Page 1
Exercise 9.3 page: 9.14
1. Which of the following statements are true?
(i) 16: 24 = 20: 30
(ii) 21: 6 = 35: 10
(iii) 12: 18 = 28: 12
(iv) 51: 58 = 85: 102
(v) 40 men: 200 men = Rs 5: Rs 25
(vi) 99 kg: 45 kg = Rs 44: Rs 20
Solution:
(i) 16: 24 = 20: 30
It can be written as
16/24 = 20/30
Dividing 16/24 by 4/4 and 20/30 by 5/5
(16/24) ÷ (4/4) = (20/30) ÷ (5/5)
On further calculation
4/6 = 4/6
We get
2/3 = 2/3
Hence, 16: 24 = 20: 30 is true.
(ii) 21: 6 = 35: 10
It can be written as
21/6 = 35/10
Dividing 21/6 by 3/3 and 35/10 by 5/5
(21/6) ÷ (3/3) = (35/10) ÷ (5/5)
On further calculation
7/2 = 7/2
Hence, 21: 6 = 35: 10 is true.
(iii) 12: 18 = 28: 12
It can be written as
12/18 = 28/12
On further calculation
6/9 ? 14/6
Hence, 12: 18 = 28: 12 is false.
(iv) 51: 58 = 85: 102
It can be written as
51/58 = 85/102
On further calculation
51/58 ? 5/6
Hence, 51: 58 = 85: 102 is false.
(v) 40 men: 200 men = Rs 5: Rs 25
Page 2
Exercise 9.3 page: 9.14
1. Which of the following statements are true?
(i) 16: 24 = 20: 30
(ii) 21: 6 = 35: 10
(iii) 12: 18 = 28: 12
(iv) 51: 58 = 85: 102
(v) 40 men: 200 men = Rs 5: Rs 25
(vi) 99 kg: 45 kg = Rs 44: Rs 20
Solution:
(i) 16: 24 = 20: 30
It can be written as
16/24 = 20/30
Dividing 16/24 by 4/4 and 20/30 by 5/5
(16/24) ÷ (4/4) = (20/30) ÷ (5/5)
On further calculation
4/6 = 4/6
We get
2/3 = 2/3
Hence, 16: 24 = 20: 30 is true.
(ii) 21: 6 = 35: 10
It can be written as
21/6 = 35/10
Dividing 21/6 by 3/3 and 35/10 by 5/5
(21/6) ÷ (3/3) = (35/10) ÷ (5/5)
On further calculation
7/2 = 7/2
Hence, 21: 6 = 35: 10 is true.
(iii) 12: 18 = 28: 12
It can be written as
12/18 = 28/12
On further calculation
6/9 ? 14/6
Hence, 12: 18 = 28: 12 is false.
(iv) 51: 58 = 85: 102
It can be written as
51/58 = 85/102
On further calculation
51/58 ? 5/6
Hence, 51: 58 = 85: 102 is false.
(v) 40 men: 200 men = Rs 5: Rs 25
It can be written as
40/200 = 5/25
We get 40/200 = 1/5 and 5/25 = 1/5
Hence, 40 men: 200 men = Rs 5: Rs 25 is true.
(vi) 99 kg: 45 kg = Rs 44: Rs 20
It can be written as
99/45 = 44/20
Dividing the fraction by 9
(99/45) ÷ (9/9) = (44/20) ÷ (9/9)
On further calculation
11/5 = 11/5
Hence, 99 kg: 45 kg = Rs 44: Rs 20 is true.
2. Find which of the following are in proportion:
(i) 8, 16, 6, 12
(ii) 6, 2, 4, 3
(iii) 150, 250, 200, 300
Solution:
(i) 8, 16, 6, 12
We know that
8: 16 = 8/16 = 1/2
6: 12 = 6/12 = 1/2
So we get 8/16 = 6/12
Therefore, 8, 16, 6, 12 are in proportion.
(ii) 6, 2, 4, 3
We know that
6: 2 = 6/2 = 3/1
4: 3 = 4/3
So we get 3/1 ? 4/3
Therefore, 6, 2, 4, 3 are not in proportion.
(iii) 150, 250, 200, 300
We know that
150: 250 = 150/250 = 3/5
200: 300 = 200/300 = 4/6 = 2/3
So we get 3/5 ? 2/3
Therefore, 150, 250, 200, 300 are not in proportion.
3. Find x in the following proportions:
(i) x: 6 = 55: 11
(ii) 18: x = 27: 3
(iii) 7: 14 = 15: x
Page 3
Exercise 9.3 page: 9.14
1. Which of the following statements are true?
(i) 16: 24 = 20: 30
(ii) 21: 6 = 35: 10
(iii) 12: 18 = 28: 12
(iv) 51: 58 = 85: 102
(v) 40 men: 200 men = Rs 5: Rs 25
(vi) 99 kg: 45 kg = Rs 44: Rs 20
Solution:
(i) 16: 24 = 20: 30
It can be written as
16/24 = 20/30
Dividing 16/24 by 4/4 and 20/30 by 5/5
(16/24) ÷ (4/4) = (20/30) ÷ (5/5)
On further calculation
4/6 = 4/6
We get
2/3 = 2/3
Hence, 16: 24 = 20: 30 is true.
(ii) 21: 6 = 35: 10
It can be written as
21/6 = 35/10
Dividing 21/6 by 3/3 and 35/10 by 5/5
(21/6) ÷ (3/3) = (35/10) ÷ (5/5)
On further calculation
7/2 = 7/2
Hence, 21: 6 = 35: 10 is true.
(iii) 12: 18 = 28: 12
It can be written as
12/18 = 28/12
On further calculation
6/9 ? 14/6
Hence, 12: 18 = 28: 12 is false.
(iv) 51: 58 = 85: 102
It can be written as
51/58 = 85/102
On further calculation
51/58 ? 5/6
Hence, 51: 58 = 85: 102 is false.
(v) 40 men: 200 men = Rs 5: Rs 25
It can be written as
40/200 = 5/25
We get 40/200 = 1/5 and 5/25 = 1/5
Hence, 40 men: 200 men = Rs 5: Rs 25 is true.
(vi) 99 kg: 45 kg = Rs 44: Rs 20
It can be written as
99/45 = 44/20
Dividing the fraction by 9
(99/45) ÷ (9/9) = (44/20) ÷ (9/9)
On further calculation
11/5 = 11/5
Hence, 99 kg: 45 kg = Rs 44: Rs 20 is true.
2. Find which of the following are in proportion:
(i) 8, 16, 6, 12
(ii) 6, 2, 4, 3
(iii) 150, 250, 200, 300
Solution:
(i) 8, 16, 6, 12
We know that
8: 16 = 8/16 = 1/2
6: 12 = 6/12 = 1/2
So we get 8/16 = 6/12
Therefore, 8, 16, 6, 12 are in proportion.
(ii) 6, 2, 4, 3
We know that
6: 2 = 6/2 = 3/1
4: 3 = 4/3
So we get 3/1 ? 4/3
Therefore, 6, 2, 4, 3 are not in proportion.
(iii) 150, 250, 200, 300
We know that
150: 250 = 150/250 = 3/5
200: 300 = 200/300 = 4/6 = 2/3
So we get 3/5 ? 2/3
Therefore, 150, 250, 200, 300 are not in proportion.
3. Find x in the following proportions:
(i) x: 6 = 55: 11
(ii) 18: x = 27: 3
(iii) 7: 14 = 15: x
(iv) 16: 18 = x: 96
Solution:
(i) x: 6 = 55: 11
It can be written as
x/6 = 55/11
We get
x/6 = 5/1
On further calculation
x = 5 (6) = 30
(ii) 18: x = 27: 3
It can be written as
18/x = 27/3
We get
18/x = 9/1
On further calculation
x = 18/9 = 2
(iii) 7: 14 = 15: x
It can be written as
7/14 = 15/x
We get
1/2 = 15/x
On further calculation
x = 15 (2) = 30
(iv) 16: 18 = x: 96
It can be written as
16/18 = x/96
We get
8/9 = x/96
On further calculation
x = 8/9 (96) = 256/3
4. Set up all proportions from the numbers 9, 150, 105, 1750.
Solution:
The proportions from the numbers are
9: 150 = 3: 50
9: 105 = 3: 35
9: 1750
150: 9 = 50: 3
150: 105 = 10: 7
150: 1750 = 3: 35
105: 9 = 35: 3
105: 150 = 7: 10
105: 1750 = 3: 50
1750: 9
1750: 150 = 35: 3
Page 4
Exercise 9.3 page: 9.14
1. Which of the following statements are true?
(i) 16: 24 = 20: 30
(ii) 21: 6 = 35: 10
(iii) 12: 18 = 28: 12
(iv) 51: 58 = 85: 102
(v) 40 men: 200 men = Rs 5: Rs 25
(vi) 99 kg: 45 kg = Rs 44: Rs 20
Solution:
(i) 16: 24 = 20: 30
It can be written as
16/24 = 20/30
Dividing 16/24 by 4/4 and 20/30 by 5/5
(16/24) ÷ (4/4) = (20/30) ÷ (5/5)
On further calculation
4/6 = 4/6
We get
2/3 = 2/3
Hence, 16: 24 = 20: 30 is true.
(ii) 21: 6 = 35: 10
It can be written as
21/6 = 35/10
Dividing 21/6 by 3/3 and 35/10 by 5/5
(21/6) ÷ (3/3) = (35/10) ÷ (5/5)
On further calculation
7/2 = 7/2
Hence, 21: 6 = 35: 10 is true.
(iii) 12: 18 = 28: 12
It can be written as
12/18 = 28/12
On further calculation
6/9 ? 14/6
Hence, 12: 18 = 28: 12 is false.
(iv) 51: 58 = 85: 102
It can be written as
51/58 = 85/102
On further calculation
51/58 ? 5/6
Hence, 51: 58 = 85: 102 is false.
(v) 40 men: 200 men = Rs 5: Rs 25
It can be written as
40/200 = 5/25
We get 40/200 = 1/5 and 5/25 = 1/5
Hence, 40 men: 200 men = Rs 5: Rs 25 is true.
(vi) 99 kg: 45 kg = Rs 44: Rs 20
It can be written as
99/45 = 44/20
Dividing the fraction by 9
(99/45) ÷ (9/9) = (44/20) ÷ (9/9)
On further calculation
11/5 = 11/5
Hence, 99 kg: 45 kg = Rs 44: Rs 20 is true.
2. Find which of the following are in proportion:
(i) 8, 16, 6, 12
(ii) 6, 2, 4, 3
(iii) 150, 250, 200, 300
Solution:
(i) 8, 16, 6, 12
We know that
8: 16 = 8/16 = 1/2
6: 12 = 6/12 = 1/2
So we get 8/16 = 6/12
Therefore, 8, 16, 6, 12 are in proportion.
(ii) 6, 2, 4, 3
We know that
6: 2 = 6/2 = 3/1
4: 3 = 4/3
So we get 3/1 ? 4/3
Therefore, 6, 2, 4, 3 are not in proportion.
(iii) 150, 250, 200, 300
We know that
150: 250 = 150/250 = 3/5
200: 300 = 200/300 = 4/6 = 2/3
So we get 3/5 ? 2/3
Therefore, 150, 250, 200, 300 are not in proportion.
3. Find x in the following proportions:
(i) x: 6 = 55: 11
(ii) 18: x = 27: 3
(iii) 7: 14 = 15: x
(iv) 16: 18 = x: 96
Solution:
(i) x: 6 = 55: 11
It can be written as
x/6 = 55/11
We get
x/6 = 5/1
On further calculation
x = 5 (6) = 30
(ii) 18: x = 27: 3
It can be written as
18/x = 27/3
We get
18/x = 9/1
On further calculation
x = 18/9 = 2
(iii) 7: 14 = 15: x
It can be written as
7/14 = 15/x
We get
1/2 = 15/x
On further calculation
x = 15 (2) = 30
(iv) 16: 18 = x: 96
It can be written as
16/18 = x/96
We get
8/9 = x/96
On further calculation
x = 8/9 (96) = 256/3
4. Set up all proportions from the numbers 9, 150, 105, 1750.
Solution:
The proportions from the numbers are
9: 150 = 3: 50
9: 105 = 3: 35
9: 1750
150: 9 = 50: 3
150: 105 = 10: 7
150: 1750 = 3: 35
105: 9 = 35: 3
105: 150 = 7: 10
105: 1750 = 3: 50
1750: 9
1750: 150 = 35: 3
1750: 105 = 50: 3
Hence, the proportions that are formed are
9: 150 :: 105: 1750
150: 9 :: 1750: 105
1750: 150 :: 105: 9
9: 105 :: 150: 1750
5. Find the other three proportions involving terms of each of the following:
(i) 45: 30 = 24: 16
(ii) 12: 18 = 14: 21
Solution:
(i) 45: 30 = 24: 16 can be written as 3: 2 in simplest form
So the other three proportions involving terms are
45: 24 = 30: 16 can be written as 15: 8 in simplest form
30: 45 = 16: 24 can be written as 2: 3 in simplest form
16: 30 = 24: 45 can be written as 8: 15 in simplest form
(ii) 12: 18 = 14: 21 can be written as 2: 3 in simplest form
So the other three proportions involving terms are
12: 14 = 18: 21 can be written as 6: 7 in simplest form
21: 18 = 14: 12 can be written as 7: 6 in simplest form
18: 12 = 21: 14 can be written as 3: 2 in simplest form
6. If 4, x, 9 are in continued proportion, find the value of x.
Solution:
We know that 4, x, 9 are in continued proportion
It can be written as
4: x :: x: 9
We get
4/x = x/9
On further calculation
x
2
= 9 (4) = 36
So we get
x = 6
7. If in a proportion, the first, second and fourth terms are 32, 112 and 217 respectively, find the third
term.
Solution:
It is given that in a proportion the first, second and fourth terms are 32, 112 and 217
Consider x as the third term
We can write it as
32: 112 :: x: 217
On further calculation
32/112 = x/217
So we get
x = 32/112 (217) = 62
Page 5
Exercise 9.3 page: 9.14
1. Which of the following statements are true?
(i) 16: 24 = 20: 30
(ii) 21: 6 = 35: 10
(iii) 12: 18 = 28: 12
(iv) 51: 58 = 85: 102
(v) 40 men: 200 men = Rs 5: Rs 25
(vi) 99 kg: 45 kg = Rs 44: Rs 20
Solution:
(i) 16: 24 = 20: 30
It can be written as
16/24 = 20/30
Dividing 16/24 by 4/4 and 20/30 by 5/5
(16/24) ÷ (4/4) = (20/30) ÷ (5/5)
On further calculation
4/6 = 4/6
We get
2/3 = 2/3
Hence, 16: 24 = 20: 30 is true.
(ii) 21: 6 = 35: 10
It can be written as
21/6 = 35/10
Dividing 21/6 by 3/3 and 35/10 by 5/5
(21/6) ÷ (3/3) = (35/10) ÷ (5/5)
On further calculation
7/2 = 7/2
Hence, 21: 6 = 35: 10 is true.
(iii) 12: 18 = 28: 12
It can be written as
12/18 = 28/12
On further calculation
6/9 ? 14/6
Hence, 12: 18 = 28: 12 is false.
(iv) 51: 58 = 85: 102
It can be written as
51/58 = 85/102
On further calculation
51/58 ? 5/6
Hence, 51: 58 = 85: 102 is false.
(v) 40 men: 200 men = Rs 5: Rs 25
It can be written as
40/200 = 5/25
We get 40/200 = 1/5 and 5/25 = 1/5
Hence, 40 men: 200 men = Rs 5: Rs 25 is true.
(vi) 99 kg: 45 kg = Rs 44: Rs 20
It can be written as
99/45 = 44/20
Dividing the fraction by 9
(99/45) ÷ (9/9) = (44/20) ÷ (9/9)
On further calculation
11/5 = 11/5
Hence, 99 kg: 45 kg = Rs 44: Rs 20 is true.
2. Find which of the following are in proportion:
(i) 8, 16, 6, 12
(ii) 6, 2, 4, 3
(iii) 150, 250, 200, 300
Solution:
(i) 8, 16, 6, 12
We know that
8: 16 = 8/16 = 1/2
6: 12 = 6/12 = 1/2
So we get 8/16 = 6/12
Therefore, 8, 16, 6, 12 are in proportion.
(ii) 6, 2, 4, 3
We know that
6: 2 = 6/2 = 3/1
4: 3 = 4/3
So we get 3/1 ? 4/3
Therefore, 6, 2, 4, 3 are not in proportion.
(iii) 150, 250, 200, 300
We know that
150: 250 = 150/250 = 3/5
200: 300 = 200/300 = 4/6 = 2/3
So we get 3/5 ? 2/3
Therefore, 150, 250, 200, 300 are not in proportion.
3. Find x in the following proportions:
(i) x: 6 = 55: 11
(ii) 18: x = 27: 3
(iii) 7: 14 = 15: x
(iv) 16: 18 = x: 96
Solution:
(i) x: 6 = 55: 11
It can be written as
x/6 = 55/11
We get
x/6 = 5/1
On further calculation
x = 5 (6) = 30
(ii) 18: x = 27: 3
It can be written as
18/x = 27/3
We get
18/x = 9/1
On further calculation
x = 18/9 = 2
(iii) 7: 14 = 15: x
It can be written as
7/14 = 15/x
We get
1/2 = 15/x
On further calculation
x = 15 (2) = 30
(iv) 16: 18 = x: 96
It can be written as
16/18 = x/96
We get
8/9 = x/96
On further calculation
x = 8/9 (96) = 256/3
4. Set up all proportions from the numbers 9, 150, 105, 1750.
Solution:
The proportions from the numbers are
9: 150 = 3: 50
9: 105 = 3: 35
9: 1750
150: 9 = 50: 3
150: 105 = 10: 7
150: 1750 = 3: 35
105: 9 = 35: 3
105: 150 = 7: 10
105: 1750 = 3: 50
1750: 9
1750: 150 = 35: 3
1750: 105 = 50: 3
Hence, the proportions that are formed are
9: 150 :: 105: 1750
150: 9 :: 1750: 105
1750: 150 :: 105: 9
9: 105 :: 150: 1750
5. Find the other three proportions involving terms of each of the following:
(i) 45: 30 = 24: 16
(ii) 12: 18 = 14: 21
Solution:
(i) 45: 30 = 24: 16 can be written as 3: 2 in simplest form
So the other three proportions involving terms are
45: 24 = 30: 16 can be written as 15: 8 in simplest form
30: 45 = 16: 24 can be written as 2: 3 in simplest form
16: 30 = 24: 45 can be written as 8: 15 in simplest form
(ii) 12: 18 = 14: 21 can be written as 2: 3 in simplest form
So the other three proportions involving terms are
12: 14 = 18: 21 can be written as 6: 7 in simplest form
21: 18 = 14: 12 can be written as 7: 6 in simplest form
18: 12 = 21: 14 can be written as 3: 2 in simplest form
6. If 4, x, 9 are in continued proportion, find the value of x.
Solution:
We know that 4, x, 9 are in continued proportion
It can be written as
4: x :: x: 9
We get
4/x = x/9
On further calculation
x
2
= 9 (4) = 36
So we get
x = 6
7. If in a proportion, the first, second and fourth terms are 32, 112 and 217 respectively, find the third
term.
Solution:
It is given that in a proportion the first, second and fourth terms are 32, 112 and 217
Consider x as the third term
We can write it as
32: 112 :: x: 217
On further calculation
32/112 = x/217
So we get
x = 32/112 (217) = 62
8. Show that the following numbers are in continued proportion:
(i) 36, 90, 225
(ii) 48, 60, 75
(iii) 16, 84, 441
Solution:
(i) 36, 90, 225
Consider the fraction 36/90
By dividing the fraction by 18
We get
36/90 = 2/5
Consider the fraction 90/225
By dividing the fraction by 45
We get
90/225 = 2/5
Hence, 36: 90 :: 90: 225.
(ii) 48, 60, 75
Consider the fraction 48/60
By dividing the fraction by 12
We get
48/60 = 4/5
Consider the fraction 60/75
By dividing the fraction by 15
We get
60/75 = 4/5
Hence, 48: 60 :: 60: 75.
(iii) 16, 84, 441
Consider the fraction 16/84
By dividing the fraction by 4
We get
16/84 = 4/21
Consider the fraction 84/441
By dividing the fraction by 21
We get
84/441 = 4/21
Hence, 16: 84 :: 84: 441.
9. The ratio of the length of a school ground to its width is 5: 2. Find its length if the width is 40 metres.
Solution:
It is given that
Ratio of length of a school ground to its width = 5: 2
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