NCERT Solutions Ganita Prakash (Part 1) Chapter 7 –1677.4 Problem Solving with Proportional Reasoning — Figure it Out

Book page 165 Updated on2026-09-05

Q1.

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

Answer

True: (i), (iv) and (vi).

Statement	Cross multiply (ad and bc)	Simplest forms	True?
(i) 4 : 7 :: 12 : 21	4 × 21 = 84, 7 × 12 = 84	4 : 7 and 4 : 7	True
(ii) 8 : 3 :: 24 : 6	8 × 6 = 48, 3 × 24 = 72	8 : 3 and 4 : 1	False
(iii) 7 : 12 :: 12 : 7	7 × 7 = 49, 12 × 12 = 144	7 : 12 and 12 : 7	False
(iv) 21 : 6 :: 35 : 10	21 × 10 = 210, 6 × 35 = 210	7 : 2 and 7 : 2	True
(v) 12 : 18 :: 28 : 12	12 × 12 = 144, 18 × 28 = 504	2 : 3 and 7 : 3	False
(vi) 24 : 8 :: 9 : 3	24 × 3 = 72, 8 × 9 = 72	3 : 1 and 3 : 1	True

Why (iii) is a trap: Reversing the terms of a ratio does not give a proportional ratio. 7 : 12 and 12 : 7 are equal only if 7 × 7 = 12 × 12, which would need 7 = 12. In general a : b :: b : a only when a = b.

Tip: In (ii) the first terms were tripled (8 → 24) but the second terms were doubled (3 → 6). Two different factors — so the statement fails, and you can see that without multiplying anything out.

Q2.

Give 3 ratios that are proportional to 4 : 9. ______ : ______ ______ : ______ ______ : ______

Answer

8 : 18, 12 : 27, 16 : 36 — and there are infinitely many more.

4 : 9 × 2 → 8 : 18
4 : 9 × 3 → 12 : 27
4 : 9 × 4 → 16 : 36

Any factor works, whole or fractional: × 10 gives 40 : 90, × 1⁄2 gives 2 : 4.5, × 25 gives 100 : 225.

Why 4 : 9 itself never changes: 4 and 9 have HCF 1, so 4 : 9 is already in its simplest form. Every ratio proportional to it must reduce back to 4 : 9 — which is exactly why the simplest form works as a fingerprint. Check any answer by cross multiplication: 4 × 18 = 72 = 9 × 8.

Q3.

Fill in the missing numbers for these ratios that are proportional to 18 : 24. 3 : ______ 12 : ______ 20 : ______ 27 : ______

Answer

First reduce: 18 : 24 = 3 : 4 (HCF = 6). Every answer must reduce to 3 : 4, so the second term is always 4⁄3 of the first.

3 : ___ → 3 × 4⁄3 = 4 → 3 : 4
12 : ___ → 12 × 4⁄3 = 16 → 12 : 16
20 : ___ → 20 × 4⁄3 = 80⁄3 = 26⅔ → 20 : 26⅔
27 : ___ → 27 × 4⁄3 = 36 → 27 : 36

Why one answer is not a whole number: Going from 3 to 20 needs the factor 20⁄3, which is not a whole number, so the second term 4 × 20⁄3 = 80⁄3 cannot be one either. Nothing is wrong — a ratio's terms are allowed to be fractions. If you would rather avoid the fraction, multiply both terms by 3 and write the same ratio as 60 : 80.

Check it yourself: 20 × 4 = 80 and 3 × 80⁄3 = 80. Equal cross products, so 3 : 4 :: 20 : 80⁄3 is correct.

Q4.

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.

Answer

A and D are similar to each other, and C and E are similar to each other. B is not similar to any of them.

Measure each rectangle's two sides with your scale — three of them are printed at a slant, so measure along the sides, not across the page. Rounded to the nearest millimetre:

Rectangle	Shorter side	Longer side	Longer : shorter	Simplest form
A	6 mm	18 mm	3.0	3 : 1
B	12 mm	18 mm	1.5	3 : 2
C	20 mm	51 mm	2.55	5 : 2
D	13 mm	39 mm	3.0	3 : 1
E	7 mm	17 mm	2.43	5 : 2

The five rectangles redrawn upright, to scale, with the sides measured to the nearest millimetre. Long side : short side is printed under each.

Why turning a rectangle does not matter: C, D and E are printed tilted, but rotating a figure moves it without changing any length. The pair of side lengths — and therefore the ratio — is exactly what it was before the turn. So compare side ratios and ignore the tilt entirely. A is 3 : 1 and so is D; C is about 5 : 2 and so is E; B at 3 : 2 has no partner in this set.

Tip: Your own measurements may be a millimetre off here and there, so read 2.43 and 2.55 as “both about 2.5”. When two ratios agree to within your measuring error, treat the rectangles as similar and, if you can, check by laying one diagonal against the other.

Q5.

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?

Answer

The printed rectangle measures about 35 mm × 21 mm, so its width : height is 5 : 3. Multiply both terms by the same factor to get as many copies as you like.

Factor	Width	Height	Ratio
× 0.4 cm (smaller)	2 cm	1.2 cm	5 : 3
the printed one	3.5 cm	2.1 cm	5 : 3
× 1.6 cm (bigger)	8 cm	4.8 cm	5 : 3

No, your classmates' rectangles will not all be the same size — and no, they are not wrong.

Why different answers are all correct: The question fixes the shape but leaves the size free. Each of you chose your own factor, so the drawings differ in size; but nobody changed the ratio, so they do not differ in shape. A question with infinitely many correct answers is not a defective question — it is a question about a family of figures rather than a single figure.

Check it yourself: Place any two of the drawings so that their bottom-left corners meet and their sides lie along each other. The two top-right corners and the shared corner will lie on one straight line. If a classmate's rectangle misses that line, its ratio is not 5 : 3.

Q6.

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.

Answer

(a) 9 : 6 = 3 : 2 (b) 16 : 12 = 4 : 3

Since the pattern repeats for ever, you do not count the whole wall. Find one repeating block, count inside it, and that ratio holds for the entire wall.

Wall (a). The block is 5 columns wide and 3 rows tall — 15 bricks. The coloured bricks form a triangle pointing down:

Top row: 3 coloured, 2 grey
Middle row: 2 coloured, 3 grey
Bottom row: 1 coloured, 4 grey
Coloured = 3 + 2 + 1 = 6, grey = 15 − 6 = 9
grey : coloured = 9 : 6 = 3 : 2

One repeating block of wall (a): 5 columns × 3 rows = 15 bricks, of which 3 + 2 + 1 = 6 are coloured and 9 are grey.

Wall (b). Here the coloured bricks outline diamonds, and the pattern repeats every 4 columns across 7 rows — 28 bricks in a block:

Rows from the top: 1, 2, 2, 2, 2, 2, 1 coloured
Coloured = 1 + 2 + 2 + 2 + 2 + 2 + 1 = 12, grey = 28 − 12 = 16
grey : coloured = 16 : 12 = 4 : 3

Wall (b): the coloured bricks outline diamonds that repeat every 4 columns (dashed line). One block is 4 columns × 7 rows = 28 bricks — 12 coloured, 16 grey.

Why one block settles the whole wall: A wall 20 blocks long has 20 × 9 grey and 20 × 6 coloured bricks in pattern (a). Both counts are multiplied by 20, which is the same factor for both terms — so the ratio is unchanged. This is the reason a repeating pattern can be described by a single ratio however long the wall runs.

Tip: Choosing where the block starts does not matter, as long as the block is one full period wide. Start counting one brick further along and you will still find 9 grey to 6 coloured in (a).

Q7.

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. Does the drawing look more realistic if the ratios are proportional? Why? Why not?

Answer

Measure with a tape and record the lengths, then reduce each pair. Here is one real set of measurements from a student about 150 cm tall:

Part	Length	Ratio asked for	Simplest form
head	22 cm	head : torso = 22 : 55	2 : 5
torso	55 cm	torso : arms = 55 : 60	11 : 12
arms	60 cm	torso : legs = 55 : 80	11 : 16
legs	80 cm	—	—

To draw the figure, choose one convenient factor and apply it to every measurement. Using 1⁄5 of the real lengths: head 4.4 cm, torso 11 cm, arms 12 cm, legs 16 cm.

Yes, the drawing looks far more realistic when the ratios are kept.

Why it happens: We recognise a human figure by the relative sizes of its parts, not by its absolute height — which is why a 5 cm sketch and a life-size poster can both look convincingly human. Enlarge just the head and keep everything else, and the two factors disagree; the figure reads as a cartoon. Keep one factor throughout and every distance in the drawing is the real distance scaled by that factor, so nothing looks out of place.

Did you know? Artists often measure a figure in “head lengths”. An adult is roughly 7 to 8 heads tall, while a small child is about 5 — and that difference in ratio, not the difference in height, is what makes a child's drawing look like a child.

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