How to Add, Subtract, Multiply and Divide Fractions

Fraction arithmetic comes down to four moves, each with one rule, and most wrong answers come from using the rule for one operation on another. Adding needs a common denominator; multiplying doesn't. Dividing turns into multiplying. Mixed numbers become improper fractions first.

This guide works one example of each, with every step written out. Each example links to the calculator with the same numbers already filled in, so you can check the working line by line or change the numbers to your own homework.

The parts of a fraction

In 5/6, the bottom number (the denominator, 6) says how many equal parts the whole is cut into. The top number (the numerator, 5) says how many of those parts you have.

Two facts do most of the work in what follows:

• Multiplying or dividing the top and bottom by the same number doesn't change the value. 3/4, 6/8 and 9/12 are the same amount, cut into more pieces.
• You can only add or subtract parts of the same size. Sixths and eighths have to be turned into the same kind of piece first.

Simplify: divide by the greatest common factor

A fraction is in lowest terms when the top and bottom share no factor except 1. To get there in one step, divide both by their greatest common factor (GCF).

Example: simplify 84/126.

To find the GCF of two numbers, the method in Euclid's Elements is still the quickest: divide the bigger by the smaller and keep the remainder, then repeat with the smaller number and the remainder until the remainder is 0.

126 = 1 × 84 + 42
 84 = 2 × 42 + 0

The last non-zero remainder, 42, is the GCF. Divide both parts by it:

$$\frac{84}{126} = \frac{84 \div 42}{126 \div 42} = \frac{2}{3}$$

You could also simplify in stages (÷2, then ÷3, then ÷7), but it's easy to stop too early and leave something like 14/21. Simplify 84/126 in the calculator.

Add and subtract: find the lowest common denominator

To add fractions with different denominators, rewrite them over a common denominator, then add the numerators. The smallest one that works is the lowest common denominator (LCD), the least common multiple of the denominators.

Example: 5/6 + 3/8.

1. Find the LCD of 6 and 8. Their multiples are 6, 12, 18, 24 and 8, 16, 24, so the LCD is 24. (Or by prime factors: 6 = 2 × 3 and 8 = 2³, so the LCD is 2³ × 3 = 24.)
2. Rewrite each fraction over 24. 24 ÷ 6 = 4, so multiply 5/6 by 4/4. 24 ÷ 8 = 3, so multiply 3/8 by 3/3.
3. Add the numerators and keep the denominator.
4. Simplify, and turn it into a mixed number if you want one.

$$\frac{5}{6} + \frac{3}{8} = \frac{20}{24} + \frac{9}{24} = \frac{29}{24} = 1\tfrac{5}{24}$$

Multiplying the two denominators (6 × 8 = 48) also gives a common denominator: 40/48 + 18/48 = 58/48. It's the same amount, but you then have to simplify 58/48 down to 29/24. The LCD saves that last step and keeps the numbers small.

This is the method the US Common Core standards set for grade 5: replace the fractions with equivalent fractions that have the same denominator, then add. Add 5/6 + 3/8.

Subtracting works the same way. For 2⅓ − 5/6, first turn the mixed number into an improper fraction (more on that below): 2⅓ = 7/3. The LCD of 3 and 6 is 6:

$$\frac{7}{3} - \frac{5}{6} = \frac{14}{6} - \frac{5}{6} = \frac{9}{6} = \frac{3}{2} = 1\tfrac{1}{2}$$

Subtract 2⅓ − 5/6.

The classic mistake is adding the tops and the bottoms: 1/2 + 1/3 = 2/5. A quick check shows it can't be right. 2/5 is less than 1/2, and adding 1/3 to a half can't make it smaller. The real answer is 3/6 + 2/6 = 5/6.

Multiply: tops times tops, bottoms times bottoms

Multiplying needs no common denominator. Multiply the numerators, multiply the denominators, then simplify.

Example: 4/9 × 3/8.

$$\frac{4}{9} \times \frac{3}{8} = \frac{4 \times 3}{9 \times 8} = \frac{12}{72} = \frac{1}{6}$$

To keep the numbers small, cancel before you multiply. Any top can cancel with any bottom: 4 and 8 share a factor of 4, and 3 and 9 share a factor of 3.

$$\frac{4}{9} \times \frac{3}{8} = \frac{4 \div 4}{9 \div 3} \times \frac{3 \div 3}{8 \div 4} = \frac{1}{3} \times \frac{1}{2} = \frac{1}{6}$$

Same answer, and nothing left to simplify at the end. It helps to read "×" as "of": 4/9 × 3/8 is "three-eighths of four-ninths". Multiply 4/9 × 3/8.

Divide: keep, change, flip

To divide by a fraction, multiply by its reciprocal (the fraction turned upside down). The rhyme is keep, change, flip: keep the first fraction, change ÷ to ×, flip the second.

Example: 2/3 ÷ 3/4.

$$\frac{2}{3} \div \frac{3}{4} = \frac{2}{3} \times \frac{4}{3} = \frac{8}{9}$$

Why flipping works. Division asks "how many of the second fit into the first?", and you can check an answer by multiplying back. The Common Core's grade 6 standard uses this very example: 2/3 ÷ 3/4 = 8/9 because 3/4 of 8/9 is 2/3. Check: 3/4 × 8/9 = 24/36 = 2/3. ✓

Another way to see it: put both over a common denominator, and dividing the fractions becomes dividing the numerators. 2/3 = 8/12 and 3/4 = 9/12, and 8 twelfths ÷ 9 twelfths = 8/9, the same answer.

A sense check: 3/4 is bigger than 2/3, so it fits in less than once, and 8/9 is just under 1. When you divide by a fraction less than 1, the answer is bigger than what you started with. Dividing by 1/2 doubles a number. Divide 2/3 ÷ 3/4.

Mixed numbers: convert first

A mixed number such as 2¼ is a whole number plus a fraction. Before multiplying or dividing, turn it into an improper fraction: multiply the whole number by the denominator, add the numerator, and keep the denominator.

$$2\tfrac{1}{4} = \frac{2 \times 4 + 1}{4} = \frac{9}{4}$$

Example: 2¼ × 1⅔.

$$2\tfrac{1}{4} \times 1\tfrac{2}{3} = \frac{9}{4} \times \frac{5}{3} = \frac{45}{12} = \frac{15}{4} = 3\tfrac{3}{4}$$

The tempting shortcut, multiplying the whole numbers and the fractions separately, gives 2 × 1 + 1/4 × 2/3 = 2⅙, which is wrong. It misses the cross terms: 2 × 2/3 and 1/4 × 1. Multiply 2¼ × 1⅔.

To turn an improper fraction back into a mixed number, divide: 29 ÷ 24 = 1 remainder 5, so 29/24 = 1 5/24. Convert 29/24 to a mixed number.

The four rules on one page

Operation	Common denominator?	Rule
Add, subtract	Yes, the LCD	Rewrite over the LCD, then add or subtract the tops
Multiply	No	Tops × tops, bottoms × bottoms (cancel first)
Divide	No	Keep, change, flip, then multiply
Simplify	n/a	Divide top and bottom by the GCF

And before any of them, turn mixed numbers into improper fractions.

Try it

The fraction calculator adds, subtracts, multiplies and divides exactly, with the LCD, the cancelling and every line of working shown, and a bar diagram of the answer. Type fractions (3/4), mixed numbers (2 1/4) or decimals. Its sub-pages cover the rest of this guide:

• Simplify fractions
• Mixed numbers and improper fractions
• Fraction to decimal, with repeating decimals marked

Sources

• Common Core State Standards for Mathematics, Grade 5, Number and Operations—Fractions: 5.NF.A.1 (adding and subtracting with unlike denominators) and 5.NF.B.4 (multiplying fractions). thecorestandards.org
• Common Core State Standards for Mathematics, Grade 6, The Number System: 6.NS.A.1 (dividing fractions by fractions, including 2/3 ÷ 3/4 = 8/9). thecorestandards.org
• Euclid, Elements, Book VII, Proposition 2: finding the greatest common measure of two numbers (D. E. Joyce's edition, Clark University). clarku.edu