In plain words
Is 3/4 − 1/2 equal to 2/2?
No. 3/4 − 1/2 = 1/4. One half is the same as two fourths, so taking it away from three fourths leaves one fourth. 2/2 is a whole, which is more than the 3/4 we started with.
- The big idea
- Subtraction takes away an amount. Use the same-size pieces for both amounts, then take away: 3/4 − 1/2 = 3/4 − 2/4 = 1/4.
- The common confusion
- To subtract fractions, subtract the top numbers and the bottom numbers, so 3/4 − 1/2 = 2/2. It often sounds like this: “(three fourths) take away (one half) is (two halves). Three minus one, four minus two.”
Why it works
Subtraction takes an amount away. 3/4 − 1/2 asks: if we have three fourths and take away one half, how much is left?
Fourths and halves are different-sized pieces, so first describe the half in fourths. One half covers exactly two fourths, so 1/2 = 2/4.
Now take away two of the three fourths. One fourth is left: 3/4 − 1/2 = 3/4 − 2/4 = 1/4.
Subtracting the top numbers and the bottom numbers gives 2/2, which is a whole. That is more than the 3/4 we started with, and taking something away can’t leave more than we had.
Adding back checks the answer: 1/4 + 1/2 = 1/4 + 2/4 = 3/4, the amount we started with.
A worked example
A glass is 3/4 full of juice and you drink half a glass. Half a glass is the same as two quarters of a glass, so one quarter of the glass is left: 3/4 − 1/2 = 1/4.
Why this confusion is so common
Subtracting the top numbers and the bottom numbers is the subtraction version of adding tops and bottoms, and it can give tidy-looking answers such as 2/2. The answer often isn’t compared with the amounts in the problem. Asking “Should there be more or less than we started with?” exposes it, because taking an amount away always leaves less.
Try this away from the screen
Fold a paper strip into fourths and color three of them. Fold a second strip of the same length in half. Lay the half on top of the colored part, lined up at one end. How many colored fourths does it hide, and how much color is still showing? Then check by adding back: the leftover and the half together cover 3/4 again.
Helpful things to say
| Instead of | Try |
|---|---|
| “Subtract the top numbers, then subtract the bottom numbers.” | “What are we taking away? Can we show that half using fourths?” |
| “That can’t be right. Try again.” | “Should we have more or less than we started with? Does 2/2 fit with that?” |
Practice questions
For grown-ups helping out: the answers are tucked away so learners can try first.
5/6 of a pizza is left. You eat 1/3 of the whole pizza. How much is left now?
Show the answer
Answer: 3/6. 1/3 of the pizza is 2 of the 6 slices, so 1/3 = 2/6. Then 5/6 − 2/6 = 3/6, which is half the pizza.
Start at 7/8 and move back 1/4. Where do you land?
Show the answer
Answer: 5/8. 1/4 is two eighths. Two eighth-spaces back from 7/8 is 5/8, so 7/8 − 1/4 = 7/8 − 2/8 = 5/8.
What is 7/10 − 1/2?
Show the answer
Answer: 2/10. 1/2 lines up with five tenths, so 1/2 = 5/10. Then 7/10 − 5/10 = 2/10.
Leo says 5/8 − 1/4 = 4/4. Can that be right?
Show the answer
Answer: No. 4/4 is one whole, which is more than 5/8, so it can’t be what is left. 1/4 is 2/8, and 5/8 − 2/8 = 3/8.
Use sixth pieces to build the answer to 1/2 − 1/6.
Show the answer
Answer: use pieces of 1/6 to make 2/6. 1/2 is three sixths. Take one sixth away and two sixths are left: 1/2 − 1/6 = 3/6 − 1/6 = 2/6.
Questions people ask
Why isn’t 3/4 − 1/2 equal to 2/2?
2/2 is one whole, which is more than the 3/4 you started with. Subtracting tops and bottoms doesn’t take an amount away. Write 1/2 as 2/4 instead; then 3/4 − 2/4 = 1/4.
Do you need a common denominator to subtract fractions?
You need same-size pieces, and a common denominator gives you them. Once both fractions use the same pieces, subtract the numerators and keep the denominator. For 2/3 − 1/6, write 2/3 in sixths: 4/6 − 1/6 = 3/6, which is 1/2.
Why does subtracting tops and bottoms give such strange answers?
Because it doesn’t take an amount away. When the second fraction is smaller and both subtractions give positive numbers, the result is always bigger than the fraction you started with: 3/4 − 1/2 would become 2/2, a whole.
How can my child check a subtraction answer?
Add the answer back. If 3/4 − 1/2 = 1/4, then 1/4 + 1/2 should make 3/4, and it does: 1/4 + 2/4 = 3/4. Also check that the answer is less than the starting amount.
Words to know
- Common denominator
- A shared bottom number, so two fractions are written with the same-size parts. Adding and subtracting need one.
- Equivalent fractions
- Fractions that name the same amount, such as 1/2, 2/4 and 4/8. They share one point on the number line.
- Whole
- The one complete thing, length or group that a fraction is a part of.
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:
- 5.NF.A.1
Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.)
- 5.NF.A.2
Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. For example, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2.
- 4.NF.B.3.a
Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.
Related slices
Adding Different Pieces
Is (one half) + (one fourth) equal to (two sixths)?
Why 1/2 + 1/4 is 3/4, not 2/6: make the pieces the same size before you count them. · About 3 minutes
Keep the Unit When Adding
Is (one fifth) + (two fifths) equal to (three tenths)?
Why 1/5 + 2/5 is 3/5, not 3/10: joining fifths gives more fifths, not smaller pieces. · About 3 minutes
Different Numbers, Same Amount
Is (one half) the same as (two fourths)?
Why 1/2, 2/4 and 4/8 are exactly the same amount, even though the numbers are different. · About 3 minutes
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