Proportional Reasoning: questions with answers

28. The second term went from 21 to 42, which is a factor of 2, and the first term must take the same factor: 14 times 2 is 28. The factor is not always a whole number, and that is no reason to abandon it -- adding the difference instead gives 35, which grows the two terms by the same AMOUNT rather than in the same proportion. Leaving 14 where it was gives 14 and changes the ratio outright, since only one term moved.

40 and 50. Multiplying both ages by 2 turns 20 into 40 and 25 into 50. The HCF of 40 and 50 is 10, so the pair reduces to 4 : 5 again. Only multiplying both terms by the same number leaves a ratio alone. Adding the same number of years to both ages does not: 25 and 30 is 5 : 6, not 4 : 5. Multiplying only the first age gives 40 and 25, which changes the relationship as well.

49 mm. The width went from 50 to 70, a factor of 7/5, and the height must be multiplied by that same factor: 35 times 7/5 is 49. Leaving the height at 35 would stretch or squash the picture exactly the way the chapter's image B is stretched. And a rectangle is not automatically the right rectangle -- B and E are both rectangles, and neither looks like A.

The width factor is 40 divided by 60, which is 2/3. The height factor is 20 divided by 40, which is 1/2. The two factors are 2/3 and 1/2. They are not equal, so the two pictures are not similar. The same 20 mm came off the width and off the height. Equal amounts taken off are not equal factors, so 2/3 and 2/3 is wrong. Working out the height factor alone and copying it for the width gives 1/2 and 1/2, which never tests the width at all.

It decreases. Increasing the number of workers makes the days needed smaller, not larger. The chapter makes the same point about Puneeth's father: riding faster gets him to Kanpur in less time. So the two quantities are not in direct proportion, and the Rule of Three does not apply here. They do change together, so the days needed cannot stay the same either.

9 hours. The distance is the same, so speed times time is the same: 30 times 12 is 360, and 360 divided by 40 is 9. Going faster cannot take longer, so 16 hours is wrong. Halving the time to 6 hours would be right only if the speed had doubled; here it went from 30 km/h to 40 km/h.

2 : 1. The subtraction leaves 40 : 20, and that reduces to 2 : 1 -- a different relationship from the 3 : 2 they started with. Taking the same amount off both terms feels even-handed, but it is not: 3 : 2 is unchanged only by multiplying or dividing BOTH terms by one number, which is why 3 : 2 is the wrong answer here. 1 : 1 is what comes of taking 20 off the first term and copying the second across, and a ratio never survives being worked on one term at a time.

3 : 2 is in its simplest form because its two terms share no factor above 1. Both terms of 15 : 10 are still multiples of 5, so that ratio can be reduced further. Both terms of 3 : 30 are still multiples of 3. There only one term was divided; a ratio is reduced by dividing BOTH terms by the same number.

7 : 3 is already in its simplest form, since 7 and 3 share no factor above 1. In 14 : 12, both terms share 2, so dividing both by 2 gives 7 : 6. The two simplest forms are 7 : 3 and 7 : 6, which are not the same ratio. The pair is tempting because both terms did grow: 7 was multiplied by 2 and 3 by 4. Two different factors do not preserve a ratio; only one factor applied to both terms leaves the relationship alone.

24 units and 40 units. The HCF of 24 and 40 is 8, and dividing both amounts by it gives 3 : 5. One ratio describes pairs of any size. The chapter's tiger images measure 60 : 40, 30 : 20 and 90 : 60, and all three say 3 : 2. 30 units and 32 units has the same gap as 3 : 5, but a ratio compares by multiplication, not by how much more. Scaling only the first amount, as in 24 units and 5 units, leaves the second amount untouched.

₹240. A kilogram is 1000 g, which is 2 times 500 g, so the price takes the same factor: ₹120 times 2 is ₹240. This is the step that makes the chapter's two teas comparable at all -- one priced by the gram, the other by the kilogram -- and until it is taken the two prices are simply not about the same thing. ₹120 is the price of 500 g and cannot also be the price of a kilogram; ₹60 comes of dividing by 1000 where the move from a larger unit to a smaller one multiplies.

The missing term is 15. Cross multiplying pairs the outer terms: 10 times 6 is 60. Dividing 60 by the inner term 4 leaves 15 exactly. The factor from 4 to 6 is not a whole number. It need not be: a fractional factor completes a proportion just as a whole one does. Copying the gap of the first ratio into the second gives 12. Scaling 4 to 6 but carrying 10 across unchanged gives 10. Both leave the two ratios unequal.

85 paise. Divide each price by its millilitres. ₹135 for 150 mL is 90 paise a millilitre. ₹320 for 400 mL is 80 paise. ₹680 for 800 mL is 85 paise. So one millilitre of the 800 mL bottle costs 85 paise. The 400 mL bottle gives a millilitre for 80 paise, which is less than 85 paise. The largest container does not automatically give the lower price for one millilitre, just as in the chapter's shampoo table.

50 and 30. A ratio of 5 : 3 asks for 8 equal groups, not 5 of them: 80 divided by 8 is 10, and the parts are 5 and 3 groups of that size. Dividing by 5 alone gives 80 and 48, which comes out in whole numbers and is still wrong -- the first share swallows the entire whole and there is nothing left for the second. 5 and 3 hands over the ratio terms themselves, and those count groups, not rupees: they say how the whole is split, never how much each person gets.

12. The two terms add to 7, so the whole splits into 7 equal groups and each is 84 divided by 7. Dividing by 4, the larger term, gives 21 -- a whole number, which is exactly why the slip survives -- but 4 is how many groups ONE share takes, not how many there are altogether. 4 itself is not an amount of anything: the terms of a ratio count groups, and the group size has to be worked out from the whole.