In-text — Proportion, Maps and Multi-term Ratios — Proportional Reasoning-2

1. 12 marksGanita Prakash Cl-8 Part 2, Section 3.1, page 55

   Idli batter is made with rice and urad dal in the ratio 2 : 1. Viswanath mixes 6 cups of rice with 3 cups of urad dal; Puneet mixes 4 cups of rice with 2 cups of urad dal. If cooked the same way, would their idlis taste the same?

   Hint. Cross-multiply the two ratios and compare the products.

   What the question is really asking. Two mixtures taste the same when the ingredients are in the same proportion, not when the amounts are equal. Viswanath uses more batter than Puneet; the question is whether the balance between rice and dal is the same.

   Cross-multiplication test. Viswanath's mixture is 6 : 3 and Puneet's is 4 : 2. Two ratios a : b and c : d are proportional when a × d = b × c: 6 × 2 = 12 3 × 4 = 12 The two products are equal, so 6 : 3 :: 4 : 2.

   A second check. Both reduce to the same simplest form: 6 : 3 = 2 : 1 (divide by 3) 4 : 2 = 2 : 1 (divide by 2) Both are the standard 2 : 1 batter.

   So the answer is yes — provided every other ingredient (salt, water, fermentation time) is also kept in proportion. Viswanath simply makes one-and-a-half times as much batter as Puneet, and scaling every ingredient by the same factor never changes the taste.

   ✦ Yes, the idlis should taste the same, since 6 × 2 = 3 × 4 = 12 shows the two ratios are proportional — both are 2 : 1.

2. 22 marksGanita Prakash Cl-8 Part 2, Section 3.2, page 56

   Many maps carry a ratio such as RF 1 : 60,00,000 in the corner. What does it mean? Convert 60,00,000 cm to kilometres and verify the conversion.

   Hint. 1 m = 100 cm and 1 km = 1000 m, so 1 km = 1,00,000 cm.

   What RF means. RF stands for Representative Fraction. It is the ratio between a distance measured on the map and the matching distance on the ground. So RF 1 : 60,00,000 says that 1 cm measured on the map stands for 60,00,000 cm on the ground.

   Note carefully: this is the geographical (straight-line) distance, not the road distance. Roads bend, so the road distance between two towns is always longer than what a map scale gives.

   The conversion, one step at a time. 100 cm = 1 m, so 60,00,000 cm = 60,00,000 ÷ 100 = 60,000 m 1000 m = 1 km, so 60,000 m = 60,000 ÷ 1000 = 60 km

   Verify by going the other way. 60 km = 60 × 1000 m = 60,000 m = 60,000 × 100 cm = 60,00,000 cm ✓

   Reading the scale in practice. Since 1 cm stands for 60 km, a length of 5 cm on this map means 5 × 60 = 300 km on the ground. A larger second number in the RF means a smaller-scale map showing more land in less detail.

   ✦ RF 1 : 60,00,000 means 1 cm on the map represents 60,00,000 cm on the ground, and 60,00,000 cm = 60 km.

3. 33 marksGanita Prakash Cl-8 Part 2, Section 3.2, pages 56-57

   Using the map in the book, find the geographical distance between Bengaluru and Chennai, and between Mangaluru and Chennai. Then try the same two pairs on maps with different scales. Do they all give approximately the same geographical distance?

   Hint. Measure with a ruler in centimetres, then multiply by the number of kilometres one centimetre represents.

   This item needs the printed map, and the copy in the chapter PDF is stamped "Map not to scale". The measurement therefore cannot be taken from the PDF, and no number is invented here. What follows is the method, which is what the exercise is testing.

   The method.

   1. Lay a ruler between the two city dots and read the map distance in centimetres — say d cm.
   2. Read the RF in the corner of the map. If it is 1 : 60,00,000, then 1 cm stands for 60 km.
   3. Multiply: geographical distance = d × 60 km.

   Worked illustration. If Bengaluru and Chennai measure 5 cm apart on a 1 : 60,00,000 map, the geographical distance is 5 × 60 = 300 km. If Mangaluru and Chennai measure about 10 cm apart on the same map, that is about 600 km. (These are illustrations of the arithmetic, not readings taken from the book's map.)

   The second part — different maps, different scales. Yes, they should all agree, to within measuring error. The ground distance between two cities is a fixed physical fact; the scale only decides how many centimetres of paper it occupies. A map at 1 : 20,00,000 shows the same pair three times further apart on the page, but 1 cm now stands for only 20 km, so

   3d cm × 20 km/cm = 60d km = d cm × 60 km/cm

   — the same answer.

   Why small differences appear anyway. A ruler read to the nearest millimetre on a small-scale map carries a large error on the ground: on a 1 : 60,00,000 map, 1 mm is already 6 km. Larger-scale maps therefore give more accurate readings. And none of these measurements will match a road-distance signboard, because roads are not straight.

   ✦ Measure the map distance in cm and multiply by the kilometres one cm represents (60 km at RF 1 : 60,00,000). All correctly-read maps give approximately the same geographical distance, whatever their scale, since the ground distance is fixed — differences come only from measuring error, and none of them equals the road distance.

4. 43 marksGanita Prakash Cl-8 Part 2, Section 3.3, page 57

   Viswanath's spice mix uses coriander seeds : red chillies : toor dal : fenugreek seeds in the ratio 8 : 4 : 2 : 1. Puneet has only 2 red chillies but wants the same taste. How much of the other ingredients should he use?

   Hint. Find the factor that takes Viswanath's chillies to Puneet's, then apply the same factor to every term.

   Find the scaling factor from the ingredient you are limited by. Viswanath uses 4 red chillies; Puneet has only 2. So Puneet has

   2 ÷ 4 = ½

   of the chillies. For the taste to be unchanged, every ingredient must be scaled by that same factor of ½ — scaling only some of them would change the proportion and therefore the taste.

   Halve each term.

   Ingredient	Viswanath	× ½	Puneet
   Coriander seeds	8 spoons		4 spoons
   Red chillies	4		2
   Toor dal	2 spoons		1 spoon
   Fenugreek seeds	1 spoon		½ spoon

   Check the proportion. Puneet's ratio is 4 : 2 : 1 : 0.5. Dividing each of Viswanath's terms by the matching term of Puneet's gives 8/4 = 4/2 = 2/1 = 1/0.5 = 2 — the same quotient every time, which is exactly the test for two multi-term ratios to be proportional. We write

   8 : 4 : 2 : 1 :: 4 : 2 : 1 : 0.5

   A note on the half spoon. Terms of a ratio do not have to be whole numbers. If half a spoon is awkward to measure, Puneet could instead double everything and make Viswanath's full batch — but he cannot round 0.5 up to 1, because that would make the mix twice as bitter with fenugreek.

   ✦ Puneet should use 4 spoons of coriander seeds, 2 red chillies, 1 spoon of toor dal and ½ spoon of fenugreek seeds — every quantity halved, giving 4 : 2 : 1 : 0.5.

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.