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

Question 1

If a: b = 5: 3, find: .

Solution 1

Question 2

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

Solution 2

Question 3

If a: b = 3: 8, find the value of .

Solution 3

Question 4

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

Solution 4

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

Question 5

Solution 5

 

Question 6

If.

Solution 6

Question 7

Find, when x² + 6y² = 5xy.

Solution 7

x² + 6y² = 5xy

Dividing both sides by y², we get,

Question 8

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

Solution 8

Given, 

Question 9

 

Solution 9

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 10

Question 11

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

Solution 11

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

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 12

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 13

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,

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 14

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

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 15

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 16

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.

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 17

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 18

(A)

(i)

 

(ii)

(B)

(i)

Question 19(i)

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

Solution 19(i)

3A = 4B = 6C

3A = 4B 

4B = 6C 

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

Question 19(ii)

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

Solution 19(ii)

Question 20

Find the compound ratio of:

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

(ii) 2a: 3b, mn: x² and x: n.

(iii) 

Solution 20

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

 

(ii) Required compound ratio = 2a mn x: 3b  x² n

 

(iii) Required compound ratio = 

Question 21

Find duplicate ratio of:

(i) 3: 4 (ii) 

Solution 21

(i) Duplicate ratio of 3: 4 = 3²: 4² = 9: 16

(ii) Duplicate ratio of 

Question 22

Find the triplicate ratio of:

(i) 1: 3 (ii) 

Solution 22

(i) Triplicate ratio of 1: 3 = 1³: 3³ = 1: 27

(ii) Triplicate ratio of 

Question 23

Find sub-duplicate ratio of:

(i) 9: 16 (ii) (x - y)⁴: (x + y)⁶

Solution 23

(i) Sub-duplicate ratio of 9: 16 = 

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

= 

Question 24

Find the sub-triplicate ratio of:

(i) 64: 27 (ii) x³: 125y³

Solution 24

(i) Sub-triplicate ratio of 64: 27 = 

(ii) Sub-triplicate ratio of x³: 125y³ = 

Question 25

Find the reciprocal ratio of:

(i) 5: 8 (ii) 

Solution 25

(i) Reciprocal ratio of 5: 8 = 

(ii) Reciprocal ratio of 

Question 26

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

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

Solution 26

Question 27

If m: n is the duplicate ratio of m + x: n + x; show that x² = mn.

Solution 27

Question 28

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

Solution 28

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 29

Reciprocal ratio of 15: 28 = 28: 15

Sub-duplicate ratio of 36: 49 = 

Triplicate ratio of 5: 4 = 5³: 4³ = 125: 64

Required compounded ratio

= 

Question 30(a)

If r² ₌pq, show that p : q is the duplicate ratio of (p + r) : (q + r).

Solution 30(a)

Question 30(b)

Solution 30(b)