Chapter 10.1 — Exercise 10.1 — Direct Proportion

Cloth by the Metre

5 metres of cloth costs ₹210. Since more cloth always costs more, length and cost are directly proportional, and x₁/y₁ = x₂/y₂ applies with x₁ = 5, y₁ = 210 throughout:

Every entry uses exactly the same relationship — cost per metre is always ₹42 — so each answer is just 42 multiplied by the new length, even though the working above frames it as a ratio rather than stating the rate outright.

Filling In an Apples Table

1 apple costs ₹8. Using x₁/y₁ = x₂/y₂ with x₁ = 1, y₁ = 8 for each new quantity:

Because the base rate here is a whole number (₹8 per apple), the ratio method collapses into plain multiplication — the fraction step barely shows, but it's the same x₁/y₁ = x₂/y₂ relationship underneath as the cloth problem above, where the numbers didn't divide so cleanly.

Proportion Working Backwards — Fewer Bags, Fewer Members

Direct proportion doesn't only handle growth — it works identically when a quantity shrinks. 48 bags of paddy cost ₹16,800; since fewer bags should cost less, the same direct-proportion test applies with the ratio running downward instead of up:

A family of 4 spends ₹2,800 a month on average; a family of 3 spends proportionally less, since the two quantities — people and expenditure — still rise and fall together even as both numbers here are decreasing rather than increasing:

A model ship's mast is scaled down from a real ship 28 m long with a 12 m mast, and the model's mast measures 9 cm. The unknown here is the model's length, not its mast height, so the ratio is set up with length as x and mast height as y throughout:

So the model ship is 21 cm long — the length and the mast height shrink by exactly the same scale factor, which is the entire point of a scale model.

A Pole, Its Shadow, and Two Related Questions in One Setup

A 5.6 m pole casts a 3.2 m shadow. Height and shadow length are directly proportional (a taller pole casts a longer shadow at the same time of day), so the same x₁/y₁ = x₂/y₂ ratio answers two different questions from one starting pair:

Notice that the same known pair (5.6 m, 3.2 m) gets reused for both questions — only which side of the ratio holds the unknown changes, depending on whether height or shadow length is what's being asked for.

Distance Problems Hiding a Unit Conversion

A truck covers 14 km in 25 minutes; how far does it travel in 5 hours at the same speed? Before the ratio can be set up, 5 hours has to become minutes, since the two time values need matching units:

A train moving at a constant 75 km/hr raises the same unit-matching issue twice over — first finding the distance covered in 20 minutes, then finding the time needed to cover 250 km:

In both problems, the proportion itself is the easy part — the actual skill being tested is noticing that hours, minutes, and kilometres can't be compared directly until they're expressed in matching units.

A Fractional Answer From Whole-Number Data

12 sheets of paper weigh 40 grams; how many sheets weigh 16⅔ kilograms? Converting the target weight to grams first — 16⅔ kg = 50/3 kg = 50000/3 grams — keeps the units consistent before the ratio is applied:

Even though the given weight arrives as an awkward mixed fraction, converting it fully to grams before doing anything else turns the rest of the calculation into ordinary arithmetic.

A Scale Ratio Read the Right Way Round

A microchip's design uses a 40:1 scale, and the design measures 18 cm. It's tempting to multiply 18 by 40, but a 40:1 scale means the design is 40 times larger than the real chip, not the other way round — so the actual chip is smaller, not bigger:

Reading a scale ratio backwards — treating an enlarged design as though it were the small original — is one of the easiest ways to get an answer that's numerically confident but pointed in entirely the wrong direction.

When the Unknowns Are Ratios, Not Numbers

The average age of a group of doctors and lawyers combined is 40; doctors alone average 35, lawyers alone average 50. Nothing here is directly proportional — instead, the total age of the combined group can be written two ways, once using the overall average and once using the two separate averages, and setting these equal reveals the ratio of one group's size to the other:

So doctors and lawyers are in the ratio 2 : 1 — a result that falls straight out of comparing two different expressions for the same total, without ever needing to know the actual head count of either group.

From Growing-Together to Growing-Apart

Every problem above involved two quantities moving in the same direction — both up, or both down. Exercise 10.2 introduces the opposite case: quantities where one rises exactly as fast as the other falls, starting with a test for telling the two kinds of proportion apart just by looking at a table of values.