Figure it Out — Ratios and Proportion — Proportional Reasoning

1. 13 marksGanita Prakash Cl-8 Part 1, Figure it Out, page 165

   Circle the 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

   Hint. Two ratios are in proportion when they simplify to the same thing — or equivalently when the cross products are equal.

   Two ratios a : b and c : d are in proportion when a/b = c/d, which is the same as the cross-product test ad = bc. Either test works; cross-multiplying avoids fractions.

   (i) 4 : 7 :: 12 : 21 — cross products 4 × 21 = 84 and 7 × 12 = 84. These agree, so the two ratios are equal ✓ TRUE (Indeed 12 : 21 is just 4 : 7 with both parts tripled.)

   (ii) 8 : 3 :: 24 : 6 — 8 × 6 = 48 but 3 × 24 = 72. FALSE The trap here is that 8 was tripled to 24 but 3 was only doubled to 6 — both parts must be multiplied by the same number.

   (iii) 7 : 12 :: 12 : 7 — 7 × 7 = 49 but 12 × 12 = 144. FALSE Reversing a ratio never preserves it unless the two parts are equal, since 7/12 and 12/7 are reciprocals and a number equals its own reciprocal only when it is 1.

   (iv) 21 : 6 :: 35 : 10 — 21 × 10 = 210 and 6 × 35 = 210. Equal ✓ TRUE (Both sides simplify to 7 : 2.)

   (v) 12 : 18 :: 28 : 12 — 12 × 12 = 144 but 18 × 28 = 504. FALSE (12 : 18 is 2 : 3, while 28 : 12 is 7 : 3.)

   (vi) 24 : 8 :: 9 : 3 — 24 × 3 = 72 and 8 × 9 = 72. Equal ✓ TRUE (Both simplify to 3 : 1.)

   The check to use in an exam: cross-multiply. It is one line, needs no simplifying, and cannot go wrong on awkward numbers.

   ✦ (i), (iv) and (vi) are true.

2. 22 marksGanita Prakash Cl-8 Part 1, Figure it Out, page 165

   Give 3 ratios that are proportional to 4 : 9.

   Hint. Multiply both parts by the same number — any number will do.

   A ratio stays the same when both parts are multiplied by the same non-zero number, since that is exactly what leaves the fraction 4/9 unchanged.

   Multiplying by 2, 3 and 4:

   Multiplier	Ratio	Check
   ×2	8 : 18	8/18 = 4/9 ✓
   ×3	12 : 27	12/27 = 4/9 ✓
   ×4	16 : 36	16/36 = 4/9 ✓

   Verifying the first by cross products: 4 × 18 = 72 and 9 × 8 = 72 ✓

   There are infinitely many correct answers, since any multiplier works — 40 : 90, 400 : 900, even 2 : 4.5. The only wrong move is to add the same number to both parts rather than multiplying: 4 + 2 : 9 + 2 gives 6 : 11, and 6/11 ≠ 4/9. Adding changes a ratio; multiplying does not.

   ✦ 8 : 18, 12 : 27 and 16 : 36 — obtained by multiplying both parts of 4 : 9 by 2, 3 and 4.

3. 33 marksGanita Prakash Cl-8 Part 1, Figure it Out, page 165

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

   Hint. Simplify 18 : 24 first — the answers are much easier from the simplest form.

   Step 1 — Simplify the given ratio. The HCF of 18 and 24 is 6, so 18 : 24 = 3 : 4. Every answer must also simplify to 3 : 4.

   Step 2 — Fill each blank by scaling from 3 : 4.

   3 : ___ — the first part is already 3, so no scaling is needed: 3 : 4

   12 : ___ — 3 has been multiplied by 4, so multiply 4 by 4 as well: 12 : 16

   20 : ___ — here 3 has been multiplied by 20/3, which is not a whole number. The second part is 4 × 20/3: 20 : 80/3 (that is, 20 : 26⅔) Check by cross products: 20 × 4 = 80 and 3 × 80/3 = 80 ✓

   27 : ___ — 3 has been multiplied by 9, so multiply 4 by 9: 27 : 36

   The one to watch is the third. Students often assume the missing number must be a whole number and force it to 26 or 27. But a proportion only requires the ratio to match, and 80/3 is a perfectly valid answer — it simply means 20 is not a whole-number multiple of 3.

   ✦ 3 : 4, 12 : 16, 20 : 80/3, 27 : 36 — every one simplifying to 3 : 4.

4. 42 marksGanita Prakash Cl-8 Part 1, Figure it Out, page 165

   Look at the rectangles A to E. Which rectangles are similar to each other? Verify by measuring the width and height with a scale and comparing their ratios.

   Hint. Similar rectangles have the same shape, so the ratio of width to height must match.

   This is a measurement activity, so the answer depends on the rectangles printed in your copy of the book. What follows is the method, which is what the question is testing.

   The test for similarity. Two rectangles are similar when they have the same shape but possibly different sizes — which happens exactly when the ratio of their corresponding sides is the same. So:

   1. Measure the width and height of each rectangle A, B, C, D and E with a ruler, in millimetres for accuracy.
   2. For each rectangle, compute the ratio width : height, and simplify it (or just work out width ÷ height as a decimal — easier to compare).
   3. Rectangles whose ratios agree are similar to one another.

   Worked illustration of the method. Suppose the measurements came out as:

   Rectangle	Width	Height	width ÷ height
   A	30 mm	20 mm	1.5
   B	45 mm	30 mm	1.5
   C	40 mm	20 mm	2.0
   D	24 mm	16 mm	1.5
   E	30 mm	30 mm	1.0

   Then A, B and D would be similar to each other (all 1.5), while C and E each stand alone.

   Why the decimal is worth using. Comparing 30 : 20 with 24 : 16 by eye is slow; comparing 1.5 with 1.5 is instant. Simplifying both to 3 : 2 works equally well.

   A caution on measuring: a 1 mm error on a 20 mm side shifts the ratio by about 5%, so measure carefully and treat ratios that agree to within a few per cent as equal.

   ✦ Measure each rectangle's width and height, compute width ÷ height for each, and the ones whose values agree are similar — since similar rectangles are exactly those with equal side ratios.

5. 52 marksGanita Prakash Cl-8 Part 1, Figure it Out, page 165

   Can you draw a smaller rectangle and a bigger rectangle with the same width-to-height ratio? 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?

   Hint. The condition fixes the shape but says nothing about the size.

   Yes — and your classmates' rectangles will almost certainly differ from yours, without any of them being wrong.

   How to draw them. Suppose the given rectangle measures 6 cm by 4 cm, a ratio of 3 : 2. • A smaller one: multiply both by ½ → 3 cm by 2 cm. • A bigger one: multiply both by 2 → 12 cm by 8 cm. Any multiplier works, so 4.5 cm by 3 cm and 9 cm by 6 cm are equally correct.

   Why everyone's answers differ. The condition given fixes only the shape, not the size. Keeping the width-to-height ratio at 3 : 2 leaves the scale completely free, and there are infinitely many choices of scale. So a classmate who drew 15 cm by 10 cm has satisfied exactly the same condition you did.

   Are they wrong? No. All such rectangles are similar to the original and to each other — same shape, different size. A question that pins down a ratio but not a length must have infinitely many answers, and every one of them is correct.

   When would they be wrong? Only if the ratio itself were altered — for instance by adding 2 cm to each side instead of multiplying. From 6 by 4 that gives 8 by 6, a ratio of 4 : 3, which is a different shape. This is the standard error: scaling means multiplying both dimensions, never adding to them.

   ✦ Yes — multiply both dimensions by any factor, e.g. 3 : 2 gives 3 cm × 2 cm and 12 cm × 8 cm. Classmates' answers differ because the ratio fixes the shape but leaves the size free, so all are similar and all are correct.

6. 63 marksGanita Prakash Cl-8 Part 1, Figure it Out, page 166

   The figure shows a small portion of a long brick wall with patterns made using coloured bricks, and each wall continues its pattern throughout. What is the ratio of grey bricks to coloured bricks in (a) and in (b)? Give the ratios in simplest form.

   Hint. Count one complete repeating block of the pattern — the ratio for the whole wall is the same as for one block.

   Note on the counts: the brick numbers below come from the printed pattern on page 166 and from the book's own answer key, since they cannot be recovered from text alone. The method, which is what the question tests, is set out in full.

   The key idea. The wall repeats the same block over and over, so the ratio across the whole wall equals the ratio within one block. You do not need to count the entire wall — just identify one complete repeating unit and count inside it.

   (a) Counting one block of the pattern: • Grey bricks: 2 + 3 + 4 = 9 • Coloured bricks: 3 + 2 + 1 = 6

   Ratio = 9 : 6. Simplify by dividing both parts by their HCF, which is 3: = 3 : 2

   (b) Counting one block: • Grey bricks: 16 • Coloured bricks: 12

   Ratio = 16 : 12. The HCF of 16 and 12 is 4: = 4 : 3

   Why simplifying matters here. Both walls contain more grey than coloured bricks, but the simplest forms make the comparison immediate: wall (a) is 3 : 2 (1.5 grey per coloured) while wall (b) is 4 : 3 (about 1.33 grey per coloured). So wall (a) has proportionally more grey, which is hard to see from 9 : 6 and 16 : 12 at a glance.

   A check worth doing: the two parts of the ratio must add to the total number of bricks in the block — 9 + 6 = 15 in (a), and 16 + 12 = 28 in (b).

   ✦ (a) 9 : 6 = 3 : 2. (b) 16 : 12 = 4 : 3. Counting one repeating block is enough, since the ratio is the same throughout the wall.

7. 72 marksGanita Prakash Cl-8 Part 1, Figure it Out, page 166

   Measure your friend's body — the lengths of their head, torso, arms and legs. Write the ratios head : torso, torso : arms and torso : legs. Now draw a figure with head, torso, arms and legs in equivalent ratios.

   Hint. Once you have the ratios, choose any convenient scale and multiply every measurement by it.

   This is a measurement and drawing activity, so your numbers will be your own. Here is the method, plus typical values so you can check your work is sensible.

   Step 1 — Measure, using the same unit throughout. Measure in centimetres: head (crown to chin), torso (shoulder to hip), arm (shoulder to fingertip), leg (hip to heel).

   Step 2 — Write each ratio and simplify. For a typical person of about 150 cm:

   Part	Length
   Head	20 cm
   Torso	50 cm
   Arms	60 cm
   Legs	75 cm

   • head : torso = 20 : 50 = 2 : 5 • torso : arms = 50 : 60 = 5 : 6 • torso : legs = 50 : 75 = 2 : 3

   Step 3 — Draw at a chosen scale. Pick a scale that fits your page — say 1 : 5, meaning every 5 cm of real length becomes 1 cm on paper. Then divide every measurement by 5: head 4 cm, torso 10 cm, arms 12 cm, legs 15 cm.

   Check the ratios survived: 4 : 10 = 2 : 5 ✓, 10 : 12 = 5 : 6 ✓, 10 : 15 = 2 : 3 ✓

   The rule that makes the drawing work. Every measurement must be divided by the same number. Dividing some by 5 and others by 4 would distort the figure — the same error as adding to a ratio instead of multiplying.

   A well-known reference point. Artists often use the guide that an adult figure is about 7 to 8 head-lengths tall. Check yours: 150 ÷ 20 = 7.5 ✓ If your figure comes out at 4 or 12 head-lengths, re-check a measurement.

   ✦ Measure all four parts in the same unit, write and simplify the three ratios, then divide every measurement by one common scale factor to draw — the ratios are preserved only if the same divisor is used throughout.

Solutions written by the tuition.in editorial team and checked against the NCERT Class 8 Mathematics textbook Ganita Prakash Part 1, Reprint 2026-27 (hegp107.pdf). Four 'Figure it Out' blocks sit on pages 165, 170, 175 and 176-178. Every numeric answer here was independently recomputed before comparison with the book's printed answer key — including the acre-to-square-feet conversions (1 acre = 43,560 sq ft), the Lilavati saffron problem, the population densities of Delhi and Mumbai, and the cupro-nickel coin costings. THREE NOTES ARE FLAGGED IN PLACE: (1) question 6 on page 165 depends on counting bricks in a printed pattern, so its counts are cited from the key rather than derived; (2) questions 4, 5 and 7 on page 165 are measurement and drawing activities with no single numeric answer — the method is given instead; (3) the printed key's working for question 2 on page 176 contains a garbled fraction ('x = 34/9'), so the solution is re-derived cleanly here while reaching the same final answer of 4 buses.. Questions are referenced from the NCERT textbook for identification.