# Chapter 7 : Ratio and Proportion (Including Properties and Uses) - Selina Solutions for Class 10 Maths ICSE

Mathematics is one of the crucial and most scoring subjects in ICSE Class 10. TopperLearning presents study materials for ICSE Class 10 Mathematics which will enable students to score well in the board examination. The syllabus includes certain challenging concepts like Value Added Tax, Ratio and Proportion, Quadric Equations among others which require effective study materials. Our study materials for ICSE Class 10 consist of video lessons, question banks, sample papers, revision notes and past year papers which improve the quality of learning.

## Chapter 7 - Ratio and Proportion (Including Properties and Uses) Excercise Ex. 7(A)

If a: b = 5: 3, find: .

If x: y = 4: 7, find the value of (3x + 2y): (5x + y).

If a: b = 3: 8, find the value of .

If (a - b): (a + b) = 1: 11, find the ratio (5a + 4b + 15): (5a - 4b + 3).

Hence, (5a + 4b + 15): (5a - 4b + 3) = 5: 1

If.

Find, when x^{2} + 6y^{2}
= 5xy.

x^{2} + 6y^{2} =
5xy

Dividing both sides by y^{2},
we get,

If the ratio between 8 and 11 is the same as the ratio of 2x - y to x + 2y, find the value of

Given,

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.

What quantity must be subtracted from each term of the ratio 9: 17 to make it equal to 1: 3?

Let x be subtracted from each term of the ratio 9: 17.

Thus, the required number which should be subtracted is 5.

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.

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.

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,

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.

According to the given information,

Increased (new) bus fare = original bus fare

(i) We have:

Increased (new) bus fare = Rs 245 = Rs 315

Increase in fare = Rs 315 - Rs 245 = Rs 70

(ii) We have:

Rs 207 = original bus fare

Original bus fare =

Increase in fare = Rs 207 - Rs 161 = Rs 46

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?

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.

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.

Let the original number of oranges and apples be 7x and 13x.

According to the given information,

Thus, the original number of oranges and apples are 7 5 = 35 and 13 5 = 65 respectively.

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?

(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

(A)

(i)

(ii)

(B)

(i)

If 3A = 4B = 6C; find A: B: C.

3A = 4B = 6C

3A = 4B

4B = 6C

Hence, A: B: C = 4: 3: 2

If 2a = 3b and 4b = 5c, find: a : c.

Find the compound ratio of:

(i) 2: 3, 9: 14 and 14: 27

(ii) 2a: 3b, mn: x^{2} and x: n.

(iii)

(i) Required compound ratio = 2 9 14: 3 14 27

(ii) Required compound ratio = 2a mn x: 3b x^{2} n

(iii) Required compound ratio =

Find duplicate ratio of:

(i) 3: 4 (ii)

(i) Duplicate ratio of 3: 4 = 3^{2}: 4^{2} = 9: 16

(ii) Duplicate ratio of

Find the triplicate ratio of:

(i) 1: 3 (ii)

(i) Triplicate ratio of 1: 3 = 1^{3}:
3^{3} = 1: 27

(ii) Triplicate ratio of

Find sub-duplicate ratio of:

(i) 9: 16 (ii) (x -
y)^{4}: (x + y)^{6}

(i) Sub-duplicate ratio of 9: 16 =

(ii) Sub-duplicate ratio of(x - y)^{4}:
(x + y)^{6}

=

Find the sub-triplicate ratio of:

(i) 64: 27 (ii) x^{3}:
125y^{3}

(i) Sub-triplicate ratio of 64: 27 =

(ii) Sub-triplicate ratio of x^{3}:
125y^{3} =

Find the reciprocal ratio of:

(i) 5: 8 (ii)

(i) Reciprocal ratio of 5: 8 =

(ii) Reciprocal ratio of

If (x + 3) : (4x + 1) is the duplicate ratio of 3 : 5, find the value of x.

If m: n is the duplicate ratio of
m + x: n + x; show that x^{2} = mn.

If (3x - 9) : (5x + 4) is the triplicate ratio of 3 : 4, find the value of x.

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.

Reciprocal ratio of 15: 28 = 28: 15

Sub-duplicate ratio of 36: 49 =

Triplicate ratio of 5: 4 = 5^{3}:
4^{3} = 125: 64

Required compounded ratio

=

If r^{2 }_{=}pq, show that p : q is the duplicate ratio of (p + r) : (q + r).

## Chapter 7 - Ratio and Proportion (Including Properties and Uses) Excercise Ex. 7(B)

Find the fourth proportional to:

(i)
1.5, 4.5 and 3.5 (ii) 3a, 6a^{2} and 2ab^{2}

(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

(i)
Let the fourth proportional to 3a, 6a^{2} and 2ab^{2} be x.

3a : 6a^{2}
= 2ab^{2} : x

3a x = 2ab^{2} 6a^{2}

3a x = 12a^{3}b^{2}

x = 4a^{2}b^{2}

Find the third proportional to:

(i)
2 and 4 (ii)
a - b and a^{2} - b^{2}

(i) Let the third proportional to 2 and 4 be x.

2, 4, x are in continued proportion.

2 : 4 = 4 : x

(ii)
Let the third proportional to a - b and a^{2} - b^{2} be x.

a - b, a^{2}
- b^{2}, x are in continued proportion.

a - b : a^{2}
- b^{2} = a^{2} - b^{2} : x

Find the mean proportional between:

(i) 6 + 3 and 8 - 4

(ii) a - b and a^{3} - a^{2}b

(i) Let the mean proportional between 6 + 3 and 8 - 4 be x.

6 + 3, x and 8 - 4 are in continued proportion.

6 + 3 : x = x : 8 - 4

x x = (6 + 3) (8 - 4)

x^{2 }= 48 + 24- 24 - 36

x^{2 }= 12

x= 2

(ii) Let the mean proportional between a - b and a^{3} - a^{2}b be x.

a - b, x, a^{3} - a^{2}b are in continued proportion.

a - b : x = x : a^{3} - a^{2}b

x x = (a - b) (a^{3} - a^{2}b)

x^{2} = (a - b) a^{2}(a - b) = [a(a - b)]^{2}

x = a(a - b)

If x + 5 is the mean proportional between x + 2 and x + 9; find the value of x.

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)

x^{2}
+ 25 + 10x = x^{2} + 2x + 9x + 18

25 - 18 = 11x - 10x

x = 7

If x^{2}, 4 and 9 are in continued proportion, find x.

What least number must be added to each of the numbers 6, 15, 20 and 43 to make them proportional?

Let the number added be x.

(6 + x) : (15 + x) :: (20 + x) (43 + x)

Thus, the required number which should be added is 3.

What least number must be subtracted from each of the numbers 7, 17 and 47 so that the remainders are in continued proportion?

Let the number subtracted be x.

(7 - x) : (17 - x) :: (17 - x) (47 - x)

Thus, the required number which should be subtracted is 2.

If
y is the mean proportional between x and z; show that xy + yz is the mean
proportional between x^{2}+y^{2} and y^{2}+z^{2}.

Since y is the mean proportion between x and z

Therefore, y^{2 }= xz

Now,
we have to prove that xy+yz is the mean proportional between x^{2}+y^{2}
and y^{2}+z^{2}, i.e.,

LHS = RHS

Hence, proved.

If q is the mean proportional between p and r, show that:

pqr
(p + q + r)^{3} = (pq + qr + rp)^{3}.

Given, q is the mean proportional between p and r.

q^{2} =
pr

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.

Let x, y and z be the three quantities which are in continued proportion.

Then, x : y :: y : z y^{2} = xz ....(1)

Now, we have to prove that

x : z = x^{2 }: y^{2}

That is we need to prove that

xy^{2 }= x^{2}z

LHS = xy^{2} = x(xz) = x^{2}z = RHS [Using (1)]

Hence, proved.

If y is the mean proportional between x and z, prove that:

Given, y is the mean proportional between x and z.

y^{2} = xz

Given four quantities a, b, c and d are in proportion. Show that:

LHS = RHS

Hence proved.

Find two numbers such that the mean mean proportional between them is 12 and the third proportional to them is 96.

Let a and b be the two numbers, whose mean proportional is 12.

Now, third proportional is 96

Therefore, the numbers are 6 and 24.

Find the third proportional to

Let the required third proportional be p.

, p are in continued proportion.

If p: q = r: s; then show that:

mp + nq : q = mr + ns : s.

Hence, mp + nq : q = mr + ns : s.

If p + r = mq and ; then prove that p : q = r : s.

Hence, proved.

## Chapter 7 - Ratio and Proportion (Including Properties and Uses) Excercise Ex. 7(C)

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.

If a : b = c : d, prove that:

(6a + 7b) (3c - 4d) = (6c + 7d) (3a - 4b).

Given, , prove that:

If ; then prove that:

x: y = u: v.

If (7a + 8b) (7c - 8d) = (7a - 8b) (7c + 8d), prove that a: b = c: d.

Given,

Applying componendo and dividendo,

Hence, a: b = c: d.

(i) If x = , find the value of:

.

(ii) If a = , find the value of:

(i) x =

(ii)

If (a + b + c + d) (a - b - c + d) = (a + b - c - d) (a - b + c - d), prove that a: b = c: d.

If , show that 2ad = 3bc.

If ; prove that: .

Given,

If a, b and c are in continued proportion, prove that:

Given, a, b and c are in continued proportion.

Using properties of proportion, solve for x:

If , prove that: 3bx^{2} - 2ax + 3b = 0.

Since,

Applying componendo and dividendo, we get,

Squaring both sides,

Again applying componendo and dividendo,

3bx^{2} + 3b = 2ax

3bx^{2} - 2ax + 3b = 0.

If , express n in terms of x and m.

Applying componendo and dividendo,

If , show that:

nx = my.

Applying componendo and dividendo,

## Chapter 7 - Ratio and Proportion (Including Properties and Uses) Excercise Ex. 7(D)

If a: b = 3: 5, find:

(10a + 3b): (5a + 2b)

Given,

If 5x + 6y: 8x + 5y = 8: 9, find x: y.

If (3x - 4y): (2x - 3y) = (5x - 6y): (4x - 5y), find x: y.

(3x - 4y): (2x - 3y) = (5x - 6y): (4x - 5y)

Find the:

(i) duplicate ratio of

(ii) triplicate ratio of 2a: 3b

(iii)
sub-duplicate ratio of 9x^{2}a^{4 }: 25y^{6}b^{2}

(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.

(i) Duplicate ratio of

(ii)
Triplicate ratio of 2a: 3b = (2a)^{3}: (3b)^{3} = 8a^{3}
: 27b^{3}

(iii)
Sub-duplicate ratio of 9x^{2}a^{4 }: 25y^{6}b^{2}
=

(iv) Sub-triplicate ratio of 216: 343 =

(v) Reciprocal ratio of 3: 5 = 5: 3

(vi) Duplicate ratio of 5: 6 = 25: 36

Reciprocal ratio of 25: 42 = 42: 25

Sub-duplicate ratio of 36: 49 = 6: 7

Required compound ratio =

Find the value of x, if:

(i) (2x + 3): (5x - 38) is the duplicate ratio of

(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.

(i) (2x + 3): (5x - 38) is the duplicate ratio of

Duplicate ratio of

(ii) (2x + 1): (3x + 13) is the sub-duplicate ratio of 9: 25

Sub-duplicate ratio of 9: 25 = 3: 5

(iii) (3x - 7): (4x + 3) is the sub-triplicate ratio of 8: 27

Sub-triplicate ratio of 8: 27 = 2: 3

What quantity must be added to each term of the ratio x: y so that it may become equal to c: d?

Let the required quantity which is to be added be p.

Then, we have:

A woman reduces her weight in the ratio 7 : 5. What does her weight become if originally it was 84 kg?

If
15(2x^{2} - y^{2}) = 7xy, find x: y; if x and y both are
positive.

15(2x^{2}
- y^{2}) = 7xy

Find the:

(i)
fourth proportional to 2xy, x^{2} and y^{2}.

(ii)
third proportional to a^{2} - b^{2} and a + b.

(iii)
mean proportional to (x - y) and (x^{3} - x^{2}y).

(i)
Let the fourth proportional to 2xy, x^{2} and y^{2} be n.

2xy: x^{2}
= y^{2}: n

2xy n = x^{2} y^{2}

n =

(ii)
Let the third proportional to a^{2} - b^{2} and a + b be n.

a^{2}
- b^{2}, a + b and n are in continued proportion.

a^{2}
- b^{2} : a + b = a + b : n

n =

(iii)
Let the mean proportional to (x - y) and (x^{3} - x^{2}y) be
n.

(x - y), n, (x^{3}
- x^{2}y) are in continued proportion

(x - y) : n =
n : (x^{3} - x^{2}y)

Find two numbers such that the mean proportional between them is 14 and third proportional to them is 112.

Let the required numbers be a and b.

Given, 14 is the mean proportional between a and b.

a: 14 = 14: b

ab = 196

Also, given, third proportional to a and b is 112.

a: b = b: 112

Using (1), we have:

Thus, the two numbers are 7 and 28.

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.

Given,

Hence, z is mean proportional between x and y.

If , find the value of .

If (4a + 9b) (4c - 9d) = (4a - 9b) (4c + 9d), prove that:

a: b = c: d.

If , show that:

(a + b) : (c + d) =

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?

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:

Thus, 6 girls are added to the council.

If 7x - 15y = 4x + y, find the value of x: y. Hence, use componendo and dividend to find the values of:

7x - 15y = 4x + y

7x - 4x = y + 15y

3x = 16y

If , use properties of proportion to find:

(i) m: n

(ii)

If x, y, z are in continued proportion, prove that

x, y, z are in continued proportion,

Therefore,

(By alternendo)

Hence Proved.

Given x =.

Use
componendo and dividendo to prove that b^{2} =.

x =

By componendo and dividendo,

Squaring both sides,

By componendo and dividendo,

b^{2} =

Hence Proved.

If , find:

Using componendo and dividendo find the value of x:

If b is the mean proportion between a and c, show that:

Given that b is the mean proportion between a and c.

If , use properties of proportion to find:

i.

ii.

From (i),

i. If x and y both are positive and (2x^{2}
- 5y^{2}): xy = 1: 3, find x: y.

ii. Find x, if.

i. (2x^{2} - 5y^{2}): xy = 1: 3

ii.

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