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Out of All Proportion · Quiz 24

Setting Up & Solving Proportions

Two ratios, one missing number
The drawing is the subject: two ratios set equal, and the two diagonals that turn out to be the same product. No creature at the masthead this time; the shape of the thing is the thing.
The Guide

Before you begin

Quiz 23 found a single rate. This quiz sets two of them equal and hunts for the number that makes them match. That is how a recipe gets doubled, a map gets read, a dose gets worked out and a paycheck gets checked. And it is where cross-multiplying arrives, which will be explained here rather than announced.

What a proportion says

A proportion is two ratios set equal to one another: AB = CD. What it means is that one ratio is the other one scaled. Not added to. Scaled: multiplied by something.

So if 3 pounds of coffee cost $13.50, then 6 pounds cost $27.00, because everything doubled. Not $16.50, which is what you get by adding three dollars because you added three pounds. That mistake has a name later in this quiz, and it is the most common thing that goes wrong here.

Setting it up is the hard part

Solving a proportion is mechanical. Writing it down correctly is where the thinking is, and there is one rule that prevents nearly every error:

The same kind of thing goes in the same place on both sides. If miles are on top on the left, miles are on top on the right. If pounds are underneath on the left, pounds are underneath on the right. Label the two rows before you write any numbers, and the setup cannot come out crossed.

A van goes 234 miles on 9 gallons. How far on 15 gallons? Miles on top, gallons underneath, both sides:

Written the other way round on one side only, as 234/9 = 15/x, it falls apart: gallons have arrived on top on the right while miles are still on top on the left, and the answer comes out at about half a mile. Turning both sides over would have been perfectly fine. It is changing one and not the other that breaks it.

Three ways to solve it, and when each is quickest

1. Find the scale factor. Look for a jump that is a whole number and use it. The van's numbers are not friendly that way, so here is a smaller proportion where the jump is easy to see:

Three goes down to twelve by multiplying by 4, so five must go down to x the same way, and x = 20. You can hunt for the factor down a column or across a row, whichever gives the friendlier number. When one is sitting there, this is the fastest route by a distance.

2. Find the unit rate. This is Quiz 23's method and it works every time. Three pounds cost $13.50, so one pound costs $4.50, so seven pounds cost 7 × 4.50 = $31.50. Slower to write, but the numbers stay meaningful the whole way, which makes mistakes easier to see.

3. Cross-multiply. The one that always works, including when the numbers are ugly and no factor is friendly. And here is why it works, which matters more than the fact that it does.

Why cross-multiplying works

You already know how to check whether two fractions are equal, because Quiz 9 did it: put them over a common denominator and compare the tops. Cross-multiplying is that, with the writing-down step skipped.

So the diagonals are not a trick. A × D and B × C are the two numerators you get when both fractions are put over the bottom B × D. Equal fractions with equal bottoms must have equal tops, and that is the whole of it.

There is a second way to see it, shorter still. A proportion says the right-hand ratio is the left-hand one scaled by some amount, so C is A times that amount and D is B times it. Then A × D and B × C are both the top, the bottom and the scale factor multiplied together, just in a different order. They cannot come out different, because multiplication does not care about order. In algebra you will meet a third version, where both sides are multiplied by BD and the denominators cancel. All three are the same fact.

The trap: scaling is not adding

If 3 pounds cost $13.50, what do 7 pounds cost? The tempting move is to notice that 7 is 4 more than 3 and add 4 dollars, giving $17.50. It is wrong, and it is wrong in a way that feels reasonable, which is why it survives. The relationship is multiplicative: one pound is $4.50, so seven pounds are $31.50, which is nearly twice the tempting answer.

The guard that catches it. Ask what happened to the first number, then ask whether the same thing happened to the second. Seven pounds is more than double three pounds, so the price must be more than double $13.50. Anything under $27 is refused before a single calculation.

The guard

Two habits. Label the rows before writing the numbers, so the units line up by construction rather than by luck. And once you have an answer, put it back into the original proportion and check that the two ratios really do come out equal. A proportion is one of the few things in arithmetic that hands you a free check every time.

Worked Examples

Three to study before you start

Example 1 · Spot the factor

Solve 312 = 5x. Before reaching for any method, look for a friendly jump. Going down the left, 3 becomes 12, which is a multiplication by 4. Whatever the left column does, the right column must do, so 5 becomes 5 × 4 = 20. Check it: 3/12 is a quarter, and 5/20 is a quarter. When a whole-number factor is sitting there, this is the fastest route by a distance, and it uses nothing but multiplication.

Example 2 · Through the unit rate

Three pounds of coffee cost $13.50. What do seven pounds cost? Set it up with dollars on top and pounds underneath on both sides: 13.50/3 = x/7. Now solve it the long way, which keeps every number meaningful. One pound costs 13.50 ÷ 3 = $4.50. Seven pounds cost 7 × 4.50 = $31.50. Guard: seven pounds is more than twice three pounds, so the answer had to clear $27, and it does.

Example 3 · When the numbers are not friendly

Solve 712 = x30. There is no whole-number factor here: 12 to 30 is two and a half, and 7 divided by 12 is not a number anybody wants to carry around. This is what cross-multiplying is for. The diagonals give 12 × x on one side and 7 × 30 = 210 on the other, so 12x = 210 and x = 17.5. Check by putting it back: 17.5/30 works out to the same as 7/12, which is a little under six tenths. All three methods would have got here; this one got here without any awkward numbers in the middle.

The Quiz · Ten Questions

Now you

Work without a calculator, on paper. Questions 4 and 7 are multiple choice: choose the one best answer. For every word problem, label the two rows before writing any numbers.

  1. Solve for x:  312 = 5x
  2. Solve for x:  x8 = 1524
  3. Three pounds of coffee cost $13.50. At the same price, what do 7 pounds cost?
  4. A van goes 234 miles on 9 gallons. Which setup will correctly find how far it goes on 15 gallons?
    • A) 2349 = 15x
    • B) 2349 = x15
    • C) 9234 = x15
    • D) x234 = 915
  5. A rice recipe uses 2 cups of rice to 3 cups of water. How much water for 5 cups of rice?
  6. On a map, 1 inch stands for 25 miles. Two towns are 3.5 inches apart on the map. How far apart are they on the ground?

ReadingA reading rest stop, for the stubborn ones.

Back to d'Orbigny's picture atlas, the one the whale on Quiz 23 came from. This time hunt for the insectes. The ants are drawn many times their real size, which is the only way anyone could have looked at them in 1841, and it is a proportion somebody had to choose.

The words are French and it does not matter. Use the arrows to turn the pages.

  1. Three pounds of coffee cost $13.50. Denise works out that 7 pounds cost $17.50, because 7 is four more than 3 so she added four dollars. What went wrong?
    • A) Nothing; $17.50 is correct.
    • B) She added where the relationship multiplies. One pound costs $4.50, so seven pounds cost $31.50.
    • C) She should have added $4.50 instead of $4, giving $18.
    • D) The problem cannot be solved without knowing the brand of coffee.
  2. Solve for x:  712 = x30
  3. Explain why cross-multiplying gives the right answer. Use the idea of a common denominator.
  4. Yuliana is paid $522 for 36 hours. At the same rate: (a) what would she earn for 42 hours? (b) how many hours would earn her $754?
Answer Key

Check your work

Open the key: after you've finished all ten
1 x = 20
The friendly route: 3 goes down to 12 by multiplying by 4, so 5 goes down to 20. Cross-multiplying gives the same thing, since 3x = 60. Check by simplifying both: 3/12 and 5/20 are each one quarter.
2 x = 5
Here the friendly factor runs across rather than down: 8 becomes 24 by multiplying by 3, so whatever x is, tripling it gives 15, and x = 5. Look for the whole-number jump wherever it happens to be sitting, down a column or along a row.
3 $31.50
One pound is 13.50 ÷ 3 = $4.50, and seven pounds are 7 × 4.50. By cross-multiplying instead: 13.50/3 = x/7 gives 3x = 94.50, so x = 31.50. Seven pounds is more than double three, so the answer had to clear $27 before any working began.
4 B.
Miles on top and gallons underneath on both sides, so 234/9 = x/15, giving 9x = 3,510 and 390 miles. A) puts gallons on top on one side only, and solving it gives 0.58, which is not a number of miles. C) also crosses the units and gives the same nonsense. D) mixes the two sides differently again and produces 140.4. Worth knowing: turning both sides upside down would have been fine, since 9/234 = 15/x also gives 390. What breaks a proportion is flipping one side and not the other.
5 7.5 cups of water
Rice on top and water underneath both sides: 2/3 = 5/x, so 2x = 15 and x = 7.5. Two and a half times the rice needs two and a half times the water, which is another way to reach the same answer and a good check on it.
6 87.5 miles
Inches on top, miles underneath: 1/25 = 3.5/x, so x = 87.5. A map scale is a unit rate wearing a legend, and every map you have ever read was solved by this proportion whether or not anybody wrote it down.
7 B.
Denise brought an adding habit into a multiplying situation, which is her pattern: in Quiz 14 she lined up the right-hand edges because that is what worked with whole numbers, and in Quiz 21 she found the right distance and dropped the sign. Each time an old habit was carried somewhere it does not hold. Here, more than doubling the coffee has to more than double the price, and her answer does not even reach $27. C) makes the same mistake with a better-looking number. The gap between $17.50 and $31.50 is most of the bill.
8 x = 17.5
No friendly factor here, so cross-multiply: 12x = 7 × 30 = 210, and x = 17.5. Put it back to check, since 17.5/30 and 7/12 both come to a little under six tenths. This is the question that shows what the third method is for.
9 Because putting both fractions over the same bottom makes their tops A × D and B × C. Equal fractions with the same bottom must have equal tops, so those two products are equal.
Any answer of that shape earns the mark. What is not wanted is "the numbers swap diagonally," because that describes the picture instead of the reason. Nothing swaps: the denominators were simply never written down.
10 (a) $609  ·  (b) 52 hours
Her rate is 522 ÷ 36 = $14.50 an hour. For (a): 522/36 = x/42, so 36x = 21,924 and x = 609. For (b) the unknown moves to the other row: 522/36 = 754/h, so 522h = 27,144 and h = 52. Same proportion, asked from both ends, which is what makes this worth being able to do. A rate you can run in either direction is a rate you can check somebody else's arithmetic with.

Reading your results

Questions The skill they test If they gave trouble
1, 2 Spotting the scale factor Look for a whole-number jump, down a column or across a row, and use it. It is the fastest route when it is there.
4 Setting up correctly Label the two rows first. The same kind of thing goes in the same place on both sides.
3, 5, 6, 10 Proportions in the world Recipes, maps and pay are all one relationship. Write it as two ratios and solve whichever way is easiest.
7 Scaling against adding The signature error of the topic. Ask what happened to the first number, then make the same thing happen to the second.
8, 9 Cross-multiplying, and why It is the common-denominator check from Quiz 9 with the writing-down step skipped.

Eight or more right: proportions are yours, and Quiz 25 turns them loose on the world. Five to seven: review the flagged rows and retake this in a few days. Fewer than five: good news: we've found the right ground to work. One habit for a week: whenever you set up a proportion, write the units beside the numbers, like miles over gallons on both sides, and only then put the figures in. Almost every wrong answer in this topic is a correct calculation performed on a crossed setup, and labels make crossing impossible. Then come back to these same ten.

Where this goes

Next in the series: Quiz 25 — Proportional Reasoning in the World. Recipes, maps, scale drawings, unit pricing, and the question of when a proportion is the right tool at all. Because sometimes it is not, and knowing the difference is worth as much as the arithmetic.

The Company · An Interlude

The claim this quiz can refute

Here is a sentence that turns up wherever ants are discussed: an ant can lift fifty times its own weight, so if a human had an ant's strength they could lift a car. The first half is roughly true for some species. The second half is false, and this quiz contains the reason.

Scaling something up does not scale everything about it by the same amount, because different quantities depend on size in different ways. Muscle strength depends on cross-section, which is an area, and areas grow as the square of the length. Weight depends on volume, and volumes grow as the cube. So when you make a creature longer, its strength and its weight do not keep pace, and the heavier one wins.

Take an ant five millimeters long and scale it to human height, 1,700 millimeters. That is a factor of 340. Its muscles gain 340 × 340, which is 115,600 times the cross-section. Its weight gains 340 × 340 × 340, which is 39,304,000 times. Divide one by the other and the strength-to-weight ratio has been cut by a factor of 340.

So the giant ant is not stronger. It is 340 times weaker for its size. An ant that lifted fifty times its weight would now manage 50 ÷ 340, which is under a sixth of its own body. It could not lift itself. It would stand up, if it could stand at all, and its legs would fail.

The same arithmetic run backwards explains the whale at the head of Quiz 23. A blue whale is enormous because water holds it up. Take that support away and the square-cube law arrives instantly: a stranded whale is crushed by its own weight, its lungs pressed flat by a body that the sea had been carrying. The ant is strong for the same reason the whale is helpless. Neither fact is about ants or whales. Both are about squares and cubes.

Notice what this does to the idea of a proportion. Everything in this quiz has been about relationships that hold when you scale: double the rice, double the water. The square-cube law is the great case where a relationship does not hold, and the reason is that two different powers are in play at once. Quiz 27 measures areas and Quiz 29 takes up exponents properly. Both of them are this paragraph, worked out slowly.

Figures used here: an ant of 5 mm and a human of 1,700 mm, and the commonly quoted fifty-to-one lifting ratio, which varies a good deal by species. The conclusion does not depend on the exact numbers. It depends only on one quantity growing as a square while the other grows as a cube.