EN हिं

NCERT Solutions Ganita Prakash (Part 1) Chapter 7 –170Trairasika — In-text Questions

Book page 167 Updated on2026-09-05

Q1.
Example 8: For the mid-day meal in a school with 120 students, the cook usually makes 15 kg of rice. On a rainy day, only 80 students came to school. How many kilograms of rice should the cook make so that the food is not wasted?
Answer

10 kg of rice.

students : rice must stay proportional
120 : 15 :: 80 : ?
Factor of change = 80⁄120 = 2⁄3
Rice = 15 × 2⁄3 = 10 kg
Why it works: The cook is really working at a fixed rate — 15 kg for 120 students is 15⁄120 = 0.125 kg per student. Fewer students do not change that rate, so 80 × 0.125 = 10 kg. Multiplying the rice by the same 2⁄3 that the students were multiplied by is a quicker way of saying the same thing.
Check it yourself: Cross multiply — 120 × 10 = 1200 and 15 × 80 = 1200. Equal, so the proportion holds.
Q2.
What is the factor of change in the first term?
Answer

2⁄3 — the number of students has fallen to two-thirds of what it usually is.

80⁄120 = 8⁄12 = 2⁄3

A factor smaller than 1 means the quantity has decreased. The second term must be multiplied by the very same 2⁄3.

Q3.
Example 9: A car travels 90 km in 150 minutes. If it continues at the same speed, what distance will it cover in 4 hours?
Answer

144 km.

4 hours = 4 × 60 = 240 minutes
150 : 90 :: 240 : x
Cross multiplying: 150 × x = 240 × 90
x = (240 × 90) ÷ 150
x = 21600 ÷ 150 = 144 km
Why time and distance may be paired this way: The speed is unchanged, so distance grows exactly as time grows — double the time, double the distance. Quantities that rise together in this way are the only ones the Rule of Three applies to.
Check it yourself: The car's speed is 90 ÷ 150 = 0.6 km per minute. In 240 minutes that is 240 × 0.6 = 144 km. Same answer by a different route.
Q4.
Is this the right way to formulate the question?
Answer

No. The statement 150 : 90 :: 4 : ? mixes units — 150 is in minutes but 4 is in hours.

Wrong: 150 min : 90 km :: 4 h : ?
Right: 4 h = 240 min, so 150 : 90 :: 240 : ?
Why the units must match: The factor of change is found by dividing the first terms, and 4 ÷ 150 is a meaningless number when one is hours and the other minutes. Once both are minutes, 240 ÷ 150 = 8⁄5 is a genuine factor, and multiplying 90 by 8⁄5 gives 144. Using the raw 4 would have given 90 × 4⁄150 = 2.4 km for four hours of driving — an answer that fails the common-sense test instantly.
Q5.
How can you find the distance covered in 240 minutes? Discuss with your classmates and find the answer using different strategies.
Answer

All three routes below give 144 km. Use whichever you find clearest.

  • Unitary method. In 1 minute the car covers 90 ÷ 150 = 0.6 km. In 240 minutes it covers 240 × 0.6 = 144 km.
  • Scaling the ratio. 240 ÷ 150 = 8⁄5, so multiply the distance by 8⁄5 as well: 90 × 8⁄5 = 144 km.
  • Splitting the time. 150 min → 90 km, so 300 min → 180 km and 30 min → 18 km. Then 240 = 300 − 60, and 60 min → 36 km, giving 180 − 36 = 144 km.
Cross multiplication, for comparison:
150 : 90 :: 240 : x ⇒ 150x = 240 × 90 ⇒ x = 144 km
Why they must agree: Every one of these methods is applying the single fact that the car covers 0.6 km each minute. Cross multiplication is not a fourth idea — it is the same equation with the division postponed to the last step.
Q6.
Example 10: A small farmer in Himachal Pradesh sells each 200 g packet of tea for ₹200. A large estate in Meghalaya sells each 1 kg packet of tea for ₹800. Are the weight-to-price ratios in both places proportional?
Answer

No, they are not proportional. First put both weights in the same unit.

Himachal: 200 g : ₹200 → HCF 200 → 1 : 1
Meghalaya: 1 kg = 1000 g, so 1000 : 800 → HCF 200 → 5 : 4
1 : 1 is not 5 : 4, so the ratios are not proportional
Why writing 1 : 800 would be wrong: The Himachal ratio was formed with the weight in grams. If Meghalaya's weight is left in kilograms, the two ratios are measuring different things and cannot be compared at all. Converting to a common unit is not a formality — it is what makes the comparison mean anything.
Check it yourself: Cross multiply the corrected ratios — 200 × 800 = 160000 while 200 × 1000 = 200000. Unequal products confirm that the ratios are not proportional.
Q7.
Which tea is more expensive? Why?
Answer

The Himachal tea is more expensive — ₹1,000 per kg against ₹800 per kg.

Meghalaya: 1 kg costs ₹800
Himachal: 200 g is 1⁄5 of a kg, so if 1 kg costs ₹x,
1⁄5 × x = 200
x = 200 × 5 = ₹1,000
Why the packet prices cannot be compared directly: ₹200 looks far less than ₹800, but the two packets are not the same size. Prices only become comparable once they are quoted for the same weight. Bringing both to “rupees per kilogram” does exactly that, and then the Himachal tea is ₹200 per kg dearer.
Did you know? The price per unit printed on a shop shelf — per kg, per litre, per 100 g — exists for precisely this reason. It converts every packet, whatever its size, to one comparable ratio.
Q8.
Activity 1: Take your favourite dish. Find out all the ingredients and their respective quantities needed to make the dish for your family. Suppose you are celebrating a festival and you want to invite 15 guests. Find out the quantities of the ingredients required to cook the same dish for them.
Answer

Write the family recipe first, then scale every ingredient by one factor. Here is a worked example for upma cooked for a family of 5, with 15 guests invited so that 20 people must be fed.

Factor of change = 20⁄5 = 4
IngredientFor 5 people× 4For 20 people
Rava (semolina)250 g250 × 41000 g = 1 kg
Water600 mL600 × 42400 mL = 2.4 L
Onion1 medium1 × 44 medium
Oil2 tablespoons2 × 48 tablespoons
Salt1 teaspoon1 × 44 teaspoons
Why every ingredient needs the same factor: The taste of a dish is fixed by the ratios between its ingredients — rava to water, salt to rava. Scale them all by 4 and every one of those ratios survives, so the dish tastes the same, only there is more of it. Quadruple the rava but merely double the water and the ratio 250 : 600 becomes 1000 : 1200, and the upma turns out dry.
Tip: If your family is 6 and you must feed 6 + 15 = 21, the factor is 21⁄6 = 3.5 — a fraction, and perfectly usable. 250 g of rava becomes 875 g.
Was this helpful? Report an error