Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses)

Selina Publishers Concise Mathematics Class 10 ICSE Solutions Chapter 7 Ratio and Proportion (Including Properties and Uses)

Ratio and Proportion Exercise 7A – Selina Concise Mathematics Class 10 ICSE Solutions

Question 1.
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 1
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 2

Question 2.
If x: y = 4: 7, find the value of (3x + 2y): (5x + y).
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 3

Question 3.
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 4
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 5

Question 4.
If (a – b): (a + b) = 1: 11, find the ratio (5a + 4b + 15): (5a – 4b + 3).
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 6

Question 5.
selina-icse-solutions-class-10-maths-ratio-proportion-7a-q5-i
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 8

Question 6.
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 9
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 10

Question 7.
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 11
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 12

Question 8.
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 13
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 14

Question 9.
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 15
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 16

Question 10.
A school has 630 students. The ratio of the number of boys to the number of girls is 3 : 2. This ratio changes to 7 : 5 after the admission of 90 new students. Find the number of newly admitted boys.
Solution:
selina-icse-solutions-class-10-maths-ratio-proportion-7a-q10

Question 11.
What quantity must be subtracted from each term of the ratio 9: 17 to make it equal to 1: 3?
Solution:
Let x be subtracted from each term of the ratio 9: 17.
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 18
Thus, the required number which should be subtracted is 5.

Question 12.
The monthly pocket money of Ravi and Sanjeev are in the ratio 5 : 7. Their expenditures are in the ratio 3 : 5. If each saves Rs. 80 every month, find their monthly pocket money.
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 19

Question 13.
The work done by (x – 2) men in (4x + 1) days and the work done by (4x + 1) men in (2x – 3) days are in the ratio 3: 8. Find the value of x.
Solution:
Assuming that all the men do the same amount of work in one day and one day work of each man = 1 units, we have,
Amount of work done by (x – 2) men in (4x + 1) days
= Amount of work done by (x – 2)(4x + 1) men in one day
= (x – 2)(4x + 1) units of work
Similarly,
Amount of work done by (4x + 1) men in (2x – 3) days
= (4x + 1)(2x – 3) units of work
According to the given information,
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 20

Question 14.
The bus fare between two cities is increased in the ratio 7: 9. Find the increase in the fare, if:
(i) the original fare is Rs 245;
(ii) the increased fare is Rs 207.
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 21

Question 15.
By increasing the cost of entry ticket to a fair in the ratio 10: 13, the number of visitors to the fair has decreased in the ratio 6: 5. In what ratio has the total collection increased or decreased?
Solution:
Let the cost of the entry ticket initially and at present be 10 x and 13x respectively.
Let the number of visitors initially and at present be 6y and 5y respectively.
Initially, total collection = 10x × 6y = 60 xy
At present, total collection = 13x × 5y = 65 xy
Ratio of total collection = 60 xy: 65 xy = 12: 13
Thus, the total collection has increased in the ratio 12: 13.

Question 16.
In a basket, the ratio between the number of oranges and the number of apples is 7: 13. If 8 oranges and 11 apples are eaten, the ratio between the number of oranges and the number of apples becomes 1: 2. Find the original number of oranges and the original number of apples in the basket.
Solution:
Let the original number of oranges and apples be 7x and 13x.
According to the given information,
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 22
Thus, the original number of oranges and apples are 7 × 5 = 35 and 13 × 5 = 65 respectively.

Question 17.
In a mixture of 126 kg of milk and water, milk and water are in ratio 5 : 2. How much water must be added to the mixture to make this ratio 3 : 2?
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 23

Question 18.
(A) If A: B = 3: 4 and B: C = 6: 7, find:
(i) A: B: C
(ii) A: C
(B) If A : B = 2 : 5 and A : C = 3 : 4, find
(i) A : B : C
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 24

Question 19(i).
If 3A = 4B = 6C; find A: B: C.
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 25

Question 19(ii).
If 2a = 3b and 4b = 5c, find: a : c.
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 26

Question 20.
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 27
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 28

Question 21.
Find duplicate ratio of:
(i) 3: 4 (ii) 3√3 : 2√5
Solution:
(i) Duplicate ratio of 3 : 4 = 32 : 42 = 9 : 16
(ii) Duplicate ratio of 3√3 : 2√5 = (3√3)² : (2√5)² = 27 : 20

Question 22.
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 29
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 30

Question 23.
Find sub-duplicate ratio of:
(i) 9: 16 (ii) (x – y)4: (x + y)6
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 31

Question 24.
Find the sub-triplicate ratio of:
(i) 64: 27 (ii) x3: 125y3
Solution:
(i) Sub-triplicate ratio of 64 : 27 = ∛64 : ∛27 = 4 : 3
(ii) Sub-triplicate ratio of x³ : 125y³ = ∛x³ : ∛125y³ = x : 5y

Question 25.
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 32
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 33

Question 26.
If (x + 3) : (4x + 1) is the duplicate ratio of 3 : 5, find the value of x.
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 34

Question 27.
If m: n is the duplicate ratio of m + x: n + x; show that x2 = mn.
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 35

Question 28.
If (3x – 9) : (5x + 4) is the triplicate ratio of 3 : 4, find the value of x.
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 36

Question 29.
Find the ratio compounded of the reciprocal ratio of 15: 28, the sub-duplicate ratio of 36: 49 and the triplicate ratio of 5: 4.
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 37

Question 30(a).
If r=pq, show that p : q is the duplicate ratio of (p + r) : (q + r).
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 38

Question 30(b).
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 39
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 39

Ratio and Proportion Exercise 7B – Selina Concise Mathematics Class 10 ICSE Solutions

Question 1.
Find the fourth proportional to:
(i) 1.5, 4.5 and 3.5 (ii) 3a, 6a2 and 2ab2
Solution:
(i) Let the fourth proportional to 1.5, 4.5 and 3.5 be x.
⇒ 1.5 : 4.5 = 3.5 : x
⇒ 1.5 × x = 3.5 4.5
⇒ x = 10.5
(ii) Let the fourth proportional to 3a, 6a2 and 2ab2 be x.
⇒ 3a : 6a2 = 2ab2 : x
⇒ 3a × x = 2ab2 6a2
⇒ 3a × x = 12a3b2
⇒ x = 4a2b2

Question 2.
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 41
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 42

Question 3.
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 43
Solution:
(i) Let the mean proportional between 6 + 3√3 and 8 – 4√3 be x.
⇒ 6 + 3√3, x and 8 – 4√3 are in continued proportion.
⇒ 6 + 3√3 : x = x : 8 – 4√3
⇒ x × x = (6 + 3√3) (8 – 4√3)
⇒ x2 = 48 + 24√3- 24√3 – 36
⇒ x2 = 12
⇒ x = 2√3
(ii) Let the mean proportional between a – b and a3 – a2b be x.
⇒ a – b, x, a3 – a2b are in continued proportion.
⇒ a – b : x = x : a3 – a2b
⇒ x × x = (a – b) (a3 – a2b)
⇒ x2 = (a – b) a2(a – b) = [a(a – b)]2
⇒ x = a(a – b)

Question 4.
If x + 5 is the mean proportional between x + 2 and x + 9; find the value of x.
Solution:
Given, x + 5 is the mean proportional between x + 2 and x + 9.
⇒ (x + 2), (x + 5) and (x + 9) are in continued proportion.
⇒ (x + 2) : (x + 5) = (x + 5) : (x + 9)
⇒ (x + 5)2 = (x + 2)(x + 9)
⇒ x2 + 25 + 10x = x2 + 2x + 9x + 18
⇒ 25 – 18 = 11x – 10x
⇒ x = 7

Question 5.
If x2, 4 and 9 are in continued proportion, find x.
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 44

Question 6.
What least number must be added to each of the numbers 6, 15, 20 and 43 to make them proportional?
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 47

Question 7(i).
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 48
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 49

Question 7(ii).
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 50
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 51

Question 7(iii).
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 52
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 53

Question 8.
What least number must be subtracted from each of the numbers 7, 17 and 47 so that the remainders are in continued proportion?
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 54

Question 9.
If y is the mean proportional between x and z; show that xy + yz is the mean proportional between x2+y2 and y2+z2.
Solution:
Since y is the mean proportion between x and z
Therefore, y2 = xz
Now, we have to prove that xy+yz is the mean proportional between x2+y2 and y2+z2, i.e.,
selina-icse-solutions-class-10-maths-ratio-proportion-ex-10b-q9
LHS = RHS
Hence, proved.

Question 10.
If q is the mean proportional between p and r, show that:
pqr (p + q + r)3 = (pq + qr + rp)3.
Solution:
Given, q is the mean proportional between p and r.
⇒ q2 = pr
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 56

Question 11.
If three quantities are in continued proportion; show that the ratio of the first to the third is the duplicate ratio of the first to the second.
Solution:
Let x, y and z be the three quantities which are in continued proportion.
Then, x : y :: y : z ⇒ y2 = xz ….(1)
Now, we have to prove that
x : z = x2 : y2
That is we need to prove that
xy2 = x2z
LHS = xy2 = x(xz) = x2z = RHS [Using (1)]
Hence, proved.

Question 12.
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 57
Solution:
Given, y is the mean proportional between x and z.
⇒ y2 = xz
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 58

Question 13.
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 59
Solution:
selina-icse-solutions-class-10-maths-ratio-proportion-ex-10b-q13

Question 14.
Find two numbers such that the mean mean proportional between them is 12 and the third proportional to them is 96.
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 62

Question 15.
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 63
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 64

Question 16.
If p: q = r: s; then show that:
mp + nq : q = mr + ns : s.
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 65

Question 17.
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 66
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 67

Question 18.
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 68
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 69

Question 19.
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 70
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 71

Question 20.
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 72
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 73

Ratio and Proportion Exercise 7C – Selina Concise Mathematics Class 10 ICSE Solutions

Question 1.
If a : b = c : d, prove that:
(i) 5a + 7b : 5a – 7b = 5c + 7d : 5c – 7d.
(ii) (9a + 13b) (9c – 13d) = (9c + 13d) (9a – 13b).
(iii) xa + yb : xc + yd = b : d.
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 74

Question 2.
If a : b = c : d, prove that:
(6a + 7b) (3c – 4d) = (6c + 7d) (3a – 4b).
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 75

Question 3.
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 76
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 77

Question 4.
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 78
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 79

Question 5.
If (7a + 8b) (7c – 8d) = (7a – 8b) (7c + 8d), prove that a: b = c: d.
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 80

Question 6.
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 82
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 83
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 84

Question 7.
If (a + b + c + d) (a – b – c + d) = (a + b – c – d) (a – b + c – d), prove that a: b = c: d.
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 85

Question 8.
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 86
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 87

Question 9.
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 88
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 89

Question 10.
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 90
Solution:
Given, a, b and c are in continued proportion.
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 91
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 92

Question 11.
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 93
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 94
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 95

Question 12.
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 96
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 97

Question 13.
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 98
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 99

Question 14.
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 100
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 101

Question 15.
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 102
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 103

Ratio and Proportion Exercise 7D – Selina Concise Mathematics Class 10 ICSE Solutions

Question 1.
If a: b = 3: 5, find:
(10a + 3b): (5a + 2b)
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 104

Question 2.
If 5x + 6y: 8x + 5y = 8: 9, find x: y.
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 105

Question 3.
If (3x – 4y): (2x – 3y) = (5x – 6y): (4x – 5y), find x: y.
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 106

Question 4.
Find the:
(i) duplicate ratio of 2√2 : 3√5
(ii) triplicate ratio of 2a: 3b
(iii) sub-duplicate ratio of 9x2a: 25y6b2
(iv) sub-triplicate ratio of 216: 343
(v) reciprocal ratio of 3: 5
(vi) ratio compounded of the duplicate ratio of 5: 6, the reciprocal ratio of 25: 42 and the sub-duplicate ratio of 36: 49.
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 107

Question 5.
Find the value of x, if:
(i) (2x + 3): (5x – 38) is the duplicate ratio of √5 : √6
(ii) (2x + 1): (3x + 13) is the sub-duplicate ratio of 9: 25.
(iii) (3x – 7): (4x + 3) is the sub-triplicate ratio of 8: 27.
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 108

Question 6.
What quantity must be added to each term of the ratio x: y so that it may become equal to c: d?
Solution:
Let the required quantity which is to be added be p.
Then, we have:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 109

Question 7.
A woman reduces her weight in the ratio 7 : 5. What does her weight become if originally it was 84 kg?
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 110

Question 8.
If 15(2x2 – y2) = 7xy, find x: y; if x and y both are positive.
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 111

Question 9.
Find the:
(i) fourth proportional to 2xy, x2 and y2.
(ii) third proportional to a2 – b2 and a + b.
(iii) mean proportional to (x – y) and (x3 – x2y).
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 112

Question 10.
Find two numbers such that the mean proportional between them is 14 and third proportional to them is 112.
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 113

Question 11.
If x and y be unequal and x: y is the duplicate ratio of x + z and y + z, prove that z is mean proportional between x and y.
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 114

Question 12.
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 115
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 116

Question 13.
If (4a + 9b) (4c – 9d) = (4a – 9b) (4c + 9d), prove that:
a: b = c: d.
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 117

Question 14.
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 118
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 119

Question 15.
There are 36 members in a student council in a school and the ratio of the number of boys to the number of girls is 3: 1. How any more girls should be added to the council so that the ratio of the number of boys to the number of girls may be 9: 5?
Solution:
Ratio of number of boys to the number of girls = 3: 1
Let the number of boys be 3x and number of girls be x.
3x + x = 36
4x = 36
x = 9
∴ Number of boys = 27
Number of girls = 9
Le n number of girls be added to the council.
From given information, we have:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 120
Thus, 6 girls are added to the council.

Question 16.
If 7x – 15y = 4x + y, find the value of x: y. Hence, use componendo and dividend to find the values of:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 121
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 122

Question 17.
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 123
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 124

Question 18.
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 125
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 126

Question 19.
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 126
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 128

Question 20.
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 129
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 130

Question 21.
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 131
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 132

Question 22.
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 133
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 134

Question 23.
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 136
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 137

Question 24.
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 138

Question 25.
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 139

Question 26.
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 140

Question 27.
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 141.
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 142

Question 28.
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 140
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 141

Question 29.
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 142
Solution:
Since, q is the mean proportional between p and r,
q2 = pr
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 143

Question 30.
If a, b and c are in continued proportion, prove that:
a: c = (a2 + b2) : (b2 + c2)
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Ratio and Proportion (Including Properties and Uses) - 144

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