Percentages & Ratio

<p>Percentages and ratios come up frequently in the 11+ exam. A percentage is a special fraction out of 100, while a ratio compares quantities. Once you see how they connect to fractions and decimals, they become much easier to handle.</p>

<h2>What Is a Percentage?</h2>

<p>"Per cent" means "out of 100." So 35% means 35 out of every 100. You can write this as the fraction 35/100 or the decimal 0.35.</p>

<h2>Converting Between Percentages, Fractions, and Decimals</h2>

<ul>

<li>Percentage → decimal: divide by 100 (move the decimal point two places left). 45% = 0.45</li>

<li>Decimal → percentage: multiply by 100. 0.72 = 72%</li>

<li>Fraction → percentage: divide numerator by denominator, then multiply by 100. 3/8 = 0.375 = 37.5%</li>

</ul>

<h2>Finding a Percentage of an Amount</h2>

<p>There are two quick methods:</p>

<p><strong>Method 1 — Divide and multiply:</strong> To find 15% of 60, find 10% first (60 ÷ 10 = 6), then 5% (half of 6 = 3), and add: 6 + 3 = 9.</p>

<p><strong>Method 2 — Convert to a decimal:</strong> 15% = 0.15, so 0.15 × 60 = 9.</p>

<h2>Percentage Increase and Decrease</h2>

<p><strong>Worked Example:</strong> A coat costs £80 and is reduced by 25%. What is the sale price?</p>

<p>25% of £80 = £80 ÷ 4 = £20. Sale price = £80 − £20 = £60.</p>

<p>Quick alternative: if the price decreases by 25%, you pay 75%. So £80 × 0.75 = £60.</p>

<h2>What Is a Ratio?</h2>

<p>A ratio compares quantities using a colon. If a drink is made with 2 parts juice to 3 parts water, the ratio is 2 : 3. The total number of parts is 2 + 3 = 5.</p>

<h2>Simplifying Ratios</h2>

<p>Divide all parts by their highest common factor. 12 : 8 → divide both by 4 → 3 : 2.</p>

<h2>Sharing in a Given Ratio</h2>

<p><strong>Worked Example:</strong> Share £56 in the ratio 3 : 5.</p>

<p>Total parts = 3 + 5 = 8. One part = £56 ÷ 8 = £7. First share = 3 × £7 = £21. Second share = 5 × £7 = £35. Check: £21 + £35 = £56. ✓</p>

<h2>Using Ratios to Find Missing Values</h2>

<p><strong>Worked Example:</strong> Red to blue beads are in the ratio 2 : 7. If there are 14 red beads, how many blue beads are there?</p>

<p>The multiplier is 14 ÷ 2 = 7. Blue beads = 7 × 7 = 49.</p>

<h2>Connecting Percentages and Ratios</h2>

<p>If a class has boys to girls in the ratio 3 : 2, boys make up 3/5 of the class = 60%, and girls make up 2/5 = 40%.</p>

<h2>Practice Tips</h2>

<ul>

<li>Build up percentages from 10% and 1% — you can find any percentage this way</li>

<li>Always check a ratio answer by adding the shares to see if they total the original amount</li>

<li>Simplify ratios before calculating — smaller numbers mean fewer mistakes</li>

<li>Memorise: 50% = 1/2, 25% = 1/4, 10% = 1/10, 20% = 1/5, 75% = 3/4</li>

</ul>