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NCERT Solutions Ganita Prakash (Part 1) Chapter 7 –1657.4 Problem Solving with Proportional Reasoning (Examples 6–7) · Filter Coffee! — In-text Questions

Book page 164 Updated on2026-09-05

Q1.
Example 6: When Neelima was 3 years old, her mother’s age was 10 times her age. What is the ratio of Neelima’s age to her mother’s age? What would be the ratio of their ages when Neelima is 12 years old? Would it remain the same?
Answer

At age 3 the ratio is 1 : 10; at age 12 it is 4 : 13. No, it does not remain the same.

Neelima 3, mother 3 × 10 = 30 → 3 : 30 = 1 : 10
12 − 3 = 9 years later, mother is 30 + 9 = 39
12 : 39, HCF = 3 → 4 : 13
Why it happens: Nine years were added to both ages. Adding the same number to both terms of a ratio does not keep the ratio — only multiplying both terms by the same factor does. For a proportional ratio the mother would have had to be 120 when Neelima turned 12, which is impossible. Ages simply do not stay in proportion; the gap of 27 years stays fixed instead.
Did you know? As both ages grow, the ratio keeps creeping towards 1 : 1. At 30 and 57 it is 10 : 19; at 60 and 87 it is 20 : 29. This is why an age gap feels enormous between a child and a parent and hardly noticeable between two adults.
Q2.
Example 7: Fill in the missing numbers for the following ratios that are proportional to 14 : 21. ______ : 42 6 : ______ 2 : ______
Answer

28 : 42, 6 : 9, 2 : 3.

14 : 21 in simplest form is 2 : 3 (HCF = 7)

___ : 42 → 42 ÷ 21 = 2, so first term = 14 × 2 = 28
6 : ___ → 6 ÷ 14 = 3⁄7, so second term = 21 × 3⁄7 = 9
2 : ___ → 14 ÷ 7 = 2, so second term = 21 ÷ 7 = 3
Check it yourself: Each answer must reduce to 2 : 3 — and 28 : 42, 6 : 9 and 2 : 3 all do. Cross multiplication is a second check: 14 × 42 = 588 = 21 × 28.
Q3.
What factor should we multiply 14 by to get 6? Can it be an integer? Or should it be a fraction?
Answer

The factor is 3⁄7 — it must be a fraction, not an integer.

14y = 6
y = 6⁄14 = 3⁄7
Why it cannot be a whole number: 6 is smaller than 14, so the factor has to be less than 1, and the only whole number less than 1 that is of any use here is 0. A factor smaller than 1 shrinks a quantity; there is nothing unusual about it. Applying the same 3⁄7 to the other term gives 21 × 3⁄7 = 9, so the ratio is 6 : 9.
Q4.
Why is this coffee stronger?
Answer

Because a larger share of the cup is decoction.

Regular: 15 mL decoction + 35 mL milk = 50 mL cup → decoction is 15⁄50 = 30%
Stronger: 20 mL decoction + 30 mL milk = 50 mL cup → decoction is 20⁄50 = 40%

Both cups hold the same 50 mL, but the second carries 5 mL more decoction and 5 mL less milk. In ratio language, 15 : 35 = 3 : 7 has become 20 : 30 = 2 : 3, and 2 : 3 has more decoction for every part of milk than 3 : 7 does.

Tip: To compare two ratios quickly, make one term match. 3 : 7 is the same as 3 : 7, and 2 : 3 is the same as 4.67 : 7 — clearly more decoction for the same milk.
Q5.
Why is this coffee lighter?
Answer

Because a smaller share of the cup is decoction.

Lighter: 10 mL decoction + 40 mL milk = 50 mL cup → decoction is 10⁄50 = 20%
Ratio 10 : 40 = 1 : 4, against the regular 15 : 35 = 3 : 7

For every 4 parts of milk this cup carries only 1 part of decoction, whereas the regular cup carries 1 part for every 2⅓ parts of milk. Less decoction per unit of milk means a lighter taste.

Why the ratio and not the amount: Manjunath could pour 100 mL of decoction and 400 mL of milk into a jug. That is much more decoction than a regular cup contains, yet the coffee is still light — because strength is decided by the ratio 1 : 4, not by the raw quantity.
Q6.
The following table shows the different ratios in which Manjunath mixes coffee decoction with milk. Write in the last column if the coffee is stronger or lighter than the regular coffee.
Answer

Reduce each row and compare it with the regular mix, 15 : 35 = 3 : 7.

Coffee decoction (mL)Milk (mL)Simplest formDecoction in the cupRegular / Strong / Light
3006001 : 2300⁄900 = 33.3%Strong
1505003 : 10150⁄650 = 23.1%Light
2004001 : 2200⁄600 = 33.3%Strong
24563 : 724⁄80 = 30%Regular
1003001 : 3100⁄400 = 25%Light
Why comparing simplest forms works: Regular coffee is 3 parts decoction to 7 parts milk, that is 30% decoction. A mix is stronger exactly when its decoction share is above 30%. Row 4 reduces to 3 : 7 itself, so 24 mL and 56 mL is just a smaller cup of ordinary filter coffee — the quantities are different but the drink is identical.
Tip: Rows 1 and 3 have different quantities (300 : 600 and 200 : 400) but the same simplest form 1 : 2, so they must taste exactly alike. That is the whole point of reducing a ratio — it strips away the size and leaves the recipe.
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