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NCERT Solutions Ganita Prakash (Part 1) Chapter 7 –172Trairasika — In-text Questions

Book page 171 Updated on2026-09-05

Q1.
Puneeth’s father went from Lucknow to Kanpur in 2 hours by riding his motorcycle at a speed of 50 km/h. If he drives at 75 km/h, how long will it take him to reach Kanpur? Can we form this problem as a proportion — 50 : 2 :: 75 : __ Would it take Puneeth’s father more time or less time to reach Kanpur? Think about it.
Answer

1 hour 20 minutes, and no — this problem cannot be written as 50 : 2 :: 75 : __ .

Distance = speed × time = 50 × 2 = 100 km
At 75 km/h: time = 100 ÷ 75 = 4⁄3 hours
4⁄3 h = 1 h + 1⁄3 × 60 min = 1 hour 20 minutes
Why the Rule of Three fails here: The Rule of Three needs two quantities that rise together. Here the distance is fixed, so a faster ride is a shorter ride — as speed goes up, time comes down. Writing 50 : 2 :: 75 : x would give x = 3 hours, saying that riding faster takes longer. What stays constant is not the ratio speed : time but the product speed × time, which is the 100 km of road between Lucknow and Kanpur.
Tip: Ask yourself before setting up any proportion: if the first quantity increases, should the second increase too? Students and rice — yes. Speed and time over a fixed distance — no. That single question tells you whether the Rule of Three applies.
Q2.
Activity 2: Go to the market and collect the prices of different sizes of shampoo containers of the same shampoo and create a table like the one given below. See if the volume of shampoo is proportional to the price. (Sachet 6 mL ₹2; Small Bottle 180 mL ₹154; Medium Bottle 340 mL ₹276; Large Bottle 1000 mL ₹540)
Answer

Collect real prices from a shop, then work out the price of one millilitre for each size — that single number makes all four containers comparable. For the sample table in the book:

ContainerVolumePricePrice ÷ volumeCost of 1 mL
Sachet6 mL₹22 ÷ 6₹0.33
Small Bottle180 mL₹154154 ÷ 180₹0.86
Medium Bottle340 mL₹276276 ÷ 340₹0.81
Large Bottle1000 mL₹540540 ÷ 1000₹0.54

The volume is not proportional to the price. If it were, every row of the last column would show the same figure; instead it runs 0.33, 0.86, 0.81, 0.54.

Why the unit rate is the right test: Two ratios are proportional exactly when they reduce to the same simplest form, and volume : price reduces to “1 mL for so many rupees”. Four different unit rates mean four different ratios, so no single proportion covers the whole table. Notice too that the rate does not even fall steadily with size here — among the three bottles it does, but the tiny sachet is the cheapest per millilitre of all. In the market you will usually find the opposite; make your own table and see.
Q3.
The ratio of the volume of a sachet to a small bottle is 6 : 180. The ratio of their prices is 2 : 154. Are these ratios proportional?
Answer

No.

Volumes: 6 : 180, HCF = 6 → 1 : 30
Prices: 2 : 154, HCF = 2 → 1 : 77
1 : 30 is not 1 : 77

Cross multiplication says the same thing: 6 × 154 = 924 while 180 × 2 = 360.

What the two numbers mean: The small bottle holds 30 times as much shampoo as the sachet, but it costs 77 times as much. For the price to have been proportional the bottle would have to cost 30 × ₹2 = ₹60. At ₹154 it is far dearer, millilitre for millilitre.
Q4.
Why do you think that the ratio of the prices is not proportional to the ratio of the volumes? Discuss the pros and cons of different size bottles for the company and for customers. For reducing ecological footprint, what would you recommend to the company and to the customer? Does the same occur for other products?
Answer

The price of a container is not only the price of what is inside it.

  • Fixed costs per pack. The bottle, the cap, the label, the printing, the filling and the transport cost roughly the same whether the pack holds 180 mL or 1000 mL. Those costs are spread over more shampoo in a big bottle, so its rate per millilitre falls.
  • Bulk discounts. A company sells a large bottle at a lower rate to persuade you to buy more at one time.
  • Affordability. A sachet can be priced low so that someone who cannot spend ₹540 today can still buy shampoo. Pricing here follows what a customer can pay, not only what the shampoo costs.
Small packsLarge packs
For the customerLow price today, easy to carry, no waste if used rarelyUsually cheaper per mL, fewer trips to the shop
For the companyReaches more buyers, but far more packaging and handling per litre soldLess packaging and handling per litre, larger sale at one go

For a smaller ecological footprint: a customer should buy the largest size they will actually finish, and refill it where refill packs are sold; a company should offer refill pouches, take back and reuse bottles, and keep the per-millilitre price of the large size genuinely lower so that the low-waste choice is also the cheap one.

Why plastic is the real cost here: Getting 1 litre of shampoo through sachets means roughly 167 sachets — 167 caps, seals and pieces of film, almost none of which is recycled, against a single bottle. The money ratio and the waste ratio are two different ratios, and they do not point the same way.
Try This: Build the same table for rice or atta at 500 g, 1 kg and 10 kg. Staples are often priced very nearly in proportion to weight, so their unit rates come out almost equal — quite unlike shampoo. Compare and discuss why.
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