2.2 Fractions, Percentages & Ratio

2.2 Fractions, Percentages & Ratio

Proportional reasoning lies at the heart of the Kent Test 11+. Questions involving fractions, decimals, percentages (FDP), and ratios test a student's ability to manipulate quantities, convert between representations, and solve real-world sharing problems.

Equivalent Fractions, Simplification & Comparison

Fractions represent equal parts of a whole quantity ($\frac{\text{numerator}}{\text{denominator}}$).

Creating Equivalent Fractions

Multiplying or dividing both numerator and denominator by the same non-zero integer changes the appearance of a fraction without altering its value:

$$\frac{a}{b} = \frac{a \times k}{b \times k}$$

Simplification (Lowest Terms)

Divide numerator and denominator by their Highest Common Factor (HCF).

• Example: Simplify $\frac{42}{70}$. $\text{HCF}(42, 70) = 14$. $\frac{42 \div 14}{70 \div 14} = \frac{3}{5}$.

Comparing and Ordering Fractions

To compare fractions with different denominators, convert them to equivalent fractions with a Lowest Common Denominator (LCD) (the LCM of the denominators).

Worked Example: Compare $\frac{5}{8}$ and $\frac{7}{12}$.

• $\text{LCM}(8, 12) = 24$.
• $\frac{5}{8} = \frac{5 \times 3}{8 \times 3} = \frac{15}{24}$.
• $\frac{7}{12} = \frac{7 \times 2}{12 \times 2} = \frac{14}{24}$.
• Since $15 > 14$, $\frac{5}{8} > \frac{7}{12}$.

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Fraction Operations & Mixed Numbers

1. Addition and Subtraction

Convert mixed numbers to improper fractions or operate on integer and fractional parts separately after finding a common denominator.

Worked Example: Calculate $2 \frac{2}{3} - 1 \frac{4}{5}$.

• Convert to improper fractions: $2 \frac{2}{3} = \frac{8}{3}$, $1 \frac{4}{5} = \frac{9}{5}$.
• Common denominator is $15$: $\frac{8 \times 5}{3 \times 5} - \frac{9 \times 3}{5 \times 3} = \frac{40}{15} - \frac{27}{15} = \frac{13}{15}$.

2. Multiplication

Multiply numerators across and denominators across. Simplify by cross-cancelling common factors before multiplying.

Worked Example: Calculate $\frac{4}{9} \times \frac{15}{16}$.

• Cross-cancel $4$ and $16$ (divide by $4$): $4 \rightarrow 1$, $16 \rightarrow 4$.
• Cross-cancel $9$ and $15$ (divide by $3$): $9 \rightarrow 3$, $15 \rightarrow 5$.
• Multiply remaining terms: $\frac{1 \times 5}{3 \times 4} = \frac{5}{12}$.

3. Division

Use the Keep-Change-Flip (KCF) rule: Keep the first fraction, Change division to multiplication, and Flip the second fraction to its reciprocal.

Worked Example: Calculate $\frac{7}{10} \div \frac{14}{15}$.

• Keep $\frac{7}{10}$, Change to $\times$, Flip to $\frac{15}{14}$.
• Expression: $\frac{7}{10} \times \frac{15}{14}$.
• Cross-cancel: $7$ and $14$ (divide by $7 \rightarrow 1$ and $2$); $10$ and $15$ (divide by $5 \rightarrow 2$ and $3$).
• Product: $\frac{1 \times 3}{2 \times 2} = \frac{3}{4}$.

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Fraction, Decimal & Percentage (FDP) Conversions

Instant recall of standard FDP benchmark conversions saves critical time during the exam:

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Percentage Calculations & Reverse Percentages

1. Building Up Percentages Non-Calculator

Decompose complex percentages into easy building blocks ($10\%, 5\%, 1\%$).

• Example: Find $37.5\%$ of $£240$.
  • Method A (FDP): $37.5\% = \frac{3}{8}$. $\frac{3}{8} \times 240 = 3 \times 30 = £90$.
  • Method B (Build-up): $10\% = 24$, $30\% = 72$, $5\% = 12$, $2.5\% = 6 \rightarrow 72 + 12 + 6 = £90$.

2. Percentage Increase and Decrease

$$\text{New Value} = \text{Original Value} \pm \text{Percentage Amount}$$

Alternatively, use multipliers (e.g., $15\%$ increase $= \times 1.15$; $20\%$ decrease $= \times 0.80$).

3. Reverse Percentages (Finding the Original)

In reverse percentage problems, the given value represents a percentage after an increase or decrease.

Worked Example: A coat is reduced by $20\%$ in a sale to $£64$. Find the original price.

• The sale price corresponds to $100\% - 20\% = 80\%$ of the original price.
• Set up equality: $80\% = £64$.
• Find $10\%$: $£64 \div 8 = £8$.
• Find $100\%$ (original price): $£8 \times 10 = £80$.

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Ratio, Proportion & Sharing Problems

Ratios compare relative quantities of two or more groups.

Standard 3-Step Sharing Method

To share a quantity in a ratio $a : b : c$:

1. Find Total Parts: $a + b + c$.
2. Find Value of One Part: $\frac{\text{Total Quantity}}{\text{Total Parts}}$.
3. Multiply Each Share: Multiply the single part value by $a, b,$ and $c$ respectively.

Difference-Based Ratio Problems

In advanced 11+ questions, the difference between two shares is given rather than the overall total.

Worked Example: Ben and Charlotte share prize money in the ratio $3 : 7$. Charlotte receives $£48$ more than Ben. What is the total prize money?

• Part difference: Charlotte has $7$ parts and Ben has $3$ parts $\rightarrow 7 - 3 = 4$ extra parts.
• Value per part: $4\text{ parts} = £48 \rightarrow 1\text{ part} = £48 \div 4 = £12$.
• Total parts: $3 + 7 = 10\text{ parts}$.
• Total prize: $10 \times £12 = £120$.