Fractions: The Four Operations Made Simple

What is a fraction?

A fraction is a part of a whole. The whole must be cut into equal pieces.

• The bottom number is the denominator. It says how many equal pieces make one whole.
• The top number is the numerator. It says how many pieces we take.

In 3/4, the whole has 4 pieces and we take 3.

A proper fraction is less than 1 (like 3/4). An improper fraction is 1 or more (like 7/4). A mixed number writes it as wholes plus a part: 7/4 = 1 3/4.

Equivalent fractions and simplest form

Multiply or divide the top and bottom by the same number and the amount does not change: 3/4 = 6/8 = 9/12.

To simplify, divide top and bottom by their highest common factor: 18/24 → divide by 6 → 3/4.

Adding, subtracting, multiplying and dividing fractions

Add and subtract

1. Make the denominators the same (use the lowest common multiple).
2. Add or subtract the numerators. Keep the denominator.
3. Simplify.

Example: 2/3 − 1/4 = 8/12 − 3/12 = 5/12.

Multiply

Multiply the tops, multiply the bottoms: 2/5 × 3/7 = 6/35. Turn mixed numbers into improper fractions first.

Divide

Keep the first fraction, Change ÷ to ×, Flip the second (its reciprocal): 3/5 ÷ 2/3 = 3/5 × 3/2 = 9/10.

Why does flipping work? Dividing by 1/4 asks how many quarters fit. There are 4 quarters in every whole, so it is the same as multiplying by 4.

Decimals and negative numbers

Decimals

A decimal is a fraction with a bottom of 10, 100, 1000…: 0.75 = 75/100 = 3/4.

• Add or subtract: line up the decimal points.
• Multiply: ignore the points, multiply, then put back as many decimal places as the two numbers had together. 0.3 × 0.4 = 0.12.
• Divide: move both points the same number of places until you divide by a whole number. 1.2 ÷ 0.3 = 12 ÷ 3 = 4.

Negative numbers

• Adding a negative is like subtracting: 5 + (−8) = −3.
• Subtracting a negative is like adding: 4 − (−6) = 10.
• Multiply or divide: same signs give a positive answer, different signs give a negative answer. (−3) × (−4) = 12; (−12) ÷ 3 = −4.

The same rules work with negative fractions: −1/2 × 3/5 = −3/10.

Order of operations, inverse operations and calculators

Order of operations (BIDMAS / BODMAS)

Do the work in this order: Brackets, Indices (powers and roots), Division and Multiplication (left to right), Addition and Subtraction (left to right).

Example: 3 + 4 × 2² = 3 + 4 × 4 = 3 + 16 = 19 (not 196).

Inverse operations

An inverse operation undoes another one: + undoes −, × undoes ÷, squaring undoes square rooting. Use it to check: if 3/5 ÷ 2/3 = 9/10, then 9/10 × 2/3 should give back 3/5. It does: 18/30 = 3/5.

Using a calculator well

• Estimate first, so you can spot a silly answer.
• Use brackets on the calculator for fraction lines: (3 + 5) ÷ (2 × 4), not 3 + 5 ÷ 2 × 4.
• Use the fraction key (a b/c) and the S⇔D key to switch between fraction and decimal.
• Use the (−) key for negatives, and round only at the end.

Key formulas and definitions

• a/b + c/b = (a + c)/b (same denominator)
• a/b + c/d = (ad + bc)/bd, then simplify
• a/b × c/d = ac/bd
• a/b ÷ c/d = a/b × d/c (keep, change, flip)
• Equivalent: a/b = (a×k)/(b×k)
• Same signs → positive; different signs → negative (× and ÷)
• BIDMAS: Brackets, Indices, Division/Multiplication, Addition/Subtraction

Worked examples

Common mistakes

• Adding the bottoms: 1/2 + 1/3 is NOT 2/5. Make the denominators the same first (3/6 + 2/6 = 5/6).
• Flipping the wrong fraction when dividing. Keep the FIRST, flip the SECOND.
• Working left to right and ignoring BIDMAS: 2 + 3 × 4 is 14, not 20.
• Thinking two negatives always make a positive. That rule is for × and ÷; −3 − 4 is −7.