Fractions explained, from pizza to percentages

A fraction is a way of writing a part of a whole. The bottom number (the denominator) says how many equal parts the whole is divided into. The top number (the numerator) says how many of those parts you have.

In 3/4, the whole is divided into four equal parts, and you have three of them.

That is the entire idea. Most fraction confusion comes from applying the wrong rule at the wrong time.

Equivalent fractions

Multiplying or dividing the numerator and denominator by the same number does not change the value.

• 1/2 = 2/4 = 3/6 = 50/100.

This is why simplifying works. Divide both parts by a common factor:

• 8/12 = (8 ÷ 4) / (12 ÷ 4) = 2/3.

A fraction is fully simplified when the numerator and denominator share no common factor other than 1.

Adding and subtracting

You can only add or subtract fractions when the denominators are the same. The denominator is the size of the pieces; you cannot add thirds and quarters directly.

1/3 + 1/4

Find a common denominator. The least common multiple of 3 and 4 is 12.

• 1/3 = 4/12
• 1/4 = 3/12
• 4/12 + 3/12 = 7/12

A common mistake is adding the tops and the bottoms: 1/3 + 1/4 ≠ 2/7. That would be adding the piece sizes, which makes no sense.

Multiplying

To multiply fractions, multiply straight across:

• 2/3 × 4/5 = (2 × 4) / (3 × 5) = 8/15.

You can simplify before multiplying to keep the numbers small:

• 2/3 × 3/4 = (2 × 3) / (3 × 4) = 6/12 = 1/2.

Notice that multiplying by a fraction less than 1 makes the result smaller. That is normal.

Dividing

To divide by a fraction, multiply by its reciprocal — flip the second fraction.

• (1/2) ÷ (3/4) = 1/2 × 4/3 = 4/6 = 2/3.

Why does flipping work? Dividing by 3/4 is the same as asking how many three-quarter pieces fit into one half. The reciprocal turns that into a multiplication.

Mixed numbers

A mixed number combines a whole number and a fraction, such as 2 1/2. To calculate, convert it to an improper fraction:

• 2 1/2 = (2 × 2 + 1) / 2 = 5/2.

Then follow the normal rules. Convert back at the end if needed.

Fractions, decimals and percentages

These are three ways to write the same relationship.

• 1/2 = 0.5 = 50%
• 1/4 = 0.25 = 25%
• 3/4 = 0.75 = 75%
• 1/5 = 0.2 = 20%

To convert a fraction to a decimal, divide the numerator by the denominator. To convert a decimal to a percentage, multiply by 100.

For percentage calculations, the percentage calculator handles the arithmetic.

Real-life uses

• Cooking: 3/4 cup plus 1/2 cup is 1 1/4 cups.
• Discounts: 1/4 off is 25% off.
• Time: 3/4 of an hour is 45 minutes.
• DIY: a 2 1/2-inch cut is 2.5 inches.
• Sharing: splitting a bill equally uses division.

Common mistakes

1. Adding denominators. Only add the tops once the bottoms match.
2. Forgetting to simplify. The answer is still correct, but simpler is easier to compare.
3. Multiplying without checking the question. “1/2 of 3/4” is multiplication; “1/2 plus 3/4” is addition.
4. Mixing units. 1/2 hour + 30 minutes is not 1/2 + 30. Convert to the same unit first.
5. Assuming a bigger denominator means a bigger fraction. 1/8 is smaller than 1/4.

The short version

Find a common denominator to add or subtract. Multiply straight across. Flip the second fraction to divide. Simplify when it helps. And always check whether the answer makes sense in the real situation.

For comparison and scaling, read ratios and proportions.