The Four Operations with Numbers and Fractions

• Step 4: Add the numerators: $\frac{35 + 21}{15} = \frac{56}{15}$

• Step 5: Final Answer: $3\,\frac{11}{15}$

Worked Example: Subtraction

Calculate $5\,\frac{1}{3} - 2\,\frac{6}{7}$

• Step 1: Convert to improper fractions: $\frac{16}{3} - \frac{20}{7}$

• Step 2: Find a common denominator: The LCM of 3 and 7 is 21.

• Step 3: Create equivalent fractions: $\frac{112}{21} - \frac{60}{21}$

• Step 4: Subtract the numerators: $\frac{112 - 60}{21} = \frac{52}{21}$

• Step 5: Final Answer: $2\,\frac{10}{21}$

Worked Example: Multiplication

Calculate $1\,\frac{1}{5} \times \frac{7}{12}$

• Step 1: Convert to improper fraction: $\frac{6}{5} \times \frac{7}{12}$

• Step 2: Cross-simplify: The 6 and 12 share a factor of 6, simplifying the calculation to $\frac{1}{5} \times \frac{7}{2}$

• Step 3: Multiply numerators and denominators: $\frac{1 \times 7}{5 \times 2}$

• Step 4: Final Answer: $\frac{7}{10}$

Worked Example: Division

Calculate $2\,\frac{2}{5} \div 1\,\frac{1}{2}$

• Step 1: Convert to improper fractions: $\frac{12}{5} \div \frac{3}{2}$

• Step 2: Apply the KFC rule: Keep the first fraction ($\frac{12}{5}$), Flip the second to find its reciprocal ($\frac{2}{3}$), and Change to multiplication ($\times$).

• Step 3: Cross-simplify and calculate: $\frac{12}{5} \times \frac{2}{3} \rightarrow \frac{4}{5} \times \frac{2}{1} = \frac{8}{5}$

• Step 4: Final Answer: $1\,\frac{3}{5}$

Formal Operations with Decimals

Why do a £1.50 coffee and a £0.25 snack cost £1.75 instead of £1.525? Because place value dictates that tenths can only be added to tenths, and hundredths to hundredths.

• For addition and subtraction, ensure strict decimal alignment so that digits of the same place value share the same column.
• Add zero placeholders to numbers with fewer decimal places to prevent alignment errors.
• When dividing by a decimal, transform the divisor into an integer by multiplying both values by the same power of 10. Do NOT write division remainders using an 'r' at GCSE level; always calculate to a full decimal or fraction.

Worked Example: Decimal Subtraction

Calculate $16 - 9.4$

• Step 1: Align and add placeholders: $16.0 - 9.4$

• Step 2: Tenths column: $0 - 4$ cannot be done. Use exchanging to move $1$ from the $6$ (units) to make it $5$. The $0$ becomes $10$.

• Step 3: Calculate tenths: $10 - 4 = 6$.

• Step 4: Units column: $5 - 9$ cannot be done. Exchange $1$ from the tens to make the units $15$. $15 - 9 = 6$.

• Step 5: Final Answer: $6.6$

Operations with Directed Numbers

Understanding how to calculate across zero is essential for navigating real-world contexts, such as when bank balances plunge from credit into overdraft.

• Directed numbers require us to track both magnitude and positive/negative signs.
• Separate the calculation from the sign logic. Perform formal written methods on the absolute values, then apply the correct sign to the result. Do NOT include negative signs inside your formal column carrying or exchanging steps.
• For mixed signs in addition/subtraction, find the difference. For numbers with the same sign (e.g., $-15.2 - 4.3$), use column addition on the magnitudes ($15.2 + 4.3 = 19.5$) and keep the original sign ($-19.5$).
• For multiplication and division: Same signs result in a positive answer; different signs result in a negative answer.

Worked Example: Formal Subtraction with Directed Integers

Calculate $-673 + 289$

• Step 1: Logic: The signs are different, so we find the difference: $-(673 - 289)$. The larger absolute magnitude is 673, so the final answer will be negative.

• Step 2: Units column: $3 - 9$ cannot be done. Use exchanging (or regrouping) to take 1 from the tens. $7$ becomes $6$; $3$ becomes $13$. $13 - 9 = 4$.

• Step 3: Tens column: $6 - 8$ cannot be done. Exchange 1 from the hundreds. $6$ becomes $5$; $6$ becomes $16$. $16 - 8 = 8$.

• Step 4: Hundreds column: $5 - 2 = 3$. The difference is $384$.

• Step 5: Final Answer: Apply the negative sign from Step 1 $\rightarrow -384$.

Worked Example: Long Multiplication with Directed Decimals

Calculate $-8.6 \times 9.4$

• Step 1: Sign rule: Different signs mean the answer is negative.

• Step 2: Formal integer method: Remove decimals to calculate $86 \times 94$.

• Step 3: Calculate rows: $86 \times 4 = 344$. Place a zero placeholder for the tens row: $86 \times 90 = 7740$.

• Step 4: Sum the rows: $344 + 7740 = 8084$.

• Step 5: Replace decimals: The question has 2 total decimal places (0.6 and 0.4). The magnitude is $80.84$.

• Step 6: Final Answer: $-80.84$

Worked Example: Bus Stop Division with Directed Decimals

Calculate $17.28 \div -3.2$

• Step 1: Sign rule: Different signs mean the answer is negative.

• Step 2: Transform divisor: Multiply both values by 10 to make the divisor an integer: $172.8 \div 32$.

• Step 3: Calculate using bus stop division: 32 into 172 goes 5 times ($32 \times 5 = 160$), with a remainder of 12.

• Step 4: Decimal and remainder: Place the decimal point in the quotient. 32 into 128 goes 4 times ($32 \times 4 = 128$), with no remainder. Magnitude is $5.4$.

• Step 5: Final Answer: $-5.4$