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Fractions

How to share a pizza fairly, and why the rules actually work.

You use fractions more than you think: half an hour, a quarter tank of gas, three-fourths of a cup of flour. This page shows what fractions mean, how to rename them, compare them, and do all four operations with them.

Along the way you’ll see why each rule works, meet a 3,500-year-old math scroll, and finish with practice problems and a game.

Jump to practicePlay Fraction Kitchen
A pizza cut into 8 equal slices with 3 slices taken, labeled 3 over 8: the 3 is the numerator, the slices taken, and the 8 is the denominator, the equal slices in the whole pizza 38 slices eatenslices in all

What is a fraction?

A fraction is a number that names part of a whole, or part of a group. It’s written as one number over another, like 3/8.

The bottom number is the denominator. It tells you how many equal parts the whole is cut into. The top number is the numerator. It tells you how many of those parts you have.

So 3/8 of a pizza means: cut the pizza into 8 equal slices and take 3. The word “equal” matters. If the slices are different sizes, they aren’t eighths.

A fraction is also a division. 3/8 means 3 ÷ 8, which is 0.375. That’s why you can never have 0 as a denominator: you can’t split something into zero parts.

A fraction bar split into 5 equal parts with 2 shaded, labeled two fifths 25 2 shaded parts out of 5 equal parts
A fraction bar for 2/5. The whole bar is 1.

Did you know?

The word “fraction” comes from the Latin fractio, “a breaking.” A fraction is literally a broken-off piece of a whole.

Equivalent fractions and simplifying

Equivalent fractions are different-looking fractions that name the same amount. Look at the bars below: 1/3 and 2/6 cover exactly the same length.

To make an equivalent fraction, multiply (or divide) the top and bottom by the same number: 1/3 = (1×2)/(3×2) = 2/6.

Three bars of equal length: one third, two sixths, and four twelfths, with the shaded part the same length in each 1/32/64/12
1/3 = 2/6 = 4/12. More, smaller pieces; same amount.

Simplifying (also called reducing, or writing in lowest terms) goes the other way. You divide the top and bottom by the greatest common factor, or GCF: the biggest whole number that divides evenly into both.

Example 1: Simplify 12/18

  1. Factors of 12: 1, 2, 3, 4, 6, 12. Factors of 18: 1, 2, 3, 6, 9, 18.
  2. The biggest one they share is 6, so the GCF is 6.
  3. 12 ÷ 6 = 2 and 18 ÷ 6 = 3, so 12/18 = 2/3.

Check: 2/3 can’t be simplified again, because 2 and 3 share no factor except 1.

Comparing fractions

Which is bigger, 3/4 or 5/7? It’s hard to tell when the pieces are different sizes. Here’s how to decide.

  • Same denominator: just compare the tops. 5/9 > 4/9, because 5 pieces beat 4 pieces of the same size.
  • Same numerator: the smaller denominator wins. 3/5 > 3/8, because fifths are bigger pieces than eighths.
  • Neither matches: rename both with a common denominator, or use the cross-multiply shortcut.

Example 2: Compare 3/4 and 5/7

  1. A common denominator is 4 × 7 = 28.
  2. 3/4 = 21/28 and 5/7 = 20/28.
  3. 21/28 > 20/28, so 3/4 > 5/7. Just barely!

Shortcut: the cross products are 3 × 7 = 21 and 5 × 4 = 20. Those are the same new numerators, which is why the shortcut works.

A number line from 0 to 1 with 5/7 marked at about 0.71 and 3/4 marked at 0.75, slightly to its right 01/21 5/73/4
On a number line, the bigger fraction sits farther right.

Adding and subtracting fractions

You can only add pieces that are the same size. 1 third plus 1 fourth isn’t “2 sevenths,” any more than 1 dollar plus 1 quarter is 2 dollars.

So first rename the fractions so they have a common denominator. The best choice is the least common denominator (LCD): the smallest number both denominators divide into. Then add or subtract the numerators and keep the denominator.

One half and one third shown as bars, then cut into sixths: three sixths plus two sixths makes five sixths += 5/61/2 = 3/61/3 = 2/6
Dashed lines cut both bars into sixths, so the pieces match: 3/6 + 2/6 = 5/6.

Example 3: 2/3 + 1/4

  1. Multiples of 3: 3, 6, 9, 12. Multiples of 4: 4, 8, 12. The LCD is 12.
  2. 2/3 = 8/12 (times 4 over 4). 1/4 = 3/12 (times 3 over 3).
  3. 8/12 + 3/12 = 11/12.

Example 4: 5/6 − 1/4

  1. The LCD of 6 and 4 is 12.
  2. 5/6 = 10/12 and 1/4 = 3/12.
  3. 10/12 − 3/12 = 7/12. Already in lowest terms.

Example 5: Pizza night

You eat 3/8 of a pizza and your friend eats 1/4. How much is gone? 1/4 = 2/8, so 3/8 + 2/8 = 5/8 of the pizza. That leaves 3/8.

Multiplying and dividing fractions

Multiplying

Good news: multiplying is the easy one. No common denominator needed. Multiply the tops, multiply the bottoms, then simplify.

The word “of” is your clue. “Half of 3/4 of a cup” means 1/2 × 3/4.

An area model: a square cut into 4 columns with 3 shaded for three fourths, and into 2 rows with 1 row marked for one half; the overlap is 3 of 8 small boxes, showing one half times three fourths equals three eighths columns: 3/4 shadedrows: 1/2 markedoverlap: 3 of 8 boxes1/2 × 3/4 = 3/8
An area model shows why you multiply both tops and both bottoms.

Example 6: 3/4 × 2/5

3 × 2 = 6 and 4 × 5 = 20, so 3/4 × 2/5 = 6/20 = 3/10 (dividing top and bottom by 2).

Example 7: Half a recipe

A recipe needs 3/4 cup of milk and you’re making half. 1/2 × 3/4 = 3/8 cup, just like the picture above.

Dividing

To divide by a fraction, multiply by its reciprocal. The reciprocal is the fraction flipped upside down: the reciprocal of 4/5 is 5/4. A number times its reciprocal always equals 1.

A memory trick some people use is “keep, change, flip”: keep the first fraction, change ÷ to ×, flip the second.

Example 8: 2/3 ÷ 4/5

  1. Flip 4/5 to get 5/4, and multiply: 2/3 × 5/4.
  2. 2 × 5 = 10 and 3 × 4 = 12, so the answer is 10/12 = 5/6.

Example 9: How many quarter-cups?

You have 3 cups of rice and a 1/4-cup scoop. 3 ÷ 1/4 = 3 × 4/1 = 12 scoops. Makes sense: each cup holds 4 quarter-cups.

Mixed numbers and improper fractions

An improper fraction has a numerator that is equal to or bigger than its denominator, like 11/4. A mixed number writes the same amount as a whole number plus a fraction, like 2 3/4.

Mixed numbers are easier to picture. Improper fractions are easier to multiply and divide. You’ll switch between them a lot.

Three circles cut into fourths: two whole circles fully shaded and a third circle with three of four parts shaded, showing 2 and 3 fourths equals 11 fourths 2 3/4= 11/4
Count the shaded quarters: 4 + 4 + 3 = 11 fourths.

Example 10: Converting both ways

  • Mixed to improper: 2 3/4 → 2 × 4 + 3 = 11, so 11/4.
  • Improper to mixed: 17/5 → 17 ÷ 5 = 3 remainder 2, so 3 2/5.

How do we know the rules work?

Rules are easier to remember when you know where they come from. Here’s the reasoning behind each one.

Why equivalent fractions work

Multiplying top and bottom by 2 is the same as multiplying by 2/2, and 2/2 = 1. Multiplying any number by 1 doesn’t change it. You’ve only cut each piece into 2 smaller pieces, so you have twice as many pieces, each half the size.

Why you need a common denominator

The denominator is like a unit, the way “inches” or “dollars” are. You can add 3 inches and 2 inches, but not 3 inches and 2 feet until you convert. Fourths and thirds are different units, so you convert both to twelfths first.

Why “flip and multiply” works

Division asks “how many of these fit in that?” How many halves fit in 3? Six, which is 3 × 2. Dividing by 1/2 gives the same answer as multiplying by 2/1. The same idea works for any fraction: dividing by a/b is the same as multiplying by b/a, because (a/b) × (b/a) = 1.

You can check any division by multiplying back: 2/3 ÷ 4/5 = 5/6, and 5/6 × 4/5 = 20/30 = 2/3. ✓

History: fractions on a 3,500-year-old scroll

One of the oldest math books in the world is the Rhind Mathematical Papyrus, an Egyptian scroll from about 1550 BCE. A scribe named Ahmes copied it from an even older text. Today most of it is in the British Museum in London.

Egyptian scribes wrote fractions in a surprising way. Apart from a special sign for 2/3, they used only unit fractions: fractions with 1 on top, like 1/2, 1/10, or 1/30. Any other amount was written as a sum of different unit fractions.

The scroll opens with a table of fractions with 2 on top. For example, it gives 2/15 as 1/10 + 1/30. (Check it: 3/30 + 1/30 = 4/30 = 2/15.)

Some of its first problems are about sharing bread among 10 people. For 6 loaves, each person gets 1/2 + 1/10 of a loaf. That equals 5/10 + 1/10 = 6/10, exactly right, and it’s easy to cut!

Six loaves drawn as bars: five cut in half and one cut into tenths, so each of 10 people gets one half plus one tenth of a loaf.
Sharing 6 loaves among 10 people the Egyptian way (an original diagram, not from the papyrus itself).

Did you know?

Fractions written as sums of different unit fractions are still called Egyptian fractions today, and mathematicians are still proving new things about them.

Practice problems

Try each one on paper first, then open the answer to check your work.

1. Simplify 15/25.

The GCF of 15 and 25 is 5. 15 ÷ 5 = 3 and 25 ÷ 5 = 5, so 3/5.

2. Which is bigger, 5/8 or 3/5?

Cross products: 5 × 5 = 25 and 3 × 8 = 24. Since 25 > 24, 5/8 is bigger (25/40 vs 24/40).

3. 2/5 + 1/3

LCD 15: 6/15 + 5/15 = 11/15.

4. 7/8 − 1/6

LCD 24: 21/24 − 4/24 = 17/24.

5. 4/9 × 3/8

12/72, and the GCF is 12, so 1/6.

6. 3/5 ÷ 9/10

3/5 × 10/9 = 30/45 = 2/3.

7. Write 3 1/3 as an improper fraction.

3 × 3 + 1 = 10, so 10/3.

8. 1 1/2 + 2 2/3

3/2 + 8/3 = 9/6 + 16/6 = 25/6 = 4 1/6.

Game: Fraction Kitchen

Ten mixed problems: simplify, add, subtract, multiply and divide. Type answers like 3/4, 2, or 1 1/2 and press Enter. Any equal value counts, but lowest terms is best. One point per correct answer. Your best score is saved only on this device. About 3 minutes.

Talk About It

  • Would you rather have 1/3 of a large pizza or 1/2 of a small one? What would you need to know to decide?
  • Why do you think Egyptian scribes liked unit fractions? What makes 1/2 + 1/10 easier to cut than 6/10?
  • Where did you see a fraction this week outside of math class?

Sources

Credits: All diagrams on this page are original SVG drawings made for Jugami. The bread-sharing diagram illustrates the papyrus’s method; it is not a copy of the papyrus.