Ratio and Proportion

# Ratio and Proportion ## Learning Objectives By the end of this lesson, you will be able to: - Understand and express ratios in their simplest form - Apply ratios to solve real-world problems involving sharing and scaling - Distinguish between direct and inverse proportion - Solve proportion problems using unitary method and algebraic approaches - Convert between ratios, fractions, and percentages ## Introduction Imagine you're making lemonade for a party. The recipe says you need 2 cups of lemon juice for every 5 cups of water. If you want to make more lemonade, how do you keep the same taste? This is where ratios come in! Ratios help us understand the relationship between quantities and maintain those relationships when we scale things up or down. Ratios and proportions are everywhere in daily life. Architects use them to create scale models of buildings, chefs use them in recipes, artists use them to mix paint colors, and currency exchanges rely on them for conversions. Understanding these concepts will not only help you in mathematics but also develop your problem-solving skills for countless real-world situations. In this lesson, we'll explore how to work with ratios, simplify them, and use them to solve practical problems. We'll also discover the difference between direct proportion (where quantities increase together) and inverse proportion (where one quantity increases as the other decreases). ## Key Concepts ### What is a Ratio? A **ratio** compares two or more quantities of the same type. It shows how much of one thing there is compared to another. Ratios can be written in three ways: - Using the word "to": 3 to 5 - Using a colon: 3:5 - As a fraction: 3/5 (though this changes the context slightly) **Example:** If a class has 12 boys and 15 girls, the ratio of boys to girls is 12:15. ### Simplifying Ratios Just like fractions, ratios should be expressed in their **simplest form** by dividing all parts by their highest common factor (HCF). **Example:** The ratio 12:15 can be simplified by dividing both numbers by 3 (the HCF): - 12 ÷ 3 = 4 - 15 ÷ 3 = 5 - Simplified ratio: **4:5** **Key Rules for Ratios:** - All parts of a ratio must be in the same units before simplifying - Ratios have no units in their final form - Order matters: 2:3 is different from 3:2 ### Sharing in a Given Ratio When dividing a quantity in a given ratio, follow these steps: 1. Add all parts of the ratio together 2. Divide the total quantity by this sum to find one part 3. Multiply to find each share **Example:** Share £60 in the ratio 2:3 - Total parts = 2 + 3 = 5 - One part = £60 ÷ 5 = £12 - First share = 2 × £12 = £24 - Second share = 3 × £12 = £36 ### Direct Proportion Two quantities are in **direct proportion** when they increase or decrease at the same rate. If one doubles, the other doubles too. **Formula:** If y is directly proportional to x, then y = kx (where k is the constant of proportionality) **Example:** If 5 pencils cost £2, then 10 pencils cost £4 (both quantities double). ### Inverse Proportion Two quantities are in **inverse proportion** when one increases while the other decreases at a reciprocal rate. **Formula:** If y is inversely proportional to x, then y = k/x or xy = k (constant) **Example:** If 4 workers can complete a job in 6 hours, then 8 workers can complete it in 3 hours (double the workers, half the time). ## Worked Examples ### Example 1: Simplifying Ratios with Units **Question:** Write the ratio 250 g to 2 kg in its simplest form. **Solution:** - **Step 1:** Convert to the same units: 2 kg = 2000 g - **Step 2:** Write as a ratio: 250:2000 - **Step 3:** Find the HCF of 250 and 2000, which is 250 - **Step 4:** Divide both by 250: 250 ÷ 250 = 1, and 2000 ÷ 250 = 8 - **Answer:** The simplest form is **1:8** ### Example 2: Solving a Direct Proportion Problem **Question:** A car travels 150 km using 12 litres of petrol. How much petrol is needed to travel 275 km at the same rate? **Solution:** - **Step 1:** Find the petrol used per km: 12 ÷ 150 = 0.08 litres per km - **Step 2:** Multiply by the new distance: 0.08 × 275 = 22 litres **Alternative method (using proportions):** - **Step 1:** Set up the proportion: 12/150 = x/275 - **Step 2:** Cross multiply: 150x = 12 × 275 - **Step 3:** Solve: 150x = 3300, so x = 3300 ÷ 150 = 22 - **Answer:** **22 litres** of petrol are needed ### Example 3: Three-Part Ratio Problem **Question:** Anna, Ben, and Chloe share £240 in the ratio 3:5:4. How much does each person receive? **Solution:** - **Step 1:** Add ratio parts: 3 + 5 + 4 = 12 parts - **Step 2:** Find the value of one part: £240 ÷ 12 = £20 - **Step 3:** Calculate each share: - Anna: 3 × £20 = £60 - Ben: 5 × £20 = £100 - Chloe: 4 × £20 = £80 - **Step 4:** Check: £60 + £100 + £80 = £240 ✓ - **Answer:** Anna receives **£60**, Ben receives **£100**, and Chloe receives **£80** ## Practice Questions **Question 1:** Simplify the ratio 48:72:60 **Question 2:** A recipe for 4 people needs 300 g of flour. How much flour is needed for 10 people? **Question 3:** Two numbers are in the ratio 5:7. If their sum is 96, find both numbers. **Question 4:** It takes 6 machines 4 hours to complete a task. How long would it take 8 machines working at the same rate? (Inverse proportion) **Question 5:** The ratio of red to blue to yellow paint is 2:3:5. If I have 40 ml of yellow paint, how much red and blue paint do I have? --- ### Answers to Practice Questions 1. **4:6:5** (dividing all by HCF of 12) 2. **750 g** (300 ÷ 4 = 75 g per person; 75 × 10 = 750 g) 3. **40 and 56** (12 parts = 96, one part = 8; 5×8=40, 7×8=56) 4. **3 hours** (6 × 4 = 24 total machine-hours; 24 ÷ 8 = 3 hours) 5. **Red: 16 ml, Blue: 24 ml** (5 parts = 40 ml, so 1 part = 8 ml; red = 2×8=16, blue = 3×8=24) ## Summary - **Ratios** compare quantities and must be in the same units before simplifying - Always simplify ratios by dividing by the **highest common factor (HCF)** - When sharing in a ratio, add all parts together, divide the total by this sum, then multiply - **Direct proportion:** quantities increase or decrease together (y = kx) - **Inverse proportion:** one quantity increases as the other decreases (xy = k) - Check your answers by ensuring all shares add up to the original total - Ratios can be extended to three or more parts using the same principles ## Exam Tips - **Always check units first:** Convert everything to the same units before starting calculations. This is one of the most common mistakes students make in ratio questions. - **Show your working clearly:** Even if you can do the calculation mentally, write down each step. Examiners award method marks, so you can still earn points even if your final answer is incorrect. Always label your steps (e.g., "Total parts = ...", "One part = ..."). - **Use the context to check reasonableness:** After solving, ask yourself if the answer makes sense. If you're sharing money and one person gets more than the total, you know something went wrong. If a recipe needs more flour for more people, your answer should be larger, not smaller.