Seahorse Series · Quiz 11 · Multiplying Fractions

The Guide

Before you begin

This quiz is about multiplying fractions, and about the word of, which is how word problems ask for it. Adding fractions (Quiz 10) made you match every piece before you could join it. Multiplying asks for none of that, and the reason is worth understanding rather than memorizing: adding asks how much in all, while multiplying asks how much "of" one thing the other thing is. Two different questions, two different machines. As in every quiz of this series: no tricks, no calculator, and whatever this shows is useful news.

The word that means multiply

In fractions, of means multiply. Half of a third. Two thirds of twelve. Three quarters of a tank. Once you hear of as an instruction, most fraction word problems announce their own operation, and you can stop guessing at which one is wanted.

The rectangle, one last time

If you did Quizzes 4, 5 and 7, you have met the rectangle three times already. Now it explains multiplication of fractions completely. Take one whole square. Cut it into thirds one way and take one of them; then cut it into halves the other way and take one of those. What survives both cuts is the answer.

Drawn to scale. Six cells were made, and exactly one survived both cuts. The bottoms multiplied because the cuts multiplied.

That picture contains the whole rule. Cutting into thirds and then into halves produces 3 × 2 = 6 cells, which is why the denominators multiply: they count the cuts. Taking 1 column and 1 row leaves 1 × 1 cells, which is why the numerators multiply: they count what you kept. So 23 × 45 = 815, straight across, no matching required.

Why adding needed matching and multiplying does not: to add, both amounts must be measured in the same unit, the way you cannot add three feet to four inches without converting first. To multiply, you are not combining two amounts at all — you are cutting one amount by the other. The second fraction is not a quantity being added in; it is an instruction about how much of the first to take.

Multiplying can make things smaller

Whole numbers taught everyone that multiplying grows things, and fractions break that promise. Half of something is less than the something. So multiplying by any fraction below 1 shrinks: 8 × 34 = 6, and 6 is less than 8. Multiplying by exactly 1 changes nothing, and only multiplying by more than 1 grows the number. This is not an exception to arithmetic; it is what of has always meant.

Whole numbers join in

Any whole number is a fraction over 1: 3 is 31. So 3 × 25 = 65, which is 115. And 23 of 12 means 23 × 121 = 243 = 8. There is a shortcut worth noticing in that last one: a third of 12 is 4, so two thirds is 8. Divide by the bottom, multiply by the top.

Simplify early, if you like

You may cancel common factors before multiplying instead of after — divide a top and a bottom by the same number — and the numbers stay friendlier. For 34 × 29: the 3 above and the 9 below share a factor of 3, and the 2 above and 4 below share a factor of 2. Canceling both leaves 12 × 13 = 16. Multiplying first gives 636, which simplifies to the same 16. Both are correct; canceling early just keeps the arithmetic small, and it is Quiz 7's factors doing the work.

Mixed numbers must be made improper first

This is the one real trap. You cannot multiply the whole parts and the fraction parts separately: 2 × 1 = 2 and ½ × ⅓ = ⅙ would give 2⅙, and that is wrong, because the whole part of each number also has to meet the fraction part of the other. Convert both to improper fractions first — a whole number and a fraction written as one fraction, as Quiz 8 shows: 2½ = 52 and 1⅓ = 43. Then 52 × 43 = 206 = 103 = 3 1/3.

The guard, adapted

Ask first whether the answer should be bigger or smaller than what you started with. Multiplying by less than one must shrink it; by more than one must grow it. That single question catches most errors on this page before any arithmetic is checked.

Worked Examples

Three to study before you start

Example 1 · Straight across

23 × 45. Guard: both are under 1, so the answer must be smaller than either. Tops: 2 × 4 = 8. Bottoms: 3 × 5 = 15. Answer 8/15, and 8 and 15 share no factor, so it is already in simplest form. Smaller than both, as promised.

Example 2 · A fraction of a whole number

34 of 8. Write 8 as 81: tops give 24, bottoms give 4, and 24 ÷ 4 = 6. Or take the shortcut: a quarter of 8 is 2, so three quarters is 6. Either way the answer is less than 8, which is what taking a part of something must do.

Example 3 · Mixed numbers, converted first

2½ × 1⅓. Convert: 52 and 43. Guard: two and a half, grown by a bit more than one, so expect something over 3. Multiply: 206, simplify to 103, convert back to 3 1/3. Had you multiplied the parts separately you would have gotten 2⅙, which the guard rejects at once.

The Quiz · Ten Questions

Now you

Work without a calculator, on paper. Give every answer in simplest form, and as a mixed number when the top outgrows the bottom. Questions 3 and 7 are multiple choice: choose the one best answer.

Multiply: 23 × 45
Multiply: 12 × 13
Without computing it exactly, what must be true of 8 × 3/4?
- A) It is greater than 8, because multiplying always makes numbers bigger.
- B) It is less than 8, because taking three quarters of something leaves less than the something.
- C) It equals 8, because the 4 and the 8 cancel.
- D) There is no way to tell without multiplying.
Multiply: 3 × 25
Find 23 of 12.
Multiply: 34 × 29

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Take a few minutes with ten of the most beautiful seahorses in the world.

What is the correct first step for 2 1/2 × 1 1/3?
- A) Multiply the whole numbers, then multiply the fractions, and put the two results together.
- B) Give both fractions a common denominator.
- C) Rewrite each mixed number as an improper fraction: 5/2 and 4/3.
- D) Add the whole numbers to the fractions.
Multiply: 2 1/2 × 1 1/3
A recipe calls for 34 cup of oil, but Rosa is making only half the recipe. How much oil does she need?
A storage room measures 12 feet by 9 1/2 feet. What is its area in square feet?

Answer Key

Check your work

Open the key: after you've finished all ten

1 8/15

Straight across: 2 × 4 = 8 above, 3 × 5 = 15 below. No common denominator was needed or wanted. If you hunted for one out of habit from Quiz 10, no harm was done to the answer, but the work was wasted: matching is for joining, not for cutting.

2 1/6

This is the figure from The Guide: three cuts one way, two the other, six cells, one survivor. Worth saying aloud once, because it sounds strange and is exactly right — half of a third is a sixth.

3 B.

The answer is 6, but the question was about reasoning, not arithmetic. A) is the whole-number promise that fractions break. C) mistakes canceling for equality, since canceling simplifies the work but never leaves the starting number untouched. D) gives up a judgment that of makes instantly. Any factor under 1 shrinks what it multiplies, and knowing that before you compute is what makes the guard useful.

4 6/5, which is 1 1/5

Write 3 as 3/1: tops give 6, bottoms give 5. Here the answer grew, because 3 is greater than 1 — the rule was never that multiplying by a fraction shrinks, but that multiplying by anything less than one shrinks. Since the top outgrew the bottom, finish with the mixed number.

5 8

Divide by the bottom, multiply by the top: a third of 12 is 4, so two thirds is 8. Or the long way, 2/3 × 12/1 = 24/3 = 8. This is the shape of nearly every discount, tip, and split bill you will ever compute, so it is worth being fast at.

6 1/6

Multiply first and you get 6/36, which simplifies to 1/6. Cancel first — the 3 against the 9, the 2 against the 4 — and you get 1/2 × 1/3 = 1/6 with almost no arithmetic. Both are right; the second is kinder to a tired hand, and it is Quiz 7's factors doing the favor. Note that this lands on the same answer as question 2, by a different road.

7 C.

A) is the real trap and the reason this question exists: it gives 2 + 1/6 = 2 1/6, and the guard rejects it at once, since two and a half grown by more than one cannot come out smaller than two and a half. B) is the common denominator, Quiz 10's tool, brought to the wrong job: matching is for adding, not for multiplying. D) confuses what a mixed number already means, since 2½ is 2 plus ½ and needs no further adding.

8 10/3, which is 3 1/3

5/2 × 4/3 = 20/6, simplified to 10/3, converted to 3 1/3. Guard: over 3, as expected. If you got 2 1/6, that is answer A from question 7 in action, and the two items are meant to be read together.

9 3/8 cup

Half of three quarters: 1/2 × 3/4 = 3/8. Halving a recipe is the most common fraction multiplication in ordinary life, and notice what halving does to the name of the piece — quarters become eighths, since cutting each piece in two doubles how many make a whole.

10 114 square feet

12 × 19/2 = 228/2 = 114. And this is the rectangle from The Guide, standing in a building: area has been multiplication since Quiz 4, and fractional sides change nothing except the arithmetic. Guard: 12 × 9 is 108 and 12 × 10 is 120, so the answer had to land between them.

Reading your results

Questions	The skill they test	If they gave trouble
1, 2, 6	Multiplying straight across	Tops together, bottoms together; the cuts multiply, so the bottoms do too.
3	Knowing which way the answer moves	Under one shrinks, over one grows; ask before you compute.
4, 5	Whole numbers and of	Put the whole number over 1, or divide by the bottom and multiply by the top.
7, 8	Mixed numbers	Convert to improper fractions first, always; parts cannot be multiplied separately.
9, 10	Multiplying in the world	Of means multiply; area has meant multiply since Quiz 4.

Eight or more right: the cutting machine is yours — on to Quiz 12. 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. Draw the unit square with your teacher and shade the two cuts by hand, three or four times with different fractions, until the overlap stops being a rule and becomes a picture. Then come back to these same ten.

The Company · An Interlude

Fellow travelers

Today's masthead abandoned photography for heraldry, and the two figures flanking the title are not seahorses as biology knows them. They are sea-horses as the heralds drew them: the forequarters of a horse, webbed forefeet, a finned mane, and a fish's tail behind. A creature assembled from parts, which suits a quiz about taking parts of parts.

A heraldic sea-horse drawn in black: a rearing horse's foreparts with webbed hooves, a spined mane, and a scaled fish tail curling beneath — The heraldic sea-horse in full: horse above, fish below, and the join treated as though nobody should find it strange. This is the figure the masthead supporters were drawn from, rendered there in gold.

A heraldic sea-horse drawn in outline only, showing the same rearing horse and fish-tail construction — And the same beast in outline, the way a herald would hand it to a painter: the shape settled, the colors left to whoever holds the arms. Two hundred years before anyone photographed a living seahorse, this is what the word pictured.

Prints only this gallery, full size. The main print button stays photograph-free to spare your ink.