Seahorse Series · Quiz 12 · Dividing Fractions

The Guide

Before you begin

Dividing fractions carries the oddest-looking rule in all of arithmetic: turn the second fraction upside down and multiply. Taught as a trick it is unforgettable and meaningless. Taught as a picture it becomes obvious, and Quiz 5 already drew half of that picture. As in every quiz of this series: no tricks, no calculator, and whatever this shows is useful news.

Division asks how many fit

Quiz 5 gave division two faces: the sharing question, and the measuring question — how many of these fit inside that? For fractions the measuring face is the useful one. 3 ÷ 1/4 asks: how many quarter-cup scoops fill three cups? Four fit in each cup, so twelve fit in three.

Drawn to scale. Nothing was shared out; the pieces were simply counted.

Notice what happened: the answer came out larger than the 3 we started with. That is not strange once the question is heard properly. Small pieces go into a quantity many times over, so dividing by anything less than one makes the answer bigger. It is the exact mirror of Quiz 11, where multiplying by less than one made things smaller.

You have always known this

Everyone accepts that dividing by 2 is the same as taking half: 10 ÷ 2 = 10 × ½. The rule you are about to learn is only that sentence read in the other direction. If dividing by 2 means multiplying by ½, then dividing by ½ means multiplying by 2. The upside-down fraction was never a trick; it is the same familiar swap, running backward.

Reciprocals

Two numbers are reciprocals when they multiply to 1: 23 and 32, since 2 × 3 above and 3 × 2 below give 66. To find one, turn the fraction over. A whole number turns over too, because 5 is 51, whose reciprocal is 15 — which is just the old fact that dividing by 5 takes a fifth.

The rule, and its name

Keep, change, flip. Keep the first fraction as it stands, change the division sign to multiplication, and flip the second. Only the second — the one doing the dividing. Then multiply straight across and simplify, as in Quiz 11.

Drawn to scale. Counted by eye: six. By rule: 3/4 × 8/1 = 24/4 = 6. The picture and the trick agree.

Why flipping works, in one line: asking how many eighths fit in something is the same as asking what you get when you take eight of it for every whole. Dividing by 18 and multiplying by 8 are two descriptions of one act, so the flip is not a trick performed on the symbols. It is a translation between two ways of saying the same sentence.

Whole numbers and mixed numbers

Whole numbers go over 1 as before: 6 ÷ 2/3 = 61 × 32 = 182 = 9. Mixed numbers must become improper fractions (single fractions with the top bigger than or equal to the bottom) before anything is flipped, exactly as in Quiz 11: 2½ ÷ 1¼ becomes 52 ÷ 54 = 52 × 45 = 2010 = 2.

The guard, mirrored

Before computing, ask which way the answer must move. Dividing by less than one makes it bigger; dividing by more than one makes it smaller; dividing by exactly one changes nothing. An answer that moves the wrong way has almost always been flipped on the wrong side.

Worked Examples

Three to study before you start

Example 1 · Keep, change, flip

23 ÷ 45. Guard: dividing by something under 1, so the answer must exceed 23. Keep 23, change the sign, flip to 54: 1012, which simplifies to 5/6. And 5/6 is indeed larger than 2/3, since in sixths they are 5 and 4.

Example 2 · Dividing by a whole number

34 ÷ 6. The 6 is 61, so flipping gives 16: 34 × 16 = 324 = 1/8. Guard: dividing by more than one, so the answer shrank, and it did. In plain words, three quarters shared among six people gives each an eighth.

Example 3 · Mixed numbers

2½ ÷ 1¼. Convert first: 52 ÷ 54. Now keep, change, flip: 52 × 45 = 2010 = 2. Read it as measuring and it makes sense on its own: one and a quarter fits into two and a half exactly twice.

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.

Divide: 3 ÷ 14
Divide: 12 ÷ 14
Without computing it exactly, what must be true of 6 ÷ 1/2?
- A) It is less than 6, because dividing always makes numbers smaller.
- B) It equals 3, because dividing by a half is the same as halving.
- C) It is greater than 6, because halves are small and many of them fit inside 6.
- D) There is no way to tell without dividing.
Divide: 23 ÷ 45
Divide: 6 ÷ 23
Divide: 34 ÷ 6

▶ VideoA video rest stop, for anyone who has kept going this far.

Take two minutes with a seahorse swimming in a tall kelp forest.

Marcus worked 23 ÷ 45 by flipping the first fraction: 32 × 45 = 1210 = 65. What went wrong?
- A) Nothing; 6/5 is correct.
- B) Only the second fraction (the divisor), the one doing the dividing, gets flipped, so it should be 2/3 × 5/4 = 5/6.
- C) Both fractions should be flipped.
- D) He should have found a common denominator first.
Divide: 2 1/2 ÷ 1 1/4
Brian has a 6-foot length of wire and needs pieces 34 of a foot long. How many pieces can he cut?
Rosa has 4 cups of rice, and each batch of her recipe uses 23 cup. How many full batches can she make?

Answer Key

Check your work

Open the key: after you've finished all ten

1 12

Four quarters fit in each whole, and there are three wholes. By rule: 3/1 × 4/1 = 12. This is the figure from The Guide, and the answer being larger than 3 is the point of the whole quiz.

2 2

How many quarters fit in a half? Two. By rule: 1/2 × 4/1 = 4/2 = 2. If you answered 1/8, you multiplied instead of dividing — an easy slip, and the guard catches it, since dividing by a quarter cannot make a half smaller.

3 C.

The answer is 12. A) is the whole-number promise that fractions break, the mirror of the one broken in Quiz 11. B) is the most interesting wrong answer: dividing by 2 is halving, but dividing by a half is the opposite move, and confusing the two is the commonest misreading on this topic. D) gives up a judgment you can make instantly by asking how many halves fit in 6.

4 5/6

Keep 2/3, change the sign, flip to 5/4: 10/12, which simplifies to 5/6. Guard: 4/5 is under one, so the answer had to exceed 2/3, and in sixths 5 beats 4.

5 9

6/1 × 3/2 = 18/2 = 9. Read as measuring: two thirds fits into 6 nine times, since three of them fill every two wholes. Guard: dividing by less than one, so the answer grew from 6, and it did.

6 1/8

The 6 is 6/1, so flipped it is 1/6: 3/4 × 1/6 = 3/24 = 1/8. Guard: dividing by more than one, so the answer shrank. In plain words, three quarters shared among six gives each an eighth. Note that questions 5 and 6 both involve 6 and a fraction, and the answers move in opposite directions — because in one the 6 is being divided and in the other it is doing the dividing.

7 B.

Flipping the wrong fraction is the signature error of this topic, and it is invisible unless you check direction: Marcus's 6/5 is bigger than 1, while the true answer 5/6 is smaller. The rule names which one moves — keep the first, flip the second — because only the second fraction, the one doing the dividing, is turned into its reciprocal. C) flips both and lands somewhere else entirely. D) brings the common denominator, Quiz 10's tool, to a job that never needed it.

8 2

Convert first: 5/2 ÷ 5/4. Then 5/2 × 4/5 = 20/10 = 2. Check it by measuring rather than by rule: one and a quarter fits into two and a half exactly twice. If you tried to divide the whole parts and the fraction parts separately, that is the same trap as Quiz 11's question 7, wearing a division sign.

9 8 pieces

6 ÷ 3/4 = 6/1 × 4/3 = 24/3 = 8. This is the measuring question in its natural habitat: how many three-quarter-foot pieces fit in six feet? Nothing is left over, which is worth noticing — when a cut list divides evenly there is no offcut, and when it doesn't, the remainder is scrap. The leftover can mean different things depending on the question, as Quiz 5 showed.

10 6 batches

4 ÷ 2/3 = 4/1 × 3/2 = 12/2 = 6. Guard: two thirds is less than a cup, so more than four batches must be possible. The word full in the question is doing Quiz 5's work again — had the answer come out with a remainder, only whole batches would count.

Reading your results

Questions	The skill they test	If they gave trouble
1, 2	Division as measuring	Ask how many fit, and count them on the bar before reaching for any rule.
3	Knowing which way the answer moves	Dividing by less than one grows the answer; dividing by 2 and dividing by a half are opposite moves.
4, 5, 6	Keep, change, flip	Whole numbers go over 1; only the second fraction turns over.
7, 8	Flipping the right one, and mixed numbers	The divisor is the one translated; convert mixed numbers before anything else.
9, 10	Dividing in the world	Cut lists and batch counts are measuring questions; watch for what the leftover means.

Eight or more right: all four operations now work in fraction country — on to Quiz 13. 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. With your teacher, take a measuring cup and fill larger amounts with a quarter-cup scoop, counting aloud. The rule will stop being strange the moment your hands have done it.

The Company · An Interlude

Fellow travelers

The masthead returns home today: our own Hippocampus kuda, alone on the burgundy as in the first three quizzes, with the twelfth topic closed and fraction country behind us. The company below comes from the film that the rest stops of earlier quizzes have pointed to.

Film frame: a leafy seadragon in luminous blue, its body striped and its leaf-shaped fins spread — The leafy seadragon, lit from the side. Count the leaf-fins if you can; the difficulty of counting them is roughly the difficulty of dividing by a fraction before you know what the question is asking.

Film frame: a weedy seadragon in warm light with a slender striped body and small paddle fins — And the weedy, in warmer water. Two cousins, one built for ornament and one for speed, both solving the same problem in different proportions.

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