How do you subtract fractions, including fractions with different denominators and mixed numbers?

To subtract fractions, give both fractions the same denominator, subtract the numerators, keep the denominator and simplify. For 3/4 − 1/6, the least common denominator is 12, so 3/4 becomes 9/12, 1/6 becomes 2/12, and 9/12 − 2/12 = 7/12. With mixed numbers, regroup one whole as a fraction when the first fraction is smaller than the second.

How do you subtract fractions with different denominators?

Fractions with different denominators are subtracted by first renaming them with a common denominator. Find the least common multiple of the denominators, multiply the top and bottom of each fraction to reach it, subtract the numerators and simplify. For 2/3 − 1/4 the common denominator is 12, so 8/12 − 3/12 = 5/12.

How do you subtract mixed numbers when the first fraction is smaller?

Mixed numbers whose first fraction is smaller are subtracted by regrouping one whole into the fraction part. In 3 1/4 − 1 3/4, the 1/4 is smaller than 3/4, so take 1 from the 3 and write it as 4/4: 3 1/4 becomes 2 5/4. Then 2 5/4 − 1 3/4 = 1 2/4, which simplifies to 1 1/2.

Do you subtract the denominators when subtracting fractions?

The denominators are not subtracted when you subtract fractions. The denominator tells you the size of the pieces, and taking pieces away does not change their size, so only the numerators are subtracted. 5/8 − 2/8 = 3/8, while subtracting the denominators would give 3/0, which has no meaning.

How do you check a fraction subtraction?

A fraction subtraction is checked by adding the answer to the number you took away. For 3/4 − 1/6 = 7/12, add 7/12 + 2/12 to get 9/12, which simplifies to 3/4, the starting number. A quick estimate with benchmark fractions such as 1/2 also catches answers that are far off.

How to Subtract Fractions With Unlike Denominators

Q: How do you subtract a fraction from a whole number?

A fraction is subtracted from a whole number by taking one whole and writing it as a fraction with the same denominator. For 5 − 2/3, write 5 as 4 3/3, then subtract: 4 3/3 − 2/3 = 4 1/3. You can also count up: from 2/3 to 1 is 1/3, and from 1 to 5 is 4, so the answer is 4 1/3.

Three quarters of a pie sit on the table, and someone eats a piece that is 1/6 of the whole pie, so how much is left? The question looks easy until the homework sheet writes it as 3/4 − 1/6 and the quarters and sixths refuse to line up. Subtracting fractions runs on one rule, same-size pieces, and this guide shows how to subtract fractions with like and unlike denominators, how to subtract fractions with whole numbers and how to subtract mixed fractions, with every step worked out.

What does it mean to subtract fractions?

Subtracting fractions means taking some pieces away from pieces of the same size, all cut from the same whole. When the pieces already match, you subtract the counts: 5/8 − 2/8 = 3/8, because five eighths take away two eighths leaves three eighths.

The denominator names the size of the pieces and the numerator counts them. Taking pieces away changes the count, so the numerator changes and the denominator stays the same.

Common Core standard 4.NF.B.3a puts it in one line: students "understand addition and subtraction of fractions as joining and separating parts referring to the same whole." Fourth graders work with like denominators, and fifth graders move to unlike ones.

Why the pieces must be the same size

Quarters and sixths cannot be subtracted directly, because a quarter is bigger than a sixth and "3 quarters minus 1 sixth" has no single unit to count in. The fix is to cut both into a smaller piece that fits each of them evenly.

Twelfths work for both: each quarter splits into 3 twelfths and each sixth splits into 2 twelfths. So 3/4 becomes 9/12, 1/6 becomes 2/12, and 9/12 − 2/12 = 7/12 of the pie is left.

Three fraction bars: 3/4 re-cut into twelfths equals 9/12, 1/6 re-cut into twelfths equals 2/12, and a bar of 12 pieces with 9 shaded and 2 crossed out leaves 7/12, so 3/4 − 1/6 = 9/12 − 2/12 = 7/12.

How to subtract fractions, step by step

Subtracting fractions takes five steps, and steps 2 and 3 are only needed when the denominators differ. OpenStax Prealgebra gives the same order: find the LCD, convert each fraction, subtract, then simplify.

Compare the denominators. If they match, as in 5/8 − 2/8, skip to step 4.
Find the least common denominator. The least common denominator (LCD) is the least common multiple of the denominators. For 4 and 6, the multiples of 4 are 4, 8, 12, 16 and the multiples of 6 are 6, 12, 18, so the LCD is 12.
Rename each fraction. Multiply the top and the bottom by the same number: 3/4 = (3 × 3)/(4 × 3) = 9/12 and 1/6 = (1 × 2)/(6 × 2) = 2/12.
Subtract the numerators and keep the denominator. 9/12 − 2/12 = 7/12.
Simplify the answer. Divide the top and bottom by their greatest common factor; 7 and 12 share only the factor 1, so 7/12 is already in simplest form.

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Subtracting fractions with different denominators: two more examples

Fractions with different denominators often have one denominator that is a multiple of the other, and then the larger denominator is the LCD. For 7/10 − 1/2, rename 1/2 as 5/10, subtract to get 2/10, and simplify to 1/5.

When the denominators share no factor, the LCD is their product. For 2/3 − 1/4, the LCD is 3 × 4 = 12, so 8/12 − 3/12 = 5/12, and the guide to GCF and LCM shows faster ways to find the least common multiple when the numbers get bigger.

Standard 5.NF.A.1 describes this exact move: "add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions." Adding uses the same renaming, and the fractions explainer for parents walks through the adding side.

Five ways to subtract fractions, compared

Five methods cover almost every fraction subtraction a grade 4–6 student meets. All of them give the same answer, and they differ in how big the numbers get on the way.

Method	How it works	Example	Best for	Watch out for
Like denominators	Subtract the numerators, keep the denominator	5/8 − 2/8 = 3/8	Grade 4 problems with matching pieces	Works only when the pieces already match
Multiply the denominators	Use b × d as the common denominator	3/4 − 1/6 = 18/24 − 4/24 = 14/24 = 7/12	Any pair of fractions, no LCM needed	Bigger numbers and an extra simplifying step
Least common denominator	Use the LCM of the denominators	3/4 − 1/6 = 9/12 − 2/12 = 7/12	The smallest numbers and the least simplifying	You need to know your multiples
Improper fractions	Turn mixed numbers into improper fractions first	13/4 − 7/4 = 6/4 = 1 1/2	Mixed numbers, no regrouping needed	Large numerators when the whole numbers are big
Counting up on a number line	Count the distance from the smaller number to the larger	2 1/4 − 1 3/4 = 1/4 + 1/4 = 1/2	Mental math and numbers that are close together	Hard when the denominators differ

Which method to pick

The least common denominator is the best default, because it keeps the numbers small and the answer often needs no simplifying. Multiplying the denominators is a safe backup when the LCM is not obvious, as long as you simplify at the end.

For mixed numbers, choose between regrouping and improper fractions by size. Small whole numbers suit improper fractions, while 12 1/5 − 9 3/5 is easier with regrouping (11 6/5 − 9 3/5 = 2 3/5) than as 61/5 − 48/5.

How to subtract mixed numbers and whole numbers

Mixed numbers are subtracted in two parts: the fractions first, then the whole numbers. For 4 3/4 − 1 1/4, the fractions give 3/4 − 1/4 = 2/4 and the wholes give 4 − 1 = 3, so the answer is 3 2/4, or 3 1/2.

Regrouping is needed when the first fraction is smaller than the second. In 3 1/4 − 1 3/4 you cannot take 3/4 from 1/4, so take one whole from the 3 and write it as 4/4, and 3 1/4 becomes 2 + 4/4 + 1/4 = 2 5/4.

Then subtract as usual: 2 5/4 − 1 3/4 = 1 2/4, and 2/4 simplifies to 1/2, so the answer is 1 1/2. The improper-fraction route agrees: 13/4 − 7/4 = 6/4 = 1 1/2.

Circles showing 3 1/4, then one whole circle re-cut into 4/4 to make 2 5/4, with one whole and three quarters crossed out, leaving 1 2/4, so 2 5/4 − 1 3/4 = 1 2/4 = 1 1/2.

Fraction regrouping follows the same idea as subtraction with regrouping in whole numbers, with one change. In place value you trade 1 ten for 10 ones; with fractions you trade 1 whole for as many pieces as the denominator says, so 4/4, 3/3 or 8/8.

Subtracting a fraction from a whole number

A fraction is subtracted from a whole number by turning one of the wholes into a fraction with the same denominator. For 5 − 2/3, write 5 as 4 3/3, then 4 3/3 − 2/3 = 4 1/3.

Mixed numbers with unlike denominators combine both moves. For 4 1/3 − 1 1/2, rename to sixths as 4 2/6 − 1 3/6; since 2/6 is smaller than 3/6, regroup to 3 8/6 − 1 3/6 = 2 5/6.

Standard 4.NF.B.3c asks fourth graders to "add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction." England's national curriculum reaches the full skill in Year 6: "add and subtract fractions with different denominators and mixed numbers, using the concept of equivalent fractions."

How to check progress with fraction subtraction

Five short problems show whether subtracting fractions has landed, one for each skill in this guide.

Like denominators: 7/9 − 4/9. The answer is 3/9, which simplifies to 1/3.
One denominator a multiple of the other: 5/6 − 1/3. Rename 1/3 as 2/6; 5/6 − 2/6 = 3/6 = 1/2.
Unlike denominators: 3/5 − 1/4. The LCD is 20, so 12/20 − 5/20 = 7/20.
A whole number: 6 − 3/5. Write 6 as 5 5/5; the answer is 5 2/5.
Regrouping: 2 1/3 − 1 2/3. Regroup to 1 4/3 − 1 2/3 = 2/3.

Ask for the working each time. A right answer with no renamed fractions on the page can be a lucky guess, and the next problem with different denominators will show it.

Check every answer by adding back

Subtraction is checked by addition: the answer plus the number you took away must equal the number you started with. For 3/4 − 1/6 = 7/12, add 7/12 + 2/12 = 9/12, and 9/12 simplifies back to 3/4.

An estimate catches the big errors before the check. Standard 5.NF.A.2 asks students to "use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers." For 3/4 − 1/6, taking a small piece from 3/4 should land between 1/2 and 3/4, and 7/12 does, because 6/12 < 7/12 < 9/12.

The What Works Clearinghouse practice guide on fractions recommends that teachers "help students understand why procedures for computations with fractions make sense." Adding back and estimating are two quick ways to show a child that the procedure gives a sensible answer.

Common mistakes when subtracting fractions

Four mistakes cause most wrong answers in fraction subtraction, and each has a short fix.

Subtracting the denominators: 3/4 − 1/6 written as (3 − 1)/(4 − 6) gives 2/−2, and 5/8 − 2/8 would give 3/0. Keep the denominator, because it names the piece size, and subtract only the numerators.
Renaming only the denominator: Writing 3/4 as 3/12 changes the amount from three quarters to one quarter. Multiply the numerator by the same number as the denominator: 3/4 = 9/12.
Borrowing 10 instead of one whole: In 3 1/4 − 1 3/4, writing 2 11/4 treats the fraction like a ones column. One whole is 4/4, so 3 1/4 becomes 2 5/4.
Swapping the fractions: In 3 1/4 − 1 3/4, flipping the fraction part to 3/4 − 1/4 gives 2 2/4, which is 1 too much. Regroup instead of swapping.

Edge case: when the second fraction is bigger

A subtraction such as 1/4 − 3/4 takes away more than you have, and the answer is a negative number: 1/4 − 3/4 = −2/4 = −1/2. Grades 4 and 5 keep the first number larger, and negative fractions arrive in grade 7.

In a word problem, a negative answer usually means the numbers were placed in the wrong order. Reread which amount is the starting amount and put it first; the overview of what math each grade should know shows where fractions sit in each school year.

Your fraction subtraction checklist and next steps

Match the pieces: check the denominators before you subtract anything.
Find the LCD: use the least common multiple of the denominators, or multiply them as a backup.
Rename both parts: multiply the numerator and the denominator by the same number.
Subtract the tops only: keep the denominator.
Regroup with wholes: trade 1 whole for 4/4, 3/3 or 8/8, never for 10.
Simplify and check: reduce the answer, then add it back to the number you took away.

Fraction subtraction rests on multiples, so fast recall of multiplication facts makes every common denominator easier to find. The subtraction worksheet for 5th graders gives printable practice with an answer key, and the math game for 5th graders page collects grade-level practice. When your child is ready for the next operation, the guide on how to divide fractions continues the series.

MathIt is a free app for iOS and Android with about 90 game types, adaptive difficulty and daily practice in 33 languages. Two to five players can also race each other in real-time battles, and 22 of the games play free in the browser with no sign-up.

Free printable worksheetSubtraction — 5th GradersWorksheet · PDF