NCERT Solutions for Class 8 Maths Ganita Prakash Chapter 7 — Figure it Out (Pages 165–167)#
Two ratios $a : b$ and $c : d$ are proportional, written $a : b :: c : d$, when both terms of one are multiplied by the same number to give the other. An easy test: write both in their simplest form and compare, or check that $a \times d = b \times c$.
1. Circle the following statements of proportion that are true. (i) 4 : 7 :: 12 : 21 (ii) 8 : 3 :: 24 : 6 (iii) 7 : 12 :: 12 : 7 (iv) 21 : 6 :: 35 : 10 (v) 12 : 18 :: 28 : 12 (vi) 24 : 8 :: 9 : 3 #
Solution
We write each ratio in its simplest form:
| Statement | First ratio | Second ratio | True? |
|---|---|---|---|
| (i) 4 : 7 :: 12 : 21 | 4 : 7 | 12 : 21 = 4 : 7 | True |
| (ii) 8 : 3 :: 24 : 6 | 8 : 3 | 24 : 6 = 4 : 1 | False |
| (iii) 7 : 12 :: 12 : 7 | 7 : 12 | 12 : 7 | False |
| (iv) 21 : 6 :: 35 : 10 | 21 : 6 = 7 : 2 | 35 : 10 = 7 : 2 | True |
| (v) 12 : 18 :: 28 : 12 | 12 : 18 = 2 : 3 | 28 : 12 = 7 : 3 | False |
| (vi) 24 : 8 :: 9 : 3 | 24 : 8 = 3 : 1 | 9 : 3 = 3 : 1 | True |
∴ Statements (i), (iv) and (vi) are true.
2. Give 3 ratios that are proportional to 4 : 9. #
Solution
We multiply both terms of $4 : 9$ by the same number:
- by 2: $8 : 18$
- by 3: $12 : 27$
- by 4: $16 : 36$
∴ Three such ratios are 8 : 18, 12 : 27 and 16 : 36.
3. Fill in the missing numbers for these ratios that are proportional to 18 : 24. 3 : __ 12 : __ 20 : __ 27 : __ #
Solution
In its simplest form, $18 : 24 = 3 : 4$. So the second term is always $\dfrac{4}{3}$ times the first:
- $3 \times \dfrac{4}{3} = 4$, giving 3 : 4
- $12 \times \dfrac{4}{3} = 16$, giving 12 : 16
- $20 \times \dfrac{4}{3} = \dfrac{80}{3}$, giving 20 : $\dfrac{80}{3}$ (that is, $26\dfrac{2}{3}$)
- $27 \times \dfrac{4}{3} = 36$, giving 27 : 36
4. Look at the following rectangles. Which rectangles are similar to each other? You can verify this by measuring the width and height using a scale and comparing their ratios. #
Solution
Rectangles are similar when the ratio of their longer side to their shorter side is the same. Measuring the sides of each rectangle (your measurements depend on the print size, but the ratios do not):
| Rectangle | Longer side : shorter side (about) |
|---|---|
| A | 3 : 1 |
| B | 3 : 2 |
| C | 5 : 2 |
| D | 3 : 1 |
| E | 5 : 2 |
∴ A and D are similar (3 : 1), and C and E are similar (5 : 2). B is not similar to any of the others. Turning a rectangle round does not change its shape, so tilted rectangles can still be similar.
5. Look at the following rectangle. Can you draw a smaller rectangle and a bigger rectangle with the same width to height ratio in your notebooks? Compare your rectangles with your classmates’ drawings. Are all of them the same? If they are different from yours, can you think why? Are they wrong? #
Solution
Measuring the rectangle, its width to height ratio is about 5 : 3. Any rectangle with this ratio has the same shape, for example:
- smaller: width 2.5 cm, height 1.5 cm
- bigger: width 10 cm, height 6 cm
Classmates’ rectangles may be of different sizes, because each person multiplied 5 : 3 by a different number. They are not wrong: as long as the width to height ratio is the same, all the rectangles are similar, the same shape in different sizes.
6. The following figure shows a small portion of a long brick wall with patterns made using coloured bricks. Each wall continues this pattern throughout the wall. What is the ratio of grey bricks to coloured bricks? Try to give the ratios in their simplest form. #
(a) The wall with red bricks
The pattern repeats every 5 bricks along the wall. In one repeating block:
| Row | Grey | Red |
|---|---|---|
| Top | 2 | 3 |
| Middle | 3 | 2 |
| Bottom | 4 | 1 |
| Total | 9 | 6 |
Grey : red $= 9 : 6 = 3 : 2$
∴ The ratio is 3 : 2.
(b) The wall with orange bricks
The pattern repeats every 4 bricks along the wall. In one repeating block of the 7 rows:
| Row | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Total |
|---|---|---|---|---|---|---|---|---|
| Grey | 3 | 2 | 2 | 2 | 2 | 2 | 3 | 16 |
| Orange | 1 | 2 | 2 | 2 | 2 | 2 | 1 | 12 |
Grey : orange $= 16 : 12 = 4 : 3$
∴ The ratio is 4 : 3.
7. Let us draw some human figures. Measure your friend’s body—the lengths of their head, torso, arms, and legs. Write the ratios as mentioned below— head : torso, torso : arms, torso : legs. Now, draw a figure with head, torso, arms, and legs with equivalent ratios as above. #
Solution
This is an activity; your answers will depend on your friend. An example:
| Part | Measured length |
|---|---|
| Head | 22 cm |
| Torso | 55 cm |
| Arms | 66 cm |
| Legs | 88 cm |
Ratios: head : torso $= 22 : 55 = 2 : 5$, torso : arms $= 55 : 66 = 5 : 6$, torso : legs $= 55 : 88 = 5 : 8$.
To draw a figure with the same ratios, divide every length by the same number. Dividing by 11 gives head 2 cm, torso 5 cm, arms 6 cm and legs 8 cm. A drawing with proportional parts looks realistic; if one part is scaled differently, the figure looks odd, like a cartoon.