Before you begin
Quizzes 23 and 24 built the tool. This one uses it on recipes, maps, floor plans and mixtures, and then does something more useful: it shows where the tool does not apply. Reaching for a proportion when the situation is not proportional is the most expensive error in this whole topic, because the arithmetic afterwards is perfect and nothing about the working looks wrong.
Two tests, and they take about three seconds
2. The zero test. If you have zero of the first thing, do you have zero of the second? No apples, no cost. No hours, no pay. If one is zero and the other is not, it is absolutely not a proportion.
The zero test is the sharper of the two, and here is the trap it catches. When Marcus was 8, his sister was 12. Marcus is 16 now, so how old is she?
A great many people go wrong here, and it is worth seeing exactly how. They notice that Marcus's age doubled, from 8 to 16, and take it that his sister's age must double too, from 12 to 24. The doubling is the loud thing in the problem and it pulls the eye.
Run the zero test instead. When Marcus was 0, on the day he was born, was his sister 0? No. She was 4. The relationship fails at zero, so it was never proportional and a proportion will not solve it. What is constant here is not the ratio but the gap: she is four years older than he is and always will be. Marcus is 16, so add 4. She is 20.
Set that beside something that really is proportional. Pay at $15 an hour: four hours earns $60, eight hours earns $120, and no hours earns nothing at all. Drawn on the same kind of axes, the difference between the two situations is visible before any arithmetic happens.
Five situations, both tests run on each
The tests are worth practicing on cases where the answer is not in doubt, so that they are automatic when it is.
| The situation | Zero test | Doubling test | |
|---|---|---|---|
| A bag of grass seed covers 250 sq ft | 0 bags cover 0 sq ft — passes | 2 bags cover 500 — passes | proportional |
| A pipe leaks 3 gallons an hour | at 0 hours, 0 gallons — passes | 2 hr is 6 gal, 4 hr is 12 — passes | proportional |
| A plumber charges $100 to show up, then $50 an hour | 0 hours still costs $100 — fails | 1 hr is $150, 2 hr is $200, not $300 — fails | not proportional |
| You plant a sapling already 4 feet tall; it grows 1 foot a year | at year 0 it is 4 feet, not 0 — fails | year 1 is 5 ft, year 2 is 6 ft, not 10 — fails | not proportional |
| Tap water at 60°F heats 20 degrees a minute | at 0 minutes it is 60°, not 0° — fails | 1 min is 80°, 2 min is 100°, not 160° — fails | not proportional |
And the age problem is the same animal. Marcus's sister had a four-year head start, exactly as the sapling had four feet. Nothing about ages makes them special; they simply belong to the family that begins somewhere other than zero.
Where proportions do apply, and they are everywhere
Recipes. A dish for 4 that uses 3 cups of stock needs 3/4 = x/10 for ten people, so 7.5 cups. Ingredients scale. Maps. A scale of one inch to 40 miles turns 5.25 inches into 210 miles. Floor plans. At a quarter inch to the foot, a room measuring 3.5 inches across is 14 feet across, because 3.5 ÷ 0.25 = 14. Fuel, wages, prices. All of them, all the time.
Mixtures, and a trap from Quiz 23
A cleaning concentrate mixes 1 part concentrate to 7 parts water. How much concentrate makes 2 gallons of finished solution? The 7 is not the total. Add the parts as Quiz 23 taught: 1 + 7 = 8, so the finished solution comes in eight equal shares and one of them is concentrate. Two gallons divided by 8 is a quarter of a gallon, which is one quart. Answering "two gallons divided by seven" would leave you mixing something slightly too strong, and on a label that says 1:7, that is what a great many people pour.
Where proportions fail, and the second way they fail
Ages fail because the relationship is additive: a constant gap rather than a constant ratio. There is a second kind of failure, and it is the one Quiz 24 ended on. Some quantities do not scale in step because they depend on size by a different power.
That is why the ant in Quiz 24 could not be scaled up into a strongman, and it is why a pot of soup for twelve does not take three times as long to cook as a pot for four. The stock scales, because stock is a quantity. The cooking time does not, because heat has to travel and that depends on the shape of the pot rather than on how much is in it.
The guard
Run the zero test before writing anything down. Then, after the answer, ask whether it is the right size: ten servings from a recipe for four should need somewhere between double and triple the stock, and 7.5 cups against the original 3 is exactly that. A proportion gives you a free check at both ends, and there is no reason not to take it.
Three to study before you start
A rice dish for 4 people uses 3 cups of stock. How much for 10? Zero test first: no people, no stock, so a proportion applies. Cups on top, people underneath, both sides: 3/4 = x/10. Cross-multiply and 4x = 30, so x = 7.5 cups. Check the size of it: ten people is two and a half times four people, and 7.5 cups is two and a half times 3. Both ends agree, which is the check a proportion always offers.
When Marcus was 8 his sister was 12. He is 16 now. How old is she? The numbers are laid out exactly like every proportion in Quiz 24, which is the danger. Run the zero test: when Marcus was 0, was his sister 0? No, she was 4. The relationship does not pass through zero, so it is not proportional, and setting up 8/12 = 16/x would give 24, which is wrong by four years. What is constant here is the gap, not the ratio. She is four years older than he is and always will be, so the answer is 20.
A concentrate says 1 : 7, meaning one part concentrate to seven parts water. You want 2 gallons of finished solution. The label describes part against part, so add them to find the whole: 1 + 7 = 8 shares. Two gallons in eight shares makes each share 2 ÷ 8 = 0.25 gallons, which is one quart of concentrate topped up with seven quarts of water. Dividing by 7 instead would give about 0.29 gallons, and you would be mixing it stronger than the label intends every time.
Now you
Work without a calculator, on paper. Questions 4 and 7 are multiple choice: choose the one best answer. Run the zero test on every word problem before you set anything up.
- A rice dish for 4 people uses 3 cups of stock. How much stock for 10 people?
- On a map, 1 inch stands for 40 miles. Two cities are 5.25 inches apart on the map. How far apart are they?
- An 18-ounce box costs $4.32. A 30-ounce box costs $6.90. Which costs less per ounce, and by how much?
- In which of these does doubling the first quantity double the second?
- A) Hours worked, and pay at a fixed hourly rate.
- B) Your age, and your older brother's age.
- C) The side of a square, and the area of that square.
- D) The time of day, and the temperature outside.
- A cleaning concentrate is mixed 1 part concentrate to 7 parts water. How much concentrate is needed to make 2 gallons of finished solution?
- A floor plan is drawn at ¼ inch to 1 foot. A room measures 3.5 inches across on the plan. How wide is the room?
ReadingA reading rest stop, for the stubborn ones.
Back once more to d'Orbigny's picture atlas, and this time look at it as a set of scale drawings. A beetle and a whale are printed the same size on the page. Somewhere behind every plate an artist had to choose a scale, and nothing tells you what it was except how the creature is labeled.
The words are French and it does not matter. Use the arrows to turn the pages.
- When Marcus was 8, his sister was 12. Marcus is now 16. He sets up 8/12 = 16/x and answers that his sister is 24. What went wrong?
- A) Nothing; 24 is correct.
- B) The relationship is not proportional. At age 0 his sister was 4, not 0, so it fails the zero test. The gap is constant, so she is 20.
- C) He should have set it up as 12/8 = x/16, which gives the right answer.
- D) Ages can never be compared using arithmetic.
- A car uses 12 gallons to travel 336 miles. How many gallons for 504 miles?
- State the zero test in your own words, then use it on one of the situations in question 4 to show why that one is not proportional.
- A pot of soup for 4 people uses 6 cups of stock and takes 30 minutes to cook. You are cooking for 12. (a) How much stock? (b) How long will it take to cook, and why?
Check your work
Open the key: after you've finished all ten
Reading your results
| Questions | The skill they test | If they gave trouble |
|---|---|---|
| 1, 2, 6, 8 | Proportions in ordinary use | Recipes, maps, plans and fuel are one relationship. Set it up with labels and solve it whichever way is easiest. |
| 3 | Comparing by unit rate | Two things cannot be compared until they stand on the same ground. Bring both to one ounce. |
| 5 | Mixtures, part against whole | A label reading 1 : 7 describes eight shares. Add the parts before dividing anything. |
| 4, 7, 9 | Knowing when the tool applies | The signature error of the topic. Zero test first, doubling test second, and only then a proportion. |
| 10 | Two quantities in one problem | Ask about each quantity separately. Some things in a situation scale and some do not. |
Eight or more right: ratio and proportion are finished, and Quiz 26 begins measurement. 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: before any word problem with two quantities in it, say out loud what happens when the first one is zero. That single question separates the problems a proportion will solve from the ones it will quietly ruin, and it costs three seconds. Then come back to these same ten.
Next in the series: Quiz 26 — Units and Conversions. Inches to feet, ounces to pounds, minutes to hours, and the metric system, which was built by people who had had enough of all the others. Every conversion is a proportion you already know how to solve, so the new work is deciding which way the number should move.
The scale drawing that cannot tell the truth
Question 2 used a map, and treated it the way this quiz treats everything: as a faithful scale drawing where one inch means forty miles wherever you measure. Over a few hundred miles that is close enough to true. Over the whole world it is impossible, and no amount of care can fix it.
The earth is a ball and a map is flat. You cannot flatten a ball without stretching it somewhere, and every world map ever printed has had to choose what to sacrifice. The one nearly everybody has seen is Mercator's, drawn in 1569 for sailors, and it sacrifices area in order to keep angles true. That was a fair bargain for navigation, because a course held at a constant compass bearing draws a straight line on it, which is exactly what a sailor needs.
The cost lands at the poles, and it is enormous. On a Mercator map, Greenland looks about the size of Africa. Greenland covers roughly 2.2 million square kilometers. Africa covers roughly 30.4 million. Do the division, which is this quiz's arithmetic:
A word about that 13.8, since this series keeps insisting that numbers be checked. Those two areas were rounded before the division: Greenland is nearer 2.17 million square kilometers and Africa nearer 30.37 million, and dividing those gives 14.0. Rounding the inputs moved the answer by about a fifth. It does not matter here, because the point is that one is roughly fourteen times the other and no rounding will rescue a map that shows them alike. But it is worth seeing happen: round early and the error travels with you into every calculation that follows.
Which puts the proportion in a strange position. Everything in this quiz has assumed that a scale drawing holds one ratio throughout, and that is what makes it readable. A Mercator map breaks that assumption, and it does not announce that it has. There is no note in the corner saying that the scale doubles by the time you reach Iceland.
People have argued about this for a long time, and the argument is not only geometric. A map that inflates the far north and shrinks the tropics is showing some places as larger than others, and after four hundred years of hanging on classroom walls, that shapes what looks big. Alternatives exist, and they trade different things away: keep the areas right and the shapes go strange, keep the shapes and the areas go wrong. You cannot have both, and that is not a failure of cartography. It is a fact about spheres.
The lesson is the one this quiz has been making all along, arriving in a place nobody expects it. Before trusting a scale, ask whether it holds everywhere. On a floor plan it does. On a recipe it does. On a map of the world it does not, and the picture is honest about angles and quietly wrong about size, and it has been telling that particular untruth to almost everyone alive.