Number
Integers, fractions, decimals, percentages, powers, roots, and standard form.
In this topic
Integers and Decimals
Integers are whole numbers — positive, negative, or zero. They form the foundation of all number work in GCSE Maths. Understanding place value is essential: each digit in a number has a value determined by its position.
When ordering decimals, compare digit by digit from left to right. For example, 0.35 is greater than 0.309 because 5 hundredths > 0 hundredths when we compare the second decimal place.
Key Points
- Place value determines the value of each digit
- Negative numbers follow rules: negative × negative = positive
- BIDMAS/BODMAS gives the order of operations
- Rounding: look at the next digit — 5 or more rounds up
Example Questions
2Put these numbers in order from smallest to largest: 0.7, 0.07, 0.71, 0.077
0.07, 0.077, 0.7, 0.71 — Compare digit by digit from the left after the decimal point.
[2 marks]
2Calculate: -3 × (-5) + 2 × (-4)
-3 × (-5) = 15, then 2 × (-4) = -8, so 15 + (-8) = 7
[2 marks]
1Round 3.0649 to 2 decimal places.
3.06 — The third decimal digit is 4, which is less than 5, so we round down.
[1 mark]
Place Value
Each position in a number represents a power of 10. Moving left, each position is worth 10 times more. Moving right past the decimal point, each position is worth 10 times less.
For example, in the number 4,572.36: - 4 is in the thousands place (4 × 1000) - 5 is in the hundreds place (5 × 100) - 7 is in the tens place (7 × 10) - 2 is in the ones place (2 × 1) - 3 is in the tenths place (3 × 0.1) - 6 is in the hundredths place (6 × 0.01)
Key Points
- Each place is 10× the place to its right
- The decimal point separates whole numbers from fractions
Example Questions
1What is the value of the digit 7 in 37,429?
7,000 (seven thousand) — the 7 is in the thousands column.
[1 mark]
Negative Numbers
Negative numbers are less than zero. They appear on the left side of a number line. Key rules:
Multiplication and Division: - Positive × Positive = Positive - Negative × Negative = Positive - Positive × Negative = Negative - Negative × Positive = Negative
Addition and Subtraction: - Adding a negative is the same as subtracting: 5 + (-3) = 5 - 3 = 2 - Subtracting a negative is the same as adding: 5 - (-3) = 5 + 3 = 8
Key Points
- Two negatives make a positive when multiplying or dividing
- Subtracting a negative = adding a positive
- On a number line, left = smaller, right = bigger
Example Questions
1The temperature at midnight was -6°C. By noon it had risen by 11°C. What was the temperature at noon?
-6 + 11 = 5°C
[1 mark]
2Calculate (-8) ÷ (-2) × (-3)
(-8) ÷ (-2) = 4, then 4 × (-3) = -12
[2 marks]
Order of Operations (BIDMAS)
BIDMAS tells you the order to perform calculations: - B: Brackets first - I: Indices (powers and roots) - D: Division } left to right - M: Multiplication } left to right - A: Addition } left to right - S: Subtraction } left to right
Division and Multiplication are equal priority — work left to right. Addition and Subtraction are equal priority — work left to right.
Key Points
- Always start with brackets
- Indices come before ×, ÷, +, -
- × and ÷ are done before + and -
- Equal priority operations: work left to right
Example Questions
2Calculate: 3 + 4 × 2 - (6 ÷ 3)
Brackets first: 6 ÷ 3 = 2. Then multiplication: 4 × 2 = 8. Then left to right: 3 + 8 - 2 = 9
[2 marks]
Fractions
A fraction represents a part of a whole. The top number (numerator) tells you how many parts you have. The bottom number (denominator) tells you how many equal parts the whole is divided into.
Equivalent fractions have the same value: 1/2 = 2/4 = 3/6. To simplify a fraction, divide both numerator and denominator by their highest common factor (HCF).
Key Points
- To add/subtract fractions, find a common denominator
- To multiply fractions, multiply numerators and denominators
- To divide by a fraction, flip it and multiply (KFC: Keep, Flip, Change)
- Mixed numbers can be converted to improper fractions
Example Questions
2Calculate 2/3 + 3/4
Common denominator = 12. So 8/12 + 9/12 = 17/12 = 1 5/12
[2 marks]
2Calculate 3/5 ÷ 2/7
Keep 3/5, flip 2/7 to 7/2, multiply: 3/5 × 7/2 = 21/10 = 2 1/10
[2 marks]
1Simplify 36/48 to its lowest terms.
HCF of 36 and 48 is 12. 36÷12 = 3, 48÷12 = 4. Answer: 3/4
[1 mark]
Adding and Subtracting Fractions
To add or subtract fractions, they must have the same denominator. Find the lowest common multiple (LCM) of the denominators.
Step 1: Find the LCM of the denominators Step 2: Convert each fraction to an equivalent fraction with the LCM as denominator Step 3: Add or subtract the numerators Step 4: Simplify if possible
Key Points
- Same denominator = just add/subtract numerators
- Different denominators = find LCM first
- Always simplify your answer
Example Questions
2Calculate 5/6 - 1/4
LCM of 6 and 4 is 12. 5/6 = 10/12, 1/4 = 3/12. 10/12 - 3/12 = 7/12
[2 marks]
Multiplying and Dividing Fractions
Multiplying fractions: Multiply numerators together, multiply denominators together, then simplify.
Dividing fractions (KFC method): - Keep the first fraction the same - Flip the second fraction (find its reciprocal) - Change the ÷ sign to ×
With mixed numbers, always convert to improper fractions first.
Key Points
- Multiply straight across: top × top, bottom × bottom
- You can cross-cancel before multiplying to keep numbers smaller
- Division = multiply by the reciprocal
Example Questions
3Calculate 2 1/3 × 1 2/5
Convert: 7/3 × 7/5 = 49/15 = 3 4/15
[3 marks]
Percentages
Percentage means 'out of 100'. To convert between fractions, decimals, and percentages: - Fraction → Decimal: divide numerator by denominator - Decimal → Percentage: multiply by 100 - Percentage → Decimal: divide by 100
Finding a percentage of an amount: 15% of 240 → 0.15 × 240 = 36
Key Points
- Percentage increase: new = original × (1 + rate/100)
- Percentage decrease: new = original × (1 - rate/100)
- Reverse percentages: to find original, divide by the multiplier
- Compound interest uses repeated percentage increase
Example Questions
2A jacket costs £85. It is reduced by 20% in a sale. What is the sale price?
£85 × 0.80 = £68
[2 marks]
2After a 15% increase, a house is worth £230,000. What was it worth before the increase?
£230,000 ÷ 1.15 = £200,000
[2 marks]
3£5000 is invested at 3% compound interest per year. What is it worth after 4 years?
£5000 × 1.03⁴ = £5000 × 1.1255... = £5627.54
[3 marks]
Percentage Change
Percentage change = (change ÷ original) × 100
This formula works for both increases and decreases. If the result is positive, it's an increase. If negative, it's a decrease.
Example: A phone was £400, now £340. Change = 400 - 340 = 60 Percentage change = (60 ÷ 400) × 100 = 15% decrease
Key Points
- Always divide by the ORIGINAL value
- State whether it's an increase or decrease
Example Questions
2A car was bought for £12,000 and sold for £9,600. Find the percentage loss.
Loss = £2,400. Percentage = (2400 ÷ 12000) × 100 = 20% loss
[2 marks]
Compound Interest and Depreciation
Compound interest means interest is added to the principal, and future interest is earned on the new total.
Formula: Final amount = P × (1 + r/100)ⁿ Where P = principal, r = rate per period, n = number of periods
For depreciation (losing value): Final amount = P × (1 - r/100)ⁿ
Key Points
- Compound = interest on interest
- Depreciation uses (1 - r/100) instead of (1 + r/100)
- n is the number of time periods, not years necessarily
Example Questions
3A car worth £18,000 depreciates by 12% each year. What is it worth after 3 years?
£18,000 × (1 - 0.12)³ = £18,000 × 0.88³ = £18,000 × 0.681472 = £12,266.50
[3 marks]
Powers and Roots
A power (index) tells you how many times to multiply a number by itself. For example, 2³ = 2 × 2 × 2 = 8.
Square root (√) is the inverse of squaring. Cube root (∛) is the inverse of cubing.
Laws of indices: - aᵐ × aⁿ = aᵐ⁺ⁿ - aᵐ ÷ aⁿ = aᵐ⁻ⁿ - (aᵐ)ⁿ = aᵐˣⁿ - a⁰ = 1 - a⁻ⁿ = 1/aⁿ - a^(1/n) = ⁿ√a
Key Points
- Anything to the power of 0 equals 1
- Negative indices mean reciprocals
- Fractional indices mean roots
- When multiplying same bases, add the powers
Example Questions
1Simplify 2³ × 2⁵
2³⁺⁵ = 2⁸ = 256
[1 mark]
2Evaluate 27^(2/3)
27^(1/3) = 3 (cube root), then 3² = 9
[2 marks]
2Simplify (3x²y)³
3³ × x²ˣ³ × y³ = 27x⁶y³
[2 marks]
Standard Form
Standard form is a way to write very large or very small numbers concisely. A number in standard form looks like: a × 10ⁿ where 1 ≤ a < 10 and n is an integer.
Large numbers have a positive power of 10: - 5,600,000 = 5.6 × 10⁶
Small numbers have a negative power of 10: - 0.00034 = 3.4 × 10⁻⁴
Key Points
- a must be between 1 and 10 (including 1, not including 10)
- Count decimal places moved = the power
- Moving left = positive power; moving right = negative power
- To add/subtract, convert to same power of 10 first
Example Questions
1Write 0.00052 in standard form.
5.2 × 10⁻⁴
[1 mark]
2Calculate (3 × 10⁴) × (2 × 10⁵). Give your answer in standard form.
3 × 2 = 6, 10⁴ × 10⁵ = 10⁹. Answer: 6 × 10⁹
[2 marks]
Ratio and Proportion
A ratio compares parts to parts. A proportion compares a part to the whole.
Simplifying ratios: divide all parts by their HCF. For example, 12:8 → 3:2 (divide by 4).
Sharing in a ratio: Add the parts, divide the total, then multiply. To share £100 in the ratio 3:2: total parts = 5, one part = £20, so 3 parts = £60 and 2 parts = £40.
Key Points
- Ratios must be in the same units before simplifying
- Direct proportion: as one increases, the other increases at the same rate
- Inverse proportion: as one increases, the other decreases
- Unitary method: find the value of 1, then scale
Example Questions
2Share £450 in the ratio 2:3:4
Total parts = 9. One part = £50. Shares: £100, £150, £200
[2 marks]
2If 5 pens cost £3.75, how much do 8 pens cost?
1 pen = £3.75 ÷ 5 = £0.75. 8 pens = 8 × £0.75 = £6.00
[2 marks]