Dividing fractions
Divide fractions (÷ a whole, ÷ a fraction); fraction as a quotient (a/b = a ÷ b).
Dividing by a fraction asks a “how many fit?” question: 6 ÷ ½ is “how many halves fit in 6?” — and the answer, 12, is bigger than you started with.
What it means
A fraction like 3/4 means three of the four equal pieces you get when you split one whole into four. Dividing asks how many of one amount fit inside another: 12 ÷ 3 = 4 because four 3s fit in 12.
Two ideas sit under this concept.
A fraction is itself a division. The bar in a/b literally means a ÷ b. So 3/4 is “3 shared between 4”, which is why 3 ÷ 4 = 3/4 = 0.75. Any division you cannot finish neatly can be written as a fraction and left exact.
Dividing by a fraction. Splitting one whole into halves gives 2 pieces, so 1 ÷ ½ = 2. Into quarters gives 4 pieces, so 1 ÷ ¼ = 4. Notice: dividing by a smaller piece gives a bigger answer, because more small pieces fit. This surprises people, but it is the same logic as 10 ÷ 2 = 5 versus 10 ÷ ½ = 20.
The shortcut for any case is “keep, change, flip”: keep the first fraction, change ÷ to ×, and flip the second one upside down (its reciprocal — the fraction turned over, so the reciprocal of 2/3 is 3/2). Then multiply. Flipping works because dividing by 2/3 is the same as multiplying by how many thirds fit in a whole.
Worked examples
Divide by a whole number — share ¾ among 2 people:
¾ ÷ 2 = ¾ × ½ = 3/8Each person gets 3/8. (Dividing by 2 is the same as multiplying by ½.)
Divide by a fraction — how many ⅓ cups fit in ⅔?
⅔ ÷ ⅓ = ⅔ × 3/1 = 6/3 = 2Two thirds hold exactly two one-third cups. You can see it: two ⅓ pieces make ⅔.
Fraction as a quotient — write 5 ÷ 8 as a fraction:
5 ÷ 8 = 5/8 = 0.625| Problem | Keep · Change · Flip | Answer |
|---|---|---|
½ ÷ ¼ | ½ × 4/1 | 2 |
¾ ÷ ½ | ¾ × 2/1 | 6/4 = 1½ |
1 ÷ ⅗ | 1 × 5/3 | 5/3 = 1⅔ |
The generative-art connection
Division by a fraction is really a counting-how-many-fit action, and that is exactly what a repeating pattern shows. Take a strip and tile it with ⅓-length pieces: the number of tiles that fill it is the quotient. Change the tile size and the count changes — dividing by a smaller unit packs in more tiles.
The Math Learning Center Fractions app lets a child drag fraction bars together and literally watch how many small pieces cover a bigger bar — the division answer appears as a count of tiles.
The nico.art dot-multiplier tool turns the same idea radial: one dot multiplies into an evenly spaced ring, and asking “how many dots fit around the circle” is a division. Splitting a full turn into 1/n slices is 1 ÷ (1/n) = n, so a finer angle gives more arms — the picture makes “divide by a smaller fraction, get a bigger number” visible in one glance.
Common misconceptions
“Dividing always makes things smaller.” True for dividing by numbers bigger than 1, false for fractions. 8 ÷ ½ = 16 because sixteen halves fit in 8. Read every division as “how many fit?” and the growth stops being mysterious.
Flipping the wrong fraction. In keep-change-flip you flip only the second fraction (the divisor), never the first. ¾ ÷ ½ becomes ¾ × 2, not 4/3 × ½.
Forgetting a whole number is a fraction. 2 is 2/1, whose flip is ½. That is why dividing by 2 is the same as halving.
What a learner should be able to solve
The benchmark for mastering this concept — the problems a child at this stage is expected to be able to work through.
-
Work out `2/3 ÷ 3/4`. Use keep-change-flip and give your answer as a fraction in its simplest form.
Answer
`2/3 ÷ 3/4 = 2/3 × 4/3 = 8/9`. Keep `2/3`, change ÷ to ×, flip `3/4` to `4/3`, then multiply top×top and bottom×bottom: `2×4 = 8`, `3×3 = 9`. `8/9` is already in simplest form (8 and 9 share no factor). Note the answer is less than 1, because a `3/4`-tile does not quite fit once inside `2/3`.
Art hook Draw a bar of length `2/3` and, above it, one loose tile of length `3/4`. Animate sliding the tile against the bar: it overhangs, so only part of it lies inside — the part that fits is `8/9` of the tile. A live readout shows 'tiles that fit = 8/9'. A slider changes the tile length and the fitted count updates.
-
How many `1/4`-litre cups can you fill from a `2 1/2`-litre jug? Write it as a division and answer.
Answer
`2 1/2 ÷ 1/4 = 5/2 × 4/1 = 20/2 = 10` cups. First turn `2 1/2` into the improper fraction `5/2`, then keep-change-flip. Ten quarter-litre cups fit.
Art hook Canvas: a tall jug drains into a row of little cups, one filling on each beat until empty. Sliders set jug volume and cup size; the number of cups filled equals the quotient. Shrink the cup and watch the row get longer.
-
Write `7 ÷ 9` as a fraction, then say whether it is bigger or smaller than 1 and why.
Answer
`7 ÷ 9 = 7/9`, which is smaller than 1 because the top (7) is less than the bottom (9): you are sharing 7 things among 9 people, so each share is under a whole.
Art hook A pie split into 9 equal slices; drag 7 slices onto a plate. The plate fills to `7/9` of a full pie — a live picture that `7 ÷ 9 = 7/9 < 1`. A ghost outline of the missing 2 slices shows how far short of a whole it is.
-
A ribbon `3/4` m long is cut into pieces each `1/8` m long. How many pieces do you get? Show your working.
Answer
`3/4 ÷ 1/8 = 3/4 × 8/1 = 24/4 = 6` pieces. Keep-change-flip, then simplify `24/4 = 6`. Six eighth-metre pieces make three quarters (since `6/8 = 3/4`).
Art hook A ribbon strip marked with faint cut lines every `1/8` m. Click to snip at each mark; a counter animates up as pieces fall away. A slider changes the piece length — smaller pieces, more cuts.
-
Without a calculator, decide which is larger: `5 ÷ 1/2` or `5 ÷ 2`. Explain your choice.
Answer
`5 ÷ 1/2 = 5 × 2 = 10`, while `5 ÷ 2 = 2.5`. So `5 ÷ 1/2` is larger. Dividing by a number smaller than 1 makes the answer bigger, because many tiny halves fit inside 5.
Art hook Split screen: on the left, five whole circles each cut into 2 halves (10 pieces glow); on the right, 5 shared into 2 groups (2.5 each). Two dot-rings grow side by side so the 'divide by a smaller piece = more pieces' contrast is instant.
Training exercises
Practice problems, easier first, that build toward the bar above. Each doubles as a seed for an interactive artwork.
-
Finish the sentence: dividing by a fraction asks a `how many ___ fit?` question. Then answer: how many halves fit in `3`?
Answer
It asks 'how many fit?'. `3 ÷ 1/2 = 6` — six halves fit in 3, since each whole holds 2 halves and there are 3 wholes.
Art hook Three circles each split into 2. Click each half to light it up; a counter tallies to 6 as you go.
-
Write each division as a fraction: (a) `4 ÷ 5` (b) `1 ÷ 3` (c) `9 ÷ 10`.
Answer
(a) `4/5` (b) `1/3` (c) `9/10`. The `÷` sign becomes the fraction bar: `a ÷ b = a/b`.
Art hook A number line from 0 to 1 with a marker you drop at each fraction. Three dots land at `4/5`, `1/3`, `9/10`, coloured by size.
-
What is the reciprocal (the flipped fraction) of each? (a) `3/5` (b) `1/6` (c) `7`.
Answer
(a) `5/3` (b) `6/1 = 6` (c) `1/7` (because `7 = 7/1`, which flips to `1/7`).
Art hook A colour wheel where each fraction sits at an angle and its reciprocal sits at the mirrored angle — flipping a fraction visibly reflects the dot across the diagonal.
-
Share `4/5` equally between 2 friends. How much does each get?
Answer
`4/5 ÷ 2 = 4/5 × 1/2 = 4/10 = 2/5`. Dividing by 2 is the same as multiplying by `1/2`. Each friend gets `2/5`.
Art hook A bar shaded `4/5` splits down the middle into two `2/5` bars that slide apart — one per friend.
-
Use keep-change-flip to work out `1/2 ÷ 1/6`. How many sixths fit in a half?
Answer
`1/2 ÷ 1/6 = 1/2 × 6/1 = 6/2 = 3`. Three sixths fit in one half (since `3/6 = 1/2`).
Art hook A bar of length `1/2` gets tiled by `1/6`-tiles; exactly 3 tiles snap into place with a chime.
-
Work out `5/6 ÷ 5`. Simplify your answer.
Answer
`5/6 ÷ 5 = 5/6 × 1/5 = 5/30 = 1/6`. Dividing by 5 cuts the `5/6` bar into 5 equal parts, each `1/6`.
Art hook A `5/6` bar sliced into 5 equal pieces; each lit piece is labelled `1/6`.
-
A pupil worked out `3/4 ÷ 1/2` like this: `3/4 ÷ 1/2 = 4/3 × 1/2 = 4/6 = 2/3`. Spot their mistake, then write the correct answer.
Answer
They flipped the wrong fraction — in keep-change-flip you flip only the divisor (the second fraction), never the first. Correct: `3/4 ÷ 1/2 = 3/4 × 2/1 = 6/4 = 3/2 = 1 1/2`.
Art hook Two 'keep-change-flip' machines side by side: the buggy one flips the first fraction (red X), the correct one flips the second (green check).
-
How many `1/3`-hour (20-minute) chunks are there in `2` hours? Write it as a fraction division and answer.
Answer
`2 ÷ 1/3 = 2 × 3/1 = 6` chunks. Each hour holds three 20-minute chunks, and there are 2 hours, so `3 × 2 = 6`.
Art hook A clock face with a wedge that sweeps `1/3` of an hour on each tap; a counter reaches 6 as two full hours are covered.
-
Work out `3/8 ÷ 3/4` and give the answer in simplest form.
Answer
`3/8 ÷ 3/4 = 3/8 × 4/3 = 12/24 = 1/2`. After keep-change-flip you can cancel: the 3s cancel and `4/8 = 1/2`.
Art hook A `3/8` bar measured against a `3/4`-tile: exactly half of the tile fits, shading in to show the `1/2` answer.
-
Reasoning: without dividing, is `1/5 ÷ 1/10` bigger or smaller than 1? Then check by working it out.
Answer
Bigger than 1, because `1/10` is smaller than `1/5`, so more than one of them fits inside. Check: `1/5 ÷ 1/10 = 1/5 × 10/1 = 10/5 = 2`.
Art hook Two nested rings — one split into 5, one into 10 — overlaid so you see two tenths cover one fifth: the '2' appears as two tenth-slices lighting up inside one fifth-slice.
-
Open-ended: find two different fraction divisions that both give the answer `4`. Show that each one works.
Answer
Many are possible. Example 1: `1 ÷ 1/4 = 1 × 4 = 4`. Example 2: `2/3 ÷ 1/6 = 2/3 × 6/1 = 12/3 = 4`. Any `a/b ÷ c/d` whose keep-change-flip product equals 4 works.
Art hook A 'target = 4' puzzle: drag two fraction tiles into a division slot; when the tiled count fills exactly 4 units, the ring blooms into 4 evenly spaced dots.
-
Challenge: `2 1/4 ÷ 3/4`. Turn the mixed number into an improper fraction first, then divide.
Answer
`2 1/4 = 9/4`. `9/4 ÷ 3/4 = 9/4 × 4/3 = 36/12 = 3`. Three three-quarter pieces fit in `2 1/4`.
Art hook A bar of length `2 1/4` tiled by `3/4`-tiles; three tiles fill it exactly, each snapping in with a pulse.