In plain words
Why does dividing by a fraction give a bigger answer?
Because dividing by a fraction asks how many of those pieces fit into an amount. Pieces smaller than one whole fit more than once into every whole, so the count is bigger than the amount: two fourths fit into one half, so 1/2 ÷ 1/4 = 2.
- The big idea
- Dividing by a fraction asks how many of those pieces fit. When the pieces are smaller than one whole, the answer is bigger than the amount: 1/2 ÷ 1/4 = 2.
- The common confusion
- Dividing always makes a number smaller, so 1/2 ÷ 1/4 must be less than 1/2. It often sounds like this: “Dividing makes things smaller, so (one half) ÷ (one fourth) is less than (one half).”
Why it works
With whole numbers, dividing usually makes a number smaller: 12 ÷ 3 = 4. So it is natural to expect every division to do the same.
Division can also ask “how many of these fit into that?” 12 ÷ 3 asks how many 3s fit into 12. In the same way, 1/2 ÷ 1/4 asks how many fourths fit into one half.
Two fourth pieces fill one half exactly, so 1/2 ÷ 1/4 = 2. The answer is bigger than the 1/2 we started with.
Whenever a positive amount is divided by a number between 0 and 1, we are fitting pieces smaller than one whole, so the count is bigger than the amount. For example, 1 ÷ 1/5 = 5 and 1/2 ÷ 1/6 = 3.
Dividing by exactly 1 leaves an amount unchanged, and dividing by a number greater than 1 makes it smaller. The size of the divisor tells us what to expect.
A worked example
You have 2 cups of rice and a scoop that holds 1/4 of a cup. Each cup is 4 scoops, so 2 cups make 8 scoops: 2 ÷ 1/4 = 8. Dividing by 1/4 gave a bigger number than 2, because the scoop is smaller than a cup.
Why this confusion is so common
Division is usually introduced as sharing: 12 apples shared among 3 friends. Sharing among more people makes each share smaller, so children conclude that division makes numbers smaller. That is true when dividing by a number greater than 1, but dividing by a number between 0 and 1 gives a bigger answer. The other meaning of division, “how many of these fit into that?”, makes this sensible, and it is the most natural way to picture dividing by a fraction.
Try this away from the screen
Fold one paper strip into fourths and cut it into its four pieces. Fold a second strip of the same length in half and color one half. How many fourth pieces fit on the colored half? Write it as 1/2 ÷ 1/4 = 2. Then ask how many fit on the whole strip, and write that as a division too.
Helpful things to say
| Instead of | Try |
|---|---|
| “Dividing makes numbers smaller.” | “How many of these pieces fit into that amount?” |
| “Flip the second fraction and multiply.” | “Before we calculate, will the answer be more or less than what we started with? Why?” |
Practice questions
For grown-ups helping out: the answers are tucked away so learners can try first.
What is 3/4 ÷ 1/8?
Show the answer
Answer: 6. 3/4 ÷ 1/8 asks how many eighths fit into 3/4. Each fourth holds 2 eighths, so three fourths hold 6 eighths: 3/4 ÷ 1/8 = 6.
What is 2 ÷ 1/3? Use the number line to help.
Show the answer
Answer: 6. 2 ÷ 1/3 asks how many thirds fit into 2. Each whole holds 3 thirds, so 2 wholes hold 6: 2 ÷ 1/3 = 6.
What is 3/4 ÷ 1/2?
Show the answer
Answer: 1 1/2. 3/4 ÷ 1/2 asks how many halves fit into 3/4. One half fits, leaving 1/4, which is half of another half. So 3/4 ÷ 1/2 = 1 1/2.
Mia says 4 ÷ 1/2 = 2. Is she right?
Show the answer
Answer: No. 4 ÷ 1/2 asks how many halves fit into 4. Each whole holds 2 halves, so 4 wholes hold 8: 4 ÷ 1/2 = 8. Cutting 4 in half is 4 ÷ 2, which is 2.
What is 3 ÷ 1/4? Each circle is one whole.
Show the answer
Answer: 12. 3 ÷ 1/4 asks how many fourths fit into 3. Each whole holds 4 fourths, so 3 wholes hold 12: 3 ÷ 1/4 = 12.
Questions people ask
Why does dividing by a fraction give a bigger number?
Dividing by a fraction asks how many of those pieces fit into an amount. Pieces smaller than one whole fit more than once into every whole, so the answer is bigger than the amount. Two fourths fit into one half, so 1/2 ÷ 1/4 = 2.
Is dividing by 1/2 the same as dividing in half?
No. Dividing in half means dividing by 2: 6 ÷ 2 = 3. Dividing by 1/2 asks how many halves fit into 6, which is 12. Mixing these up is very common, for adults as well as children.
Where does “flip and multiply” come from?
Dividing by 1/4 asks how many fourths fit, and every whole holds 4 fourths. So dividing by 1/4 gives the same answer as multiplying by 4: 1/2 ÷ 1/4 = 1/2 × 4 = 2. The rule works, and the “how many fit?” picture shows why.
Does dividing by a fraction always make the answer bigger?
Only when the fraction is less than one and the amount is positive. Dividing by a fraction greater than one, such as 3/2, makes the answer smaller: 3 ÷ 3/2 = 2.
Words to know
- Unit fraction
- A fraction with 1 on top, such as 1/2, 1/3 or 1/8. It is one equal part of the whole.
- Whole
- The one complete thing, length or group that a fraction is a part of.
- Mixed number
- A whole number and a fraction written together, such as 1 2/4. It means one whole and two fourths more.
Curriculum connections
We don’t assume any particular curriculum. For teachers in the United States, here’s how this slice lines up with the Common Core State Standards:
- 6.NS.A.1
Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi?
- 5.NF.B.7.b
Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 × (1/5) = 4.
Related slices
Multiplication Can Make Smaller
Why does multiplying by a fraction make a number smaller?
Why 1/2 × 1/2 is 1/4: multiplying by a fraction less than one takes part of an amount. · About 4 minutes
The Bigger Denominator Trap
Is (one sixth) bigger than (one eighth)?
Why 1/8 is smaller than 1/6, even though 8 is bigger than 6. · About 3 minutes
More Than One Whole
Can a fraction be bigger than 1?
Why 6/4 makes sense: a fraction can equal one whole or go beyond it. · About 3 minutes
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