Four Operations
The Four Operations
The four arithmetic operations — addition, subtraction, multiplication and division — are the building blocks of all mathematics. At GCSE level you must apply them confidently to integers (whole numbers, positive and negative), decimals, proper and improper fractions, and mixed numbers. Equally important is a firm grasp of place value: knowing precisely what each digit in a number represents.
Place Value
Place value determines the size of each digit based on its position within a number. Moving one place to the left multiplies the value by 10; moving one place to the right divides it by 10. The decimal point separates the whole number part from the fractional part.
| Hundreds | Tens | Ones | . | Tenths | Hundredths | Thousandths |
|---|---|---|---|---|---|---|
| 3 | 2 | 4 | . | 7 | 5 | 6 |
| \( 3 \times 100 \) | \( 2 \times 10 \) | \( 4 \times 1 \) | \( \frac{7}{10} \) | \( \frac{5}{100} \) | \( \frac{6}{1000} \) |
The number above is 324.756. Each digit contributes its face value multiplied by its positional value. When adding or subtracting decimals, always align the decimal points in a column to avoid place value errors.
Working with Negative Numbers
Adding a negative number is equivalent to subtracting its positive counterpart, and subtracting a negative is equivalent to adding. These two rules resolve most errors with negatives in addition and subtraction:
\[ a + (-b) = a - b \qquad a - (-b) = a + b \]For multiplication and division, the sign of the result depends on whether the two numbers share the same sign or have different signs:
| First Number | Operation | Second Number | Result Sign | Example |
|---|---|---|---|---|
| positive | \( \times \) or \( \div \) | positive | positive | \( 4 \times 3 = +12 \) |
| positive | \( \times \) or \( \div \) | negative | negative | \( 4 \times (-3) = -12 \) |
| negative | \( \times \) or \( \div \) | positive | negative | \( (-4) \times 3 = -12 \) |
| negative | \( \times \) or \( \div \) | negative | positive | \( (-4) \times (-3) = +12 \) |
A simple memory rule: same signs give positive, different signs give negative.
Worked Examples
Select a tab to see the approach for each number type.
Integers and Negatives
Addition and subtraction:
\[ 3 + (-8) = 3 - 8 = -5 \] \[ -5 - (-3) = -5 + 3 = -2 \]Multiplication (both negative): Same signs, so the result is positive.
\[ (-4) \times (-6) = +24 \]Division (different signs): Different signs, so the result is negative.
\[ (-20) \div 4 = -5 \]Decimals
Addition — align decimal points:
Calculate \( 3.47 + 1.6 \):
\[ 3.47 + 1.60 = 5.07 \]Write \(1.6\) as \(1.60\) (same number of decimal places) before adding column by column.
Multiplication — count decimal places:
Calculate \( 0.7 \times 0.4 \):
Ignore the decimals and multiply: \( 7 \times 4 = 28 \). Count the total decimal places in both numbers: \(1 + 1 = 2\). Insert the decimal point 2 places from the right:
\[ 0.7 \times 0.4 = 0.28 \]Fractions — All Four Operations
Addition — common denominator (LCM of 4 and 5 is 20):
\[ \frac{3}{4} + \frac{2}{5} = \frac{15}{20} + \frac{8}{20} = \frac{23}{20} = 1\frac{3}{20} \]Subtraction — common denominator (LCM of 6 and 4 is 12):
\[ \frac{5}{6} - \frac{1}{4} = \frac{10}{12} - \frac{3}{12} = \frac{7}{12} \]Multiplication — multiply numerators together and denominators together:
\[ \frac{3}{4} \times \frac{2}{5} = \frac{3 \times 2}{4 \times 5} = \frac{6}{20} = \frac{3}{10} \]Division — keep the first fraction, change \( \div \) to \( \times \), flip the second fraction (keep, change, flip):
\[ \frac{3}{4} \div \frac{2}{5} = \frac{3}{4} \times \frac{5}{2} = \frac{15}{8} = 1\frac{7}{8} \]Mixed Numbers
Always convert to improper fractions first, then operate, then convert back if required.
Converting a mixed number: Multiply the whole number by the denominator and add the numerator.
\[ 2\frac{3}{4} = \frac{(4 \times 2) + 3}{4} = \frac{11}{4} \]Addition example: \( 2\frac{3}{4} + 1\frac{1}{3} \)
\[ \frac{11}{4} + \frac{4}{3} = \frac{33}{12} + \frac{16}{12} = \frac{49}{12} = 4\frac{1}{12} \]Multiplication example: \( 1\frac{1}{2} \times 2\frac{1}{3} \)
\[ \frac{3}{2} \times \frac{7}{3} = \frac{21}{6} = \frac{7}{2} = 3\frac{1}{2} \]Deep Dive: Why Does Dividing by a Fraction Flip It?
The "keep, change, flip" rule for dividing fractions is not arbitrary — it follows directly from the definition of division. Dividing by a number is the same as multiplying by its reciprocal (its multiplicative inverse). The reciprocal of \( \frac{a}{b} \) is \( \frac{b}{a} \), because \( \frac{a}{b} \times \frac{b}{a} = 1 \).
So \( \frac{3}{4} \div \frac{2}{5} \) is asking "how many lots of \( \frac{2}{5} \) fit into \( \frac{3}{4} \)?" This is the same as \( \frac{3}{4} \times \frac{5}{2} \), the reciprocal of the divisor. The rule always works — and it applies equally to whole numbers, since the reciprocal of 3 is \( \frac{1}{3} \).
Key Takeaways
- When adding or subtracting decimals, always align the decimal points before calculating column by column.
- When multiplying decimals, multiply as integers and then insert the decimal point by counting the total decimal places in both numbers.
- Same signs give a positive result when multiplying or dividing; different signs give a negative result.
- To add or subtract fractions, find a common denominator first; to multiply, go straight across; to divide, keep, change, flip.
- Always convert mixed numbers to improper fractions before carrying out any operation.
- Place value is the foundation — each digit's position tells you its actual value in the number.