Figure it Out — Dividing a Whole in a Given Ratio — Proportional Reasoning-2

1. 13 marksGanita Prakash Cl-8 Part 2, Figure it Out, page 60

   A cricket coach schedules practice sessions with activities in the ratio warm-up/cool-down : batting : bowling : fielding :: 3 : 4 : 3 : 5. If each session is 150 minutes long, how much time is spent on each activity?

   Hint. Add the terms to find how many equal parts the session is cut into, then find the size of one part.

   Count the parts first. 3 + 4 + 3 + 5 = 15 parts

   Find one part. The whole session of 150 minutes is shared into 15 equal parts, so 150 ÷ 15 = 10 minutes per part

   Multiply each term by 10.

   Activity	Parts	Time
   Warm-up / cool-down	3	3 × 10 = 30 min
   Batting	4	4 × 10 = 40 min
   Bowling	3	3 × 10 = 30 min
   Fielding	5	5 × 10 = 50 min

   Check. 30 + 40 + 30 + 50 = 150 minutes ✓ — the parts must add back to the whole, and this check catches almost every arithmetic slip in ratio-sharing questions.

   Reading the answer. Fielding gets the largest share because 5 is the largest term, and warm-up and bowling get equal time because both terms are 3. The ratio, not the total, decides who gets more.

   ✦ Warm-up/cool-down 30 minutes, batting 40 minutes, bowling 30 minutes, fielding 50 minutes.

2. 23 marksGanita Prakash Cl-8 Part 2, Figure it Out, page 60

   A school library has books in the ratio Odiya : Hindi : English :: 3 : 2 : 1. If the library has 288 Odiya books, how many Hindi and English books does it have?

   Hint. Here the total is not given — one term is. Work out the value of one part from that term instead.

   Notice what is different here. The previous question gave the whole and asked for the parts. This one gives one part and asks for the others, so the total has to be found on the way rather than used at the start.

   Find the size of one part from the Odiya books. Odiya corresponds to 3 parts, and 3 parts = 288 books, therefore 1 part = 288 ÷ 3 = 96 books

   Apply that to the other terms. Hindi = 2 parts = 2 × 96 = 192 books English = 1 part = 1 × 96 = 96 books

   Check the ratio. 288 : 192 : 96, and dividing all three by their HCF 96 gives 3 : 2 : 1 ✓

   The total, for interest. 288 + 192 + 96 = 576 books in the library — which is 6 parts, as 3 + 2 + 1 = 6 predicts.

   ✦ The library has 192 Hindi books and 96 English books.

3. 34 marksGanita Prakash Cl-8 Part 2, Figure it Out, page 60

   I have 100 coins in the ratio ₹10 coins : ₹5 coins : ₹2 coins : ₹1 coins :: 4 : 3 : 2 : 1. How much money do I have in coins?

   Hint. Share the 100 coins first. Only then convert each pile to rupees — the ratio counts coins, not rupees.

   Step 1 — share the coins, not the money. The ratio describes how many coins of each kind there are, so the 100 is divided in the ratio 4 : 3 : 2 : 1. 4 + 3 + 2 + 1 = 10 parts 100 ÷ 10 = 10 coins per part

   Coin	Parts	Number of coins
   ₹10	4	40
   ₹5	3	30
   ₹2	2	20
   ₹1	1	10

   Check: 40 + 30 + 20 + 10 = 100 coins ✓

   Step 2 — now convert each pile to rupees. 40 × ₹10 = ₹400 30 × ₹5 = ₹150 20 × ₹2 = ₹40 10 × ₹1 = ₹10

   Step 3 — add. 400 + 150 + 40 + 10 = ₹600

   Where this question traps people. It is tempting to divide ₹100 in the ratio 4 : 3 : 2 : 1 straight away. But the 100 counts coins, and the value ratio is completely different: the four piles are worth 400 : 150 : 40 : 10, which simplifies to 40 : 15 : 4 : 1 — nothing like 4 : 3 : 2 : 1. Read what the ratio is counting before you use it.

   ✦ ₹600 in all — 40 coins of ₹10, 30 of ₹5, 20 of ₹2 and 10 of ₹1.

4. 43 marksGanita Prakash Cl-8 Part 2, Figure it Out, page 60

   Construct a triangle with sidelengths in the ratio 3 : 4 : 5. Will all the triangles drawn with this ratio of sidelengths be congruent to each other? Why or why not?

   Hint. A ratio fixes the shape of the triangle but says nothing about how big it is.

   Constructing one. Choose any value for one part — say 1 cm. Then draw a triangle with sides 3 cm, 4 cm and 5 cm, using a compass: draw the 5 cm base, swing an arc of 3 cm from one end and an arc of 4 cm from the other, and join the crossing point to both ends. The three sides satisfy the triangle inequality (3 + 4 = 7 > 5), so the construction closes.

   You will also notice something: 3² + 4² = 9 + 16 = 25 = 5², so by the Baudhāyana-Pythagoras relation this is a right-angled triangle. Every triangle with sides in the ratio 3 : 4 : 5 is right-angled.

   Will they all be congruent? No.

   Congruent means identical in size as well as shape. But the ratio 3 : 4 : 5 does not fix any actual length — it only fixes how the three lengths compare. Taking one part to be 1 cm, 2 cm or 10 cm gives

   3 cm, 4 cm, 5 cm 6 cm, 8 cm, 10 cm 30 cm, 40 cm, 50 cm

   and these are all genuinely different triangles. The last has sides ten times as long as the first, so they cannot be laid on top of each other.

   What they are instead. They all have exactly the same three angles, so they are the same shape at different sizes — similar, not congruent. Enlarging a photograph is the everyday version of the same idea.

   ✦ You can construct such a triangle (for example 3 cm, 4 cm, 5 cm, which is right-angled), but they are not all congruent — the ratio fixes the shape and the angles, not the size, so 3-4-5, 6-8-10 and 30-40-50 are all different triangles of the same shape.

5. 53 marksGanita Prakash Cl-8 Part 2, Figure it Out, page 60

   Can you construct a triangle with sidelengths in the ratio 1 : 3 : 5? Why or why not?

   Hint. Try to close the triangle: will the two shorter sides reach across the longest one?

   Take the smallest case first. With one part = 1 unit, the sides would be 1, 3 and 5.

   Lay the longest side, 5 units, on the table. Now try to bridge it with the other two: they are 1 and 3 units, and together they stretch only

   1 + 3 = 4 units

   which is less than 5. The two shorter sides cannot meet — they fall short by a whole unit, and the triangle never closes.

   Scaling does not rescue it. Any triangle with sides in the ratio 1 : 3 : 5 has sides k, 3k, 5k for some positive k. Then k + 3k = 4k, and 4k < 5k for every k > 0. The shortfall grows with k rather than disappearing, so no triangle with these sides exists at any size.

   The rule behind it. In any triangle, the sum of two sides must be greater than the third — the triangle inequality. A straight path between two points is the shortest one, so going round via the third vertex must be longer.

   Contrast this with Example 5 in the same section. There the ratio 1 : 3 : 5 was applied to the angles, and it worked perfectly: 20°, 60° and 100°, adding to 180°. Angles in a ratio only need to sum to 180°; sides in a ratio must additionally satisfy the triangle inequality. The same three numbers behave completely differently depending on what they are measuring.

   ✦ No. The sides would be k, 3k and 5k, and k + 3k = 4k is less than 5k for every k, so the triangle inequality fails and the triangle cannot be constructed at any size.

Solutions written by the tuition.in editorial team and checked against the NCERT Class 8 Mathematics textbook Ganita Prakash Part 2, Reprint 2026-27 (hegp203.pdf), where this is Chapter 3 (pages 55-69) — the tenth chapter of the Class 8 course. Like the rest of Part 2, this PDF carries NO printed answer key, so every answer was derived from first principles and independently recomputed in Python before being written. TWO FIGURE-ONLY ITEMS WERE RESOLVED BY MEASURING THE RENDERED PAGE rather than guessing: (1) the eight sleep ring charts on page 67 were rendered at 300 dpi and the shaded arc of each ring measured by sampling 3600 points around its circumference — the eight measured fractions map one-to-one onto the eight values the question offers (giraffe 2.96 -> 2.5, elephant 3.84 -> 3.5, human 7.78 -> 8, dog 10.07 -> 10.5, cat 12.99 -> 13, squirrel 15.08 -> 15, python 18.09 -> 18, bat 20.16 -> 20), which is what fixes the assignment; (2) the transport pie chart on page 68 prints only four of its five angles, so the page was rendered and the labels read off before deducing Car = 30 degrees. ONE ITEM IS FLAGGED AS UNANSWERABLE FROM THE PDF: the map distances in section 3.2, because the chapter PDF stamps its map 'Map not to scale' - the method is given and no distance is invented. ONE APPARENT SLIP IN THE BOOK IS FLAGGED: page 68 question 5(iii) asks how many children use 'taxis', but taxi is not one of the five modes in the pie chart and the five slices already total 360 degrees, so the answer as printed is zero; the solution says so and notes that 'two-wheeler' (36 children) was probably intended.. Questions are referenced from the NCERT textbook for identification.