Number: Fractions, Decimals & Percentages

A fraction represents a part of a whole. It has two parts: the numerator (top number) tells you how many parts you have, and the denominator (bottom number) tells you how many equal parts the whole is split into. So 3/4 means the whole is divided into 4 equal parts and you have 3 of them. Fractions can be proper (numerator < denominator, like 3/4), improper (numerator > denominator, like 7/4), or mixed numbers (like 1 and 3/4). You should be able to convert freely between these forms.

To convert a fraction to a decimal, divide the numerator by the denominator. For example, 3/4 = 3 ÷ 4 = 0.75. Some fractions give terminating decimals (they stop, like 1/4 = 0.25) and some give recurring decimals (they repeat forever, like 1/3 = 0.333..., written as 0.3 with a dot over the 3). To convert a terminating decimal back to a fraction, write it over the correct power of 10. For example, 0.35 = 35/100 = 7/20. Always simplify by dividing by the highest common factor (HCF).

Percent means 'out of 100'. So 35% means 35 out of 100, which is 35/100 = 7/20 as a fraction and 0.35 as a decimal. To convert a percentage to a decimal, divide by 100 (move the decimal point 2 places left). To convert back, multiply by 100. Knowing this triangle - fraction ↔ decimal ↔ percentage - is essential. Examiners love questions that start with one form and ask for another.

Three types appear in every exam. First: find a percentage of an amount. Example: 35% of £80 = 0.35 × 80 = £28. Second: percentage increase or decrease. A coat costs £120 and rises by 15%. New price = 120 × 1.15 = £138. Multiplying by 1.15 is the same as adding 15%. For a 20% decrease, multiply by 0.80. Third: reverse percentage. A price after a 20% increase is £96. Original = 96 ÷ 1.20 = £80. Never just subtract the percentage from the final price - that is a very common error.

Adding and subtracting fractions: find a common denominator first. For example, 1/3 + 1/4: the common denominator is 12. So 4/12 + 3/12 = 7/12. Multiplying fractions: multiply numerators together and denominators together. So 2/3 × 3/5 = 6/15 = 2/5. Dividing fractions: keep the first fraction, change the division to multiplication, and flip the second fraction. So 2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6. Remember: dividing by a fraction smaller than 1 makes the answer bigger - this trips many students up.