One-page revision sheet

• The denominator says how many equal parts make the whole; the numerator says how many you have.
• Multiply or divide both parts by the same number to get an equivalent fraction. Adding does not work.
• To simplify, divide both parts by their HCF — one step, not several.
• To compare, rewrite both over a common denominator, then compare the numerators.
• The lowest common denominator is the LCM of the two denominators.
• An improper fraction is at least 1; a mixed number shows the whole part separately.
• 7/4 → 7 ÷ 4 = 1 remainder 3 → 1 3/4. Going back: 1 × 4 + 3 = 7.
• Never compare numerators until the denominators match — that is the commonest mistake in this unit.

Key words: numerator  ·  denominator  ·  equivalent fractions  ·  simplest form  ·  improper fraction  ·  mixed number  ·  common denominator

• You can only add or subtract fractions when the denominators match — the parts must be the same size.
• Find the lowest common denominator: the LCM of the two denominators.
• Rewrite each fraction over that denominator, then add or subtract the numerators only.
• The denominator does not change when you add. 2/7 + 3/7 = 5/7, not 5/14.
• Simplify the answer, and write an improper answer as a mixed number if the question is in mixed numbers.
• For mixed numbers, either add the parts separately or convert to improper fractions first.
• For subtracting mixed numbers, converting to improper fractions avoids borrowing — it is the safer route.
• Estimate first: 1/2 + 1/3 must be more than 1/2, so an answer of 2/5 is wrong before you check anything else.

Key words: common denominator  ·  lowest common denominator  ·  equivalent fraction  ·  mixed number  ·  simplify

• To multiply fractions: multiply the numerators, multiply the denominators.
• No common denominator is needed — that step belongs to adding and subtracting.
• The word of means multiply: 1/2 of 3/4 is 1/2 × 3/4.
• Multiplying by a fraction less than 1 makes a number smaller. That is not a mistake.
• Cancel first where you can — it keeps the numbers small and the simplifying easy.
• To multiply by a whole number, divide by the denominator then multiply by the numerator: 3/4 × 8 = 8 ÷ 4 × 3 = 6.
• For mixed numbers, convert to improper fractions before multiplying.
• Check by size: 3/4 of 8 must be less than 8 but more than half of it.

Key words: product  ·  of  ·  cancel  ·  square

• The reciprocal of a fraction is that fraction upside down. The reciprocal of 3/5 is 5/3.
• A number times its reciprocal is 1.
• To divide by a fraction, multiply by its reciprocal: keep, flip, multiply.
• Dividing by a fraction less than 1 makes the answer bigger — you are asking how many small pieces fit in.
• Dividing by a whole number n is the same as multiplying by 1/n.
• A whole number is a fraction over 1, so the reciprocal of 4 is 1/4.
• For mixed numbers, convert to improper fractions before dividing.
• Check by size: 6 ÷ 1/2 must be more than 6, because halves are small.

Key words: reciprocal  ·  divide  ·  invert

• Look at a calculation before you start it — the order is your choice.
• Cancel first when multiplying, so the numbers stay small and the answer is already simplified.
• For a fraction of a quantity, divide first, then multiply: 3/4 of 48 is 48 ÷ 4 × 3, not 48 × 3 ÷ 4.
• In an addition, reorder to pair fractions that make a whole: 1/4 + 1/2 + 3/4 becomes 1 + 1/2.
• Spot reciprocals: 5/8 × 8/5 = 1, no working needed.
• Addition and multiplication may be reordered freely; subtraction and division may not.
• Estimating first tells you roughly what to expect, so a slip stands out.
• The efficient route is not a different method — it is the same method, done in a kinder order.

Key words: cancel  ·  reciprocal  ·  commutative  ·  efficient