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

Question 1

If a: b = 3: 5, find:

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

Solution 1

Given, 

Question 2

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

Solution 2

Question 3

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

Solution 3

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

Question 4

Find the:

(i) duplicate ratio of 

(ii) triplicate ratio of 2a: 3b

(iii) sub-duplicate ratio of 9x²a⁴ : 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 4

(i) Duplicate ratio of 

(ii) Triplicate ratio of 2a: 3b = (2a)³: (3b)³ = 8a³ : 27b³

(iii) Sub-duplicate ratio of 9x²a⁴ : 25y⁶b² = 

(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 = 

Question 5

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.

Solution 5

(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

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 6

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

Then, we have:

Question 7

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

Solution 7

Question 8

If 15(2x² - y²) = 7xy, find x: y; if x and y both are positive.

Solution 8

15(2x² - y²) = 7xy

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 9

(i) Let the fourth proportional to 2xy, x² and y² be n.

 2xy: x² = y²: n

 2xy  n = x² y²

 n =

(ii) Let the third proportional to a² - b² and a + b be n.

 a² - b², a + b and n are in continued proportion.

 a² - b² : a + b = a + b : n

 n =

(iii) Let the mean proportional to (x - y) and (x³ - x²y) be n.

 (x - y), n, (x³ - x²y) are in continued proportion

 (x - y) : n = n : (x³ - x²y)

Question 10

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

Solution 10

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.

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 11

Given, 

Hence, z is mean proportional between x and y.

Question 12

If , find the value of .

Solution 12

Question 13

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

a: b = c: d.

Solution 13

Question 14

If , show that:

(a + b) : (c + d) = 

Solution 14

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 15

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.

Question 16

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

Solution 16

7x - 15y = 4x + y

7x - 4x = y + 15y

3x = 16y

 

Question 17

If , use properties of proportion to find:

(i) m: n

(ii) 

Solution 17

Question 18

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

Solution 18

 x, y, z are in continued proportion,

 

Therefore,

 (By alternendo)

Hence Proved.

Question 19

Given x =.

Use componendo and dividendo to prove that b² =.

Solution 19

x = 

By componendo and dividendo,

Squaring both sides,

By componendo and dividendo,

b² = 

Hence Proved.

Question 20

If , find:

Solution 20

Question 21

Using componendo and dividendo find the value of x:

Solution 21

Question 22

Solution 22

Question 23

 

Solution 23

Question 24

Solution 24

Question 25

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

Solution 25

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

Question 26

 

If , use properties of proportion to find: 

Solution 26

i.

ii.

From (i),

Question 27

i. If x and y both are positive and (2x² - 5y²): xy = 1: 3, find x: y.

ii. Find x, if.

Solution 27

i. (2x² - 5y²): xy = 1: 3

ii.