigated again.	7. He investigated again because no ailment was found.

Exercise 2 (Textual)

Transform the following:

1. She could not prove her innocence. (Change into a Complex sentence)
2. I sold my lame horse. (Change into a Complex sentence)
3. Here comes a girl in red. (Change into a Complex sentence)
4. You can join duty as you are well now. (Change into a Compound sentence)
5. I knew her to be intelligent. (Change into a Complex sentence)
6. The doctor is hopeful of her recovery. (Change into a Complex sentence)
7. If she does not weep, she will die. (Change into a Compound sentence)
8. He was fined because of his absence. (Change into a Simple Sentence)
9. Seeing the signal, the troops marched out. (Change into a Complex sentence)
10. You must encourage him, he is sure to lose. (Change into a Compound Sentence)
Answer:
She could not prove that she was innocent.
I sold my horse that was lame.
Here comes a girl who is in red.
You are well and you can join duty.
I knew that she was intelligent.
The doctor hopes that she will recover.
She must weep or she will die.
He was fined for his absence.
The troops marched out as they got the signal.
You must encourage him or he will lose.

Exercise 3 (Textual)

Transform the following sentences:

1. Girish found a bag and a pen. (Change into Compound)
2. At last the spider reached the cobweb.
3. Seeing a tiger coming, he ran away.
4. Inspite of his wealth he is unhappy. (Change into Compound)
Answer:
1. Girish found not only a bag but a pen also.
2. The spider made the last attempt and reached the cobweb.
3. He saw a tiger and ran away.
4. He is unhappy though he is wealthy.

Exercise 4 (Textual)

Change the following sentences from Simple to Compound:

1. Having taken food, he went to school.
2. The sun having risen, the fog disappeared.
3. Having finished his task, he went to school.
4. Going to Adelaide, Anushka saw the match.
5. He laboured hard to succeed.
Answer:
1. He took food and went to school.
2. The sun rose and the fog disappeared.
3. He finished his task and went to school.
4. Anushka went to Adelaide and saw the match.
5. He laboured hard for he wanted to succeed.

Exercises 5

Change the following Compound sentences into Simple sentences:

1. He took out his gun and set out for hunting.
2. He met the Principal and came back.
3. He saw the bird and shot it.
4. He was too poor, so he could not pay the debt.
5. He was very weak, so he couldn’t walk.
Answer:
1. Taking out his gun, he set out for hunting.
2. After meeting the Principal, he came back.
3. Seeing the bird, he shot it.
4. He was too poor to pay the debt.
5. He was too weak to walk.

Exercise 6

Change the following Simple sentences into Complex sentences:

1. Reema did not admit her fault.
2. A poor man enjoys peace of mind.
3. He is too weak to run.
4. He hoped to win the prize.
5. Some trees fold their leaves at night.
6. This is not the way to clean things.
7. You cannot solve a simple problem.
8. The news spread like wild fire.
9. I could see no means of making tea.
Answer:
1. Reema did not admit that she was at fault.
2. A man who is poor enjoys peace of mind.
3. He is so weak that he cannot run.
4. He hoped that he would win the prize.
5. There are some trees that fold their leaves at night.
6. This is not the way how things are cleaned.
7. You cannot solve a problem that is simple.
8 The news spread as if it were wild fire.
9. I could see no means by which tea could be made.

Exercise 7

Change the following into Complex sentences:

1. To our surprise we found that these were no other than pelicans.
2. The island seemed to be divided into squares.
3. We turned to look in the direction pointed out.
4. He would take the rule to remeasure.
5. He stared at the approaching young be beauty.
6. They held out a ball like a pineapple.
7. I do not expect to see him back.
8. A man in a surging crowd lifted up the child.
9. They had come upon a footpath while in a field.
10. I like doing a little job of this sort.
Answer:
1. We were surprised to find that these were no other than pelicans.
2. It seemed that the island was divided into squares.
3. We looked in the direction which was pointed out.
4. He would re-measure with the rule which he had taken.
5. He stared at the young beauty that was approaching.
6. They held out a ball which was like a pineapple.
7. I do not expect that I will see him back.
8. A man in the crowd that was surging lifted up the child.
9. They had come upon a footpath which was in a field.
10. I like doing a little job which is of this sort.

Exercise 8

Do as directed:

1. We go to college to receive education. (Change into Complex)
2. O what a fall was there, my countrymen ! (Change into Assertive)
3. He is a great fool. (Change into Exclamatory)
4. I reached college in time (Change into Interrogative)
5. Why don’t you work hard ? (Change into Assertive)
6. This news is too good to be true. (Remove ‘too’)
7. As soon as I reached the college, the bejl rang. (Change into Negative)
8. Birds do not fly as fast as the aeroplane. (Change into Comparative)
9. Where did you see him ? (Change the Voice)
10. The teacher said to me, “Do you want to sing ?” (Change the Narration)
Answer:
1. We go to college so that we may receive education.
2. My countrymen, it was a very nasty fall.
3. What a fool he is !
4. Did you reach college late ?
5. You should work hard.
6. This news is so good that it cannot be true.
7. No sooner did I reach the college than the bell rang.
8. Aeroplane flies faster than birds.
9. Where was he seen by you ?
10. The teacher asked me if I wanted to sing.

Exercise 9

Do as directed:

1. It is too cold for me to go out. (Remove ‘too’)
2. None but the brave could accomplish this task. (Change into affirmative)
3. As soon as he saw the ghost, he started shuddering with fear. (Rewrite by using ‘No sooner’)
4. Why waste time in loitering outside the cinema ? (Change into Assertive)
5. Man is a wonderful piece of work. (Change into Exclamatory)
6. His hard work brought him a brilliant success. (Change into Complex sentence)
7. You ought to have helped your friend. (Change the voice)
8. The beggar said to me, “Please give me some money.” (Change the Narration)
9. This boy is more intelligent than any other student in the class. (Change the degree of Comparison)
10. Unless you are not careful, you will run into debt. (Correct the sentence)
Answer:
1. It is so cold that I cannot go out.
2. Only the brave could accomplish this task.
3. No sooner did he see the ghost, than he started shuddering with fear.
4. Don’t waste time in loitering outside the cinema.
5. What a piece of work is man !
6. He worked so hard that it brought him a brilliant success.
7. Your friend ought to have been helped by you.
8. The beggar requested me to give him some money.
9. This boy is the most intelligent student in the class.
10. Unless you are careful, you will run into debt.

Exercise 10

Do as directed:

1. These mangoes are too cheap to be good. (Remove ‘too’)
2. I saw a wounded bird. (Change into Complex sentence)
3. Honey is made by bees. (Change the Voice)
4. I have promises to keep. (Separate it in two sentences)
5. He is sometimes foolish. (Change into Negative)
6. No sooner did he see the lion than he took to he heels.
7. Can money buy health ? (Rewrite using ‘as soon as’)
8. They say honesty is the best policy. (Rewrite as ‘Statement’)
9. She is better than any other dancer in the college. (Change the Voice)
10. She left no plan untried. (Change into Superlative)
Answer:
These mangoes are so cheap that they cannot be good. I saw a bird which was wounded.
Bees make honey.
I have promises. I have to keep them.
He is not always wise.
As soon as he saw the lion, he took to his heels. Money cannot buy health.
It is said that honesty is the best policy.
She is the best dancer in the college.
She tried every plan.

Exercise 11

Do as directed:

1. As soon as the young man sensed trouble, he disappeared. (Rewrite by using ‘No sooner’)
2. The officer was so busy yesterday that he was not able to receive his visitors. (Use ‘too’)
3. Isn’t he exceptionally handsome ? (Change into assertive)
4. A soldier and afraid of bullets ! (Change into assertive)
5. My friend went to England to attain higher education. (Change into complex)
6. The patient had died before the doctor had arrived. (Correct the sentence)
7. She said to me, “Let us go to Shimla for holidaying.” (Change the narration)
8. You should look up all these words in a dictionary. (Change the voice)
9. Mount Everest is the highest peak in the world. (Change the degree of comparison)
10. Is there anything greater than the love of mother ? (Change into negative)
Answer:
1. No sooner did the young man sense trouble, than he disappeared.
2. The officer was too busy yesterday to receive his visitors.
3. He is exceptionally handsome.
4. It is disgraceful that a soldier should be afraid of bullets.
5. My friend went to England so that he might attain higher education.
6. The patient had died before the doctor arrived.
7. She suggested to me that we should go to Shimla for holidaying.
8. All these words should be looked up in a dictionary by you.
9. No other peak in the world is as high as Mount Everest.
10. There is nothing greater than the love of mother.

Exercise 12

Transform the following sentences as directed:

1. This tree is too high for me to climb. (Remove ‘too’)
2. America is the richest country in the world. (Change the degree of comparison)
3. You will have to clear the arrears. (Change the voice)
4. He said to me, “What were you doing at this time yesterday ?” (Change the narration)
5. Who does not want to be rich ? (Change into affirmative)
6. Can money buy health ? (Change into assertive)
7. He went to the hospital to consult the doctor. (Change into complex form)
8. Had you not helped me, I ………… (Complete the sentence)
9. As soon as the Minister rose to speak, the public started shouting at him. (Use ‘No sooner’)
10. Until he does not work hard, he will not pass. (Correct the sentence)
Answer:
1. This tree is so high that I cannot climb it.
2. No other country in the world is richer than America.
3. The arrears will have to be cleared by you.
4. He asked me what I was doing at that time the previous day.
5. Everybody wants to be rich.
6. Money cannot buy health.
7. He went to the hospital so that he might consult the doctor.
8. Had you not helped me, I would have been ruined.
9. No sooner did the Minister rise to speak, than the public started shouting at him.
10. Unless he works hard, he will not pass.

Exercise 13

Transform the following sentences as directed:

1. His application had been rejected. (Rewrite in the negative form)
2. When the peasants heard this they were much alarmed. (Rewrite in a simple sentence)
3. None but a fool could do it. (Change into affirmative)
4. She would utter the prettiest of thoughts. (Change the degree of comparison)
5. I am awfully glad you came. (Rewrite using too)
6. As soon as he saw the thief, he started shuddering with fear. (Rewrite by using ‘No sooner’)
7. There is no place like home. (Change into interrogative)
8. Did you see her ? (Change the voice)
9. The beggar said to me, “Please give me some money.” (Change the narration)
10. It is very nice of him that he is here today. (Change into exclamatory)
Answer:
1. His application had not been accepted.
2. The peasants were very much alarmed on hearing this.
3. Only a fool could do it.
4. She could utter prettier thoughts than others.
5. I am too glad that you came.
6. No sooner did he see the thief than he started shuddering with fear.
7. Is there any place like home ?
8. Was she seen by you ?
9. The beggar requested me to give him some money.
10. How nice of him to be here today !

Miscellaneous Exercises on All Types of Transformation

Exercise 1

Transform the following sentences as directed:

1. As soon as Sir Roger came home he called for wax candles. (Use no sooner …….. than)
2. It is probable that he will come back. (Change into a simple sentence)
3. He ran away or they would have killed him. (Change into a complex sentence)
4. What would I not give to make you happy ?) (Change into an assertive sentence)
5. Chennai is one of the largest cities in India. (Use the comparative degree of large)
Answer:
1. No sooner did Sir Roger come home than he called for wax candles.
2. Probably he will come back.
3. If he had not run away, they would have killed him.
4. I would give you anything to make you happy.
5. Chennai is larger than many other cities in India.

Exercise 2

Transform the following sentences as directed:

1. He is too good to deceive any one. (Remove too)
2. He ate twenty sandwiches in ten minutes. (Change into passive voice)
3. What a beautiful picture it is ! (Change into an assertive sentence)
4. Can this news ever be true ? (Change into an assertive sentence)
5. Calcutta (Kolkata) is the largest city in India. (Change the degree of comparison)
6. Ajay is one of the best boys of the class. (Change the degree of comparison)
Answer:
1. He is so good that he cannot deceive any one.
2. Twenty sandwiches were eaten by him in ten minutes.
3. It is a very beautiful picture.
4. This news can never be true.
5. Calcutta (Kolkata) is larger than any other city in India.
6. Ajay is better than many other boys in the class.

Exercise 3

Transform the following sentences as directed:

1. He is the brightest boy of the class. (Change the degree of comparison)
2. Are you a god that no one should oppose you ? (Change into the assertive form)
3. Don’t permit any one to leave before time. (Change into the passive voice)
4. There was no one greater than Napoleon in his day. (Change into affirmative)
5. This news is too good to be true. (Remove too)
6. There is no one that does not like him. (Change into the affirmative)
7. What a beautiful sunset ! (Change into the assertive form)
Answer:
1. No other boy of the class is as bright as he.
2. You are not a god that no one should oppose you.
3. No one should be permitted to leave before time.
4. Napoleon was greater than all in his day.
5. This news is so good that it cannot be true.
6. Everyone likes him.
7. It is a very beautiful sunset.

Exercise 4

Rewrite the following sentences as directed:

1. To eat too much makes one fat. (Remove too)
2. I have told you a thousand times not to do it. (Use already).
3. He is so honest that he will not accept a bribe. (Use too)
4. The judge suspected that the witness had been bribed. (Change into active form)
5. One more word, and I will send you out of the room. (Use if)
6. How sad was the sight of the deserted city ! (Transform to assertive)
7. Very few countries are as hot as India. (Change to comparative)
8. They will look after you well. (Change into passive form)
Answer:
1. To eat in excess makes one fat.
2. I have already told you not to do it.
3. He is too honest to accept a bribe.
4. The judge suspected that someone bribed the witness.
5. If you utter one more word, I will send you out of the room.
6. The sight of the deserted city was Very sad.
7. Very few countries are hotter than India.
8. You will be looked after well by them.

Exercise 5

Rewrite the following sentences as directed:

1. This news is too good to be true. (Remove too)
2. She is so intelligent that she will understand it. (Use too)
3. Prevention is better than cure. (Change the degree of comparison)
4. As soon as he saw his friend’s burnt house, he burst into tears. (Use no sooner …………. than)
5. I shall remember your kindness. (Use the negative without changing the meaning)
6. He does not like you. (Use a tag question)
7. No one can serve two masters. (Change into interrogative form)
8. A sailor and afraid of storm ! (Change into assertive form)
Answer:
1. This news is so good that it cannot be true.
2. She is too intelligent not to understand it.
3. Cure is not as good as prevention.
4. No sooner did he see his friends burnt house, than he burst into tears.
5. I shall never forget your kindness.
6. Does he ?
7. Can anyone serve two masters ?
8. It is a shame for a sailor to be afraid of storm.

Exercise 6

Rewrite the following sentences as directed:

1. He succeeded in everything that he attempted. (As a negative sentence)
2. Everyone has heard of Columbus. (As an interrogative sentence)
3. Give him a share of your cake. (Use the word in italics in the passive voice)
4. We cannot dispense with food and water. (Use the adjective form of the word in italics)
5. I am very pleased to accept your invitation. (Use the noun form of the word in italics)
6. That play was published after the death of its author. (Substitue a single word for the words in italics)
Answer:
1. He failed in everything that he attempted.
2. Who has not heard of Columbus ?
3. Let a share of your cake be taken by him.
4. Food and water cannot be regarded as dispensable by us.
5. I feel a great pleasure to accept your invitation.
6. That play was published posthumously.

Exercise 7

Rewrite the following sentences as directed:

1. Get out of my room and let me do my work. (Make the sentence a bit polite)
2. She spoke very gently to explain her point of view. (Use the adjective form of the italicized word)
3. The distinguished visitor received a warm welcome at the airport. (Use the adverbial form of the word in italics)
Answer:
1. Kindly allow me to do my work in the room.
2. She explained her point of view in a gentle way.
3. The distinguished visitor was welcomed warmly at the airport.

Exercise 8

Rewrite the following sentences as directed:
1. He is so weak that he cannot sit in bed. (Use too to)
2. He used force to turn the beggar out of his house. (Use the adverb form of force)
3. I am glad that my intention to become a soldier has received your assent. (Use the verb form of assent)
4. If only I could meet him and give him proper advice! (Change from an exclamatory into an assertive sentence)
5. I cannot refuse you anything. (Turn into a positive sentence)
6. Your lack of manners has shocked me much. (Change the voice)
Answer:
1. He is too weak to sit in bed.
2. He turned the beggar forcibly out of his house.
3. I am glad that you have assented to my intention to become a soldier.
4. I long to meet him and give him proper advice.
5. Iam willing to give you anything.
6. I have been shocked much by your lack of manners.

Exercise 9

Rewrite the following sentences as directed:

1. He is in the habit of grumbling. (Rewrite using used to)
2. He forcibly made his way through the crowd. (Substitute the verb form for forcibly)
3. My mother did not give me permission to swim. (Rewrite using let)
4. We will have to come again. (Rewrite using must)
5. He is so weak that he cannot walk. (Rewrite using too)
6. The battery is strong and should last twenty-four hours. (Rewrite using enough)
Answer:
1. He is used to grumbling.
2. He forced his way through the crowd.
3. My mother did not let me swim.
S 4. We must come again.
5. He is too weak to walk.
6. The battery is strong enough to last twenty-four hours.

Exercise 10

Rewrite the following sentences as directed:

1. We live in the same house but we do not like each other. (Rewrite this sentence using although)
2. Tom as well as John was in high spirits. (Rewrite using both)
3. I told you to work hard. (Change into interrogative)
4. On hearing the post office, he rushed to the news of his success. (Rewrite this in the proper order so as to make sense)
Answer:
1. Although we live in the same house, yet we do not like each other.
2. Both Tom and John were in high spirits.
3. What did I tell you ?
4. On hearing the ngws of his success, he rushed to the post office.

Additional Exercises (Solved) Based on Grammar Items

Exercise 1

Read the following sentences and identify them as simple or compound or complex sentences.
1. Our college students went to Agra and saw the Taj.
2. She studied hard yet could not pass.
3. Life is real, life is earnest.
4. We must run or we will miss the train.
5. Nalini dances on the stage.
6. Rewa who passed by me is a charming girl.
7. Girls are dancing.
8. The Principle imposed a heavy fine on the ill-mannered and abusive student.
9. This is Sanderson High School where I studied for four years.
10. We shall exchange pleasantries when we reach home.
Answer:
1. Compound
2. Compound
3. Compound
4. Compound
5. Simple
6. Complex
7. Simple
8. Simple.
9. Complex
10. Complex.

Transform the following sentences directed:

1. Who trusts a liar ? (Change into assertive)
2. Who does not love his country ? (Change into assertive)
3. What if he is divorced ? (Change into assertive)
4. When can truth die ? (Change into assertive)
5. Who wants to be miserable ? (Change into assertive)
6. Who wants to be poor ? (Change into assertive)
7. Everyone knows him. (Change into assertive)
8. Everyone wants to be a millionaire ? (Interrogative)
9. Everyone loves his motherland (Interrogative)
10. Nobody wants to lose: (Interrogative)
11. What a lovely scene ! (Assertive)
12. O for a glass of wine ! (Assertive)
13. Oh what a fall forth party ! (Assertive)
14. How pleasant the weather is ! (Assertive)
15. How shameful for an India’s opposition leader to use a fake clip to defame the government of his own country. (Assertive)
Answer:
1. Nobody trusts a liar.
2. Everybody loves his country.
3. It does not matter if he is divorced.
4. Truth can never die.
5. Nobody wants to be miserable.
6. Nobody wants to be poor.
7. There is nobody who does not know him.
8. There is nobody who does not want to be a millionaire.
9. There is no one who does not love his motherland.
10. Is there anybody who wants to lose ?
11. It is a very lovely scene.
12. I long for a glass of wine.
13. It is a nasty fall for the party.
14. The weather is very pleasant.
15. It is very shameful for an India’s opposition leader to use a government of his own country.

Exercise 3

Transform the following assertive sentences into negative:

1. He is a good person.
2. Health is wealth.
3. Look before you leap.
4. Do attend my birthday party.
5. Take exercise regularly.
Answer:
1. He is not a bad person.
2. Is not health also a wealth ?
3. Do not leap before you look.
4. Do not fail to attend my party.
5. You should not take exercise daily.