RS Aggarwal 2019,2020 solution class 7 chapter 8 Ratio and Proportion Exercise 8B

Exercise 8B

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Question 1:

Show that 30, 40, 45, 60 are in proportion.

Answer 1:

We have:

Product of the extremes = 30 $\times$ 60 = 1800
Product of the means = 40 $\times$ 45 = 1800
Product of extremes = Product of means

Hence, 30 : 40 :: 45 : 60

Question 2:

Show that 36, 49, 6, 7 are not in proportion.

Answer 2:

We have:
Product of the extremes = 36 $\times$ 7 = 252
Product of the means = 49 $\times$ 6 = 294
Product of the extremes $\ne$ Product of the means

Hence, 36, 49, 6 and 7 are not in proportion.

Question 3:

If 2 : 9 :: x : 27, find the value of x.

Answer 3:

Product of the extremes = 2 $\times$ 27 = 54
Product of the means  = 9 $\times$ x = 9x

Since 2 : 9 :: x : 27, we have:
Product of the extremes = Product of the means
⇒ 54 = 9x
⇒ x = 6

Question 4:

If 8 : x :: 16 : 35, find the value of x.

Answer 4:

Product of the extremes = 8 $\times$ 35 = 280
Product of the means = 16 $\times$ x = 16x

Since 8 : x :: 16 : 35, we have:
Product of the extremes = Product of the means
⇒ 280 = 16x
⇒ x = 17.5

Question 5:

If x : 35 :: 48 : 60, find the value of x.

Answer 5:

Product of the extremes = x $\times$ 60 = 60x
Product of the means = 35 $\times$ 48 = 1680

Since x : 35 :: 48 : 60, we have:
Product of the extremes = Product of the means
⇒ 60x= 1680
⇒ x = 28

Question 6:

Find the fourth proportional to the numbers:

(i) 8, 36, 6
(ii) 5, 7, 30
(iii) 2.8, 14, 3.5

Answer 6:

(i) Let the fourth proportional be x.
Then, 8 : 36 :: 6 : x

8 $\times x = 36 \times 6$                                   [Product of extremes = Product of means]
⇒ 8x = 216
⇒ x = 27
Hence, the fourth proportional is 27.

(ii) Let the fourth proportional be x.
Then, 5 : 7 :: 30 : x
⇒ $5 \times x = 7 \times 30$                                      [Product of extremes = Product of means]
⇒ 8x = 216
⇒ 5x = 210
⇒ x = 42

Hence, the fourth proportional is 42.

(iii) Let the fourth proportional be x.
Then, 2.8 $\times x =14 \times 3.5$                                [Product of extremes = Product of means]
⇒ 8x = 216
⇒ 2.8x = 49
⇒ x = 17.5
Hence, the fourth proportional is 17.5.

Question 7:

If 36, 54, x are in continued proportion, find the value of x.

Answer 7:

36, 54 and x are in continued proportion.
Then, 36 : 54 :: 54 : x
⇒ $36 \times x =54 \times 54$                                [Product of extremes = Product of means]
⇒ 36x = 2916
⇒ x = 81

Question 8:

If 27, 36, x are in continued proportion, find the value of x.

Answer 8:

27, 36 and x are in continued proportion.
Then, 27 : 36 :: 36 : x
⇒ $27\times x = 36 \times 36$         [Product of extremes = Product of means]
⇒ 27x = 1296
⇒ x = 48

Hence, the value of x is 48.

Question 9:

Find the third proportional to:

(i) 8 and 12
(ii) 12 and 18
(iii) 4.5 and 6

Answer 9:

(i)  Suppose that x is the third proportional to 8 and 12.
Then, 8 :12 :: 12 : x
⇒ 8 $\times x = 12 \times 12$                                            (Product of extremes = Product of means )
⇒ 8x = 144
⇒ x = 18

Hence, the required third proportional is 18.

(ii) Suppose that x is the third proportional to 12 and 18.
Then, 12 : 18 :: 18 : x
⇒ $12 \times x = 18 \times 18$                                       (Product of extremes = Product of means )
⇒ 12x = 324
⇒ x = 27

Hence, the third proportional is 27.

(iii) Suppose that x is the third proportional to 4.5 and 6.
Then, 4.5 : 6:: 6 : x
⇒ $4.5 \times x= 6 \times 6$                                        (Product of extremes = Product of means )
⇒ 4.5x = 36
⇒ x = 8

Hence, the third proportional is 8.

Question 10:

If the third proportional to 7 and x is 28, find the value of x.

Answer 10:

The third proportional to 7 and x is 28.
Then, 7 : x :: x : 28
⇒ 7 $\times$ 28 = $x^2$           (Product of extremes = Product of means)
⇒ x = 14

Question 11:

Find the mean proportional between:

(i) 6 and 24
(ii) 3 and 27
(iii) 0.4 and 0.9

Answer 11:

(i)  Suppose that x is the mean proportional.
 
Then, 6 : x :: x : 24

⇒ $6 \times 24 = x \times x$                                         (Product of extremes = Product of means)
⇒ $x^{2 } = 144$
⇒ x = 12

Hence, the mean proportional to 6 and 24 is 12.

(ii)  Suppose that x is the mean proportional.

Then, 3 : x :: x : 27
$$\begin{gathered} \Rightarrow 3 \times 27 = x \times x \\ \Rightarrow x^2 = 81 \end{gathered}$$                                        (Product of extremes =Product of means)
⇒ x = 9

Hence, the mean proportional to 3 and 27 is 9.

(iii)  Suppose that x is the mean proportional.

Then, 0.4 : x :: x : 0.9

$$\begin{gathered} \Rightarrow 0.4 \times 0.9 = x \times x \\ \Rightarrow x^2 = 0.36 \end{gathered}$$                              (Product of extremes =Product of means)
⇒x = 0.6

Hence, the mean proportional to 0.4 and 0.9 is 0.6.

Question 12:

What number must be added to each of the numbers 5, 9, 7, 12 to get the numbers which are in proportion?

Answer 12:

Suppose that the number is x.
 
Then, (5 + x) : ( 9 + x) :: (7 + x) : (12 + x)

$$\begin{gathered} \Rightarrow (5 + x) \times (12 + x) = (9 + x) \times (7 + x) (\mathrm{Product} \mathrm{of} \mathrm{extremes} = \mathrm{Product} \mathrm{of} \mathrm{means}) \\ \Rightarrow 60 +5 x + 12 x + x^2 = 63 + 9x + 7x + x^2 \\ \Rightarrow 60 + 17x = 63 + 16x \\ \Rightarrow x = 3 \end{gathered}$$

Hence, 3 must be added to each of the numbers: 5, 9, 7 and 12, to get the numbers which are in proportion.

Question 13:

What number must be subtracted from each of the numbers 10, 12, 19, 24 to get the numbers which are in proportion?

Answer 13:

Suppose that x is the number that is to be subtracted.

Then, (10 − x) : (12 − x) :: (19 − x) : (24 − x)

$$\begin{gathered} \Rightarrow (10- x) \times (24 - x) =(12 - x) \times (19 - x) (\mathrm{Product} \mathrm{of} \mathrm{extremes} =\mathrm{Product} \mathrm{of} \mathrm{means}) \\ \Rightarrow 240 - 10x -24x + x^2 = 228 - 12x -19x + x^2 \\ \Rightarrow 240 - 34x = 228 - 31x \\ \Rightarrow 3x = 12 \\ \Rightarrow x = 4 \end{gathered}$$
.
Hence, 4 must be subtracted from each of the numbers: 10, 12, 19 and 24, to get the numbers which are in proportion.

Question 14:

The scale of a map is 1 : 5000000. What is the actual distance between two towns, if they are 4 cm apart on the map?

Answer 14:

Distance represented by 1 cm on the map = 5000000 cm = 50 km

Distance represented by 3 cm on the map = 50 $\times$ 4 km = 200 km

∴ The actual distance is 200 km.

Question 15:

At a certain time a tree 6 m high casts a shadow of length 8 metres. At the same time a pole casts a shadow of length 20 metres. Find the height of the pole.

Answer 15:

(Height of tree) : (height of its shadow) = (height of the pole) : (height of its shadow)

Suppose that the height of pole is x cm.

Then, 6 : 8 = x : 20

⇒ x = $\frac{6\times 20}{8} = 15$
∴ Height of the pole = 15 cm