Proportional Reasoning. Grade 7 Math: Proportional Reasoning - Solving Direct Proportion Problems Subtopic Navigator Introduction: Understanding Direct Proportion What is Direct Proportion? The Constant of Proportionality (k) Setting Up Proportion Equations Cross-Multiplication Method Unit Rate Method Solving Missing Values Real-World Applications Checking Your Solutions Practice Exercises Conclusion Learning Objectives Define and identify direct proportion relationships Understand the constant of proportionality (k) Set up and solve proportion equations Use cross-multiplication to solve proportions Apply unit rate method to solve problems Solve for missing values in proportion tables Apply proportional reasoning to real-world situations Check and verify proportional solutions Solving Direct Proportion Problems Have you ever wondered how bakers know how much flour to use when doubling a recipe? Or how mapmakers determine distances? Or how stores calculate prices for different quantities? All these situations use proportional reasoning! When two quantities change at the same rate, they are in direct proportion. As one increases, the other increases at the same rate. Today, we'll learn how to identify, set up, and solve direct proportion problems - a skill you'll use in cooking, shopping, traveling, and many other everyday situations. Key Terms: Proportion: An equation that states two ratios are equal Direct Proportion: A relationship where two quantities increase or decrease at the same rate Ratio: A comparison of two quantities using division Constant of Proportionality (k): The constant value of the ratio between two proportional quantities Cross-Multiplication: A method to solve proportions by multiplying diagonally Unit Rate: A rate with a denominator of 1 Direct Proportion Relationship: y = kx where k is constant As x increases → y increases proportionally As x decreases → y decreases proportionally Proportional reasoning is one of the most important math skills you'll use throughout your life. Architects use it to create scale models, chefs use it to adjust recipes, and shoppers use it to find the best deals. By the end of this lesson, you'll be able to solve any direct proportion problem with confidence! What is Direct Proportion? Two quantities are in direct proportion when they increase or decrease at the same rate. If you double one quantity, the other doubles too. If you triple one, the other triples. The ratio between the two quantities always stays the same. Identifying Direct Proportion: Look for these characteristics: 1. As one quantity increases, the other increases 2. As one quantity decreases, the other decreases 3. The ratio y/x is always constant 4. The graph is a straight line through the origin (0,0) Example 1: Lemonade Recipe A lemonade recipe uses 3 lemons for 2 cups of water. Is this a direct proportion? If you need 6 cups of water, how many lemons do you need? Solution: First, check if it's direct proportion: Lemons : Water = 3 : 2 = 1.5 lemons per cup If we double water to 4 cups: Lemons = 1.5 × 4 = 6 lemons ✓ If we triple water to 6 cups: Lemons = 1.5 × 6 = 9 lemons ✓ The ratio stays constant at 1.5 lemons/cup For 6 cups water: Lemons = 1.5 × 6 = 9 lemons Check: 3 lemons for 2 cups = 9 lemons for 6 cups (tripled both) Example 2: Car Travel A car travels 120 miles in 2 hours. Is distance proportional to time? How far will it travel in 5 hours? Solution: Distance : Time = 120 : 2 = 60 miles per hour If time doubles to 4 hours: Distance = 60 × 4 = 240 miles ✓ This is direct proportion - as time increases, distance increases at constant rate For 5 hours: Distance = 60 × 5 = 300 miles Note: Speed is the constant of proportionality (k = 60 mph) Common Mistake: Incorrect: Thinking all increasing relationships are proportional Correct: Only relationships with constant ratio are proportional Remember: Check if y/x is always the same! Practice Questions If 4 apples cost ₦2, is cost proportional to number of apples? A printer prints 15 pages in 3 minutes. Is this direct proportion? If 5 pencils weigh 30 grams, how much would 8 pencils weigh? A recipe uses 2 cups flour for 12 cookies. How much flour for 30 cookies? Is age proportional to height? Explain why or why not. The Constant of Proportionality (k) The constant of proportionality, represented by k, is the constant value you get when you divide y by x in a direct proportion relationship. It's the unit rate that connects the two quantities. Finding k: k = y ÷ x (for any pair of values) Once you find k, you can find any missing value: y = k × x or x = y ÷ k Example 1: Finding k from a Table Complete the table and find the constant of proportionality: Hours worked: 2, 4, 6, ? Money earned: ₦30, ₦60, ₦90, ₦150 Solution: First, check if it's proportional: 30 ÷ 2 = 15 60 ÷ 4 = 15 90 ÷ 6 = 15 ✓ Constant k = ₦15 per hour For ₦150 earned: Hours = 150 ÷ 15 = 10 hours Check: 10 × 15 = 150 ✓ Example 2: Using k to Solve Problems The weight of water is directly proportional to its volume. If 3 liters weigh 3 kg, how much do 8 liters weigh? Solution: First find k: k = weight ÷ volume = 3 ÷ 3 = 1 kg/liter For 8 liters: Weight = k × volume = 1 × 8 = 8 kg Note: k = 1 because 1 liter of water weighs 1 kg (this is actually true!) Principles Practice If y = 24 when x = 6, find k. What is y when x = 10? 5 books cost ₦35. What is k? How much do 8 books cost? A machine makes 48 toys in 3 hours. Find k. How many toys in 7 hours? If k = 2.5 and x = 8, find y. If k = 0.75 and y = 12, find x. Setting Up Proportion Equations A proportion is an equation that shows two ratios are equal. We can write proportions in several ways, but they all mean the same thing. Writing Proportion Equations: 1. As two equal ratios: $frac{a}{b} = frac{c}{d}$ 2. Using colons: a : b = c : d 3. As a fraction equation Remember: In direct proportion, $frac{y_1}{x_1} = frac{y_2}{x_2}$ Example 1: Map Scale On a map, 1 cm represents 5 km. How many km does 4.5 cm represent? Solution: Set up the proportion: $frac{1 text{ cm}}{5 text{ km}} = frac{4.5 text{ cm}}{x text{ km}}$ This reads: "1 cm is to 5 km as 4.5 cm is to x km" The ratios must be equal because it's direct proportion Solve: $1/5 = 4.5/x$ Cross multiply: $1 times x = 5 times 4.5$ $x = 22.5$ km Important: Keep units consistent in each ratio! Example 2: Baking Cookies A cookie recipe uses 2 eggs for 24 cookies. How many eggs for 90 cookies? Solution: Set up proportion: $frac{2 text{ eggs}}{24 text{ cookies}} = frac{x text{ eggs}}{90 text{ cookies}}$ This reads: "2 eggs for 24 cookies equals x eggs for 90 cookies" $2/24 = x/90$ Simplify first: $1/12 = x/90$ Cross multiply: $1 times 90 = 12 times x$ $90 = 12x$ $x = 90 ÷ 12 = 7.5$ eggs Note: You can't use half an egg in reality, so you'd use 8 eggs Example 3: Paint Mixture To make orange paint, mix 3 parts red to 2 parts yellow. How much yellow for 15 parts red? Solution: Proportion: $frac{3 text{ red}}{2 text{ yellow}} = frac{15 text{ red}}{x text{ yellow}}$ $3/2 = 15/x$ Cross multiply: $3x = 2 times 15$ $3x = 30$ $x = 10$ parts yellow Note: Ratios stay constant in mixtures Common Mistake: Incorrect: $frac{2}{24} = frac{90}{x}$ (mixing up which goes where) Correct: $frac{2}{24} = frac{x}{90}$ (same items in same positions) Always set up so matching items are in same position in each fraction! Application Practice If 3 meters of fabric cost ₦12, set up proportion for 7 meters. 5 bags of chips cost ₦8.75. Set up proportion for 9 bags. A car uses 4 liters for 50 km. Set up proportion for 125 km. 8 workers build a wall in 6 days. Set up proportion for 12 workers. 3 cups of sugar for 48 cookies. Set up proportion for 80 cookies. Cross-Multiplication Method Cross-multiplication is the most common method for solving proportions. When two fractions are equal, the product of the numerator of the first and denominator of the second equals the product of the denominator of the first and numerator of the second. Cross-Multiplication Rules: • If $frac{a}{b} = frac{c}{d}$, then $a times d = b times c$ • Multiply diagonally across the equal sign • Solve the resulting equation • Always check your answer makes sense Example 1: Solving with Cross-Multiplication Solve: $frac{3}{5} = frac{x}{20}$ Solution: Cross multiply: $3 times 20 = 5 times x$ $60 = 5x$ Divide both sides by 5: $x = 60 ÷ 5 = 12$ Visual: Multiply 3 (top left) × 20 (bottom right) = 5 (bottom left) × x (top right) Example 2: Shopping Problem 4 notebooks cost ₦7. How much do 9 notebooks cost? Solution: $frac{4}{7} = frac{9}{x}$ Cross multiply: $4x = 7 times 9$ $4x = 63$ $x = 63 ÷ 4 = 15.75$ Answer: ₦15.75 Rule: Cost/notebook ratio stays constant Example 3: Fraction Proportion Solve: $frac{2}{3} = frac{8}{x}$ Solution: $2x = 3 times 8$ $2x = 24$ $x = 12$ Rule: Cross-multiplication works with any numbers Example 4: Decimal Proportion Solve: $frac{1.5}{4} = frac{3}{x}$ Solution: $1.5x = 4 times 3$ $1.5x = 12$ $x = 12 ÷ 1.5 = 8$ Important: Works with decimals too! Technique Practice Solve: $frac{2}{7} = frac{x}{21}$ Solve: $frac{5}{8} = frac{15}{x}$ Solve: $frac{3}{x} = frac{9}{12}$ Solve: $frac{x}{6} = frac{10}{15}$ Solve: $frac{2.5}{3} = frac{5}{x}$ Unit Rate Method The unit rate method involves finding the value for one unit first, then multiplying to find the value for any number of units. This method is intuitive and works well for many real-world problems. Common Proportion Formulas: Direct Proportion: $y = kx$ where k is constant Unit Rate Formula: k = y ÷ x (find cost/unit, speed, etc.) Missing Value: $y_2 = k times x_2$ or $x_2 = y_2 ÷ k$ Ratio Form: $frac{y_1}{x_1} = frac{y_2}{x_2}$ Product Form: $y_1 times x_2 = x_1 times y_2$ Example 1: Price Comparison 6 oranges cost ₦4.50. How much for 10 oranges? Formula: $frac{text{cost}_1}{text{quantity}_1} = frac{text{cost}_2}{text{quantity}_2}$ Solution: Method 1 - Unit rate: Cost per orange = ₦4.50 ÷ 6 = ₦0.75 For 10 oranges: 10 × ₦0.75 = ₦7.50 Method 2 - Proportion: $frac{4.50}{6} = frac{x}{10}$ $4.50 times 10 = 6 times x$ $45 = 6x$ $x = 7.50$ Check: Both methods give same answer ✓ Example 2: Speed Calculation A train travels 280 km in 4 hours. How far in 7 hours? Formula: distance = speed × time Solution: Find speed (unit rate): 280 ÷ 4 = 70 km/h Distance in 7 hours: 70 × 7 = 490 km Alternative: $frac{280}{4} = frac{x}{7}$ → 280×7 = 4x → 1960 = 4x → x = 490 km Units: km/h × h = km (hours cancel out) Example 3: Recipe Adjustment A cake recipe needs 3 cups flour for 8 servings. How much flour for 20 servings? Formula: $frac{text{flour}_1}{text{servings}_1} = frac{text{flour}_2}{text{servings}_2}$ Solution: Unit rate: 3 ÷ 8 = 0.375 cups per serving For 20 servings: 0.375 × 20 = 7.5 cups Check with proportion: $frac{3}{8} = frac{x}{20}$ → 3×20 = 8x → 60 = 8x → x = 7.5 Units: cups/serving × servings = cups Method Practice 8 pencils cost ₦3.20. Find cost of 15 pencils using unit rate. A car goes 360 km on 30 liters. How far on 45 liters? 5 workers complete job in 12 days. How long for 8 workers? (Careful!) 3 kg apples cost ₦7.50. Find cost of 5.5 kg. A pump fills tank in 15 minutes. How much in 1 hour? Solving Missing Values Sometimes proportion problems give you three values and ask for the fourth. We can solve these by setting up the proportion correctly and using any of our methods. Steps for Missing Values: 1. Identify the two quantities in proportion 2. Set up ratio for known pair 3. Set up ratio for pair with missing value 4. Make ratios equal (proportion) 5. Solve for missing value 6. Check if answer makes sense Example 1: Missing in Middle If 4 : 6 = 10 : x, find x. Solution: Write as fractions: $frac{4}{6} = frac{10}{x}$ Cross multiply: $4x = 6 times 10$ $4x = 60$ $x = 15$ Check: 4:6 = 2:3 and 10:15 = 2:3 ✓ Example 2: Similar Triangles In similar triangles, corresponding sides are proportional. If triangle A has sides 3, 4, 5 and triangle B has corresponding sides 6, 8, x, find x. Solution: Set up proportion: $frac{3}{5} = frac{6}{x}$ (using smallest and largest sides) $3x = 5 times 6$ $3x = 30$ $x = 10$ Check: All sides doubled (3→6, 4→8, 5→10) ✓ Example 3: Map Distance On a map scale 1:50,000, two towns are 8 cm apart. What is actual distance? Solution: Scale means 1 cm on map = 50,000 cm actual Proportion: $frac{1}{50000} = frac{8}{x}$ $x = 8 times 50000 = 400,000$ cm Convert to km: 400,000 cm = 4 km Answer: 4 km apart Example 4: Mixed Units If 2.5 inches = 6.35 cm, how many cm in 10 inches? Solution: $frac{2.5}{6.35} = frac{10}{x}$ $2.5x = 6.35 times 10$ $2.5x = 63.5$ $x = 63.5 ÷ 2.5 = 25.4$ cm Note: 1 inch = 2.54 cm exactly Verification Practice If 5:8 = 15:x, find x. If 3/x = 12/20, find x. If 7:9 = x:36, find x. If 2.5:4 = 10:x, find x. If x:15 = 8:12, find x. Real-World Applications Direct proportion appears everywhere in daily life. Let's solve some practical problems you might encounter. Example 1: Best Buy Comparison Store A: 3 kg sugar for ₦4.50. Store B: 5 kg sugar for ₦7.25. Which is better value? Solution: Find unit price for each: Store A: ₦4.50 ÷ 3 = ₦1.50 per kg Store B: ₦7.25 ÷ 5 = ₦1.45 per kg Store B is cheaper per kg Price for 10 kg: Store A: 10 × ₦1.50 = ₦15.00 Store B: 10 × ₦1.45 = ₦14.50 You save ₦0.50 at Store B for 10 kg Example 2: Gasoline Cost A car's gas mileage is 25 miles per gallon. Gas costs ₦3.20 per gallon. How much for a 300-mile trip? Solution: Gallons needed: 300 ÷ 25 = 12 gallons Cost: 12 × ₦3.20 = ₦38.40 Alternative proportion: $frac{25}{3.20} = frac{300}{x}$ but careful! Not direct between miles and cost Better: Miles and gallons are proportional, gallons and cost are proportional Example 3: Currency Exchange 1 US dollar = 0.85 Euros. How many Euros for ₦150? Solution: $frac{1}{0.85} = frac{150}{x}$ $x = 150 times 0.85 = 127.50$ Euros Units: Exchange rate is constant (proportional) Example 4: Scale Model A model car is built at scale 1:24. The real car is 4.8 m long. How long is model? Analysis: • Scale 1:24 means 1 unit on model = 24 units on real • Model is smaller than real car • Lengths are proportional • If real is 4.8 m = 480 cm Solution: $frac{1}{24} = frac{x}{480}$ $24x = 480$ $x = 20$ cm model length Application Practice Which is better: 500g for ₦2.50 or 750g for ₦3.60? If 1 British pound = ₦1.25, how many pounds for ₦100? A recipe for 4 uses 250g pasta. How much for 7 people? Scale 1:100, model building is 15cm tall. Real height? Car: 400km on 32L. Trip is 550km. Gas needed? Checking Your Solutions Always check your proportion solutions! A wrong answer might mean you set up the proportion incorrectly or made a calculation error. Checking Strategies: Ratio Check: Both ratios should simplify to same value Cross Product Check: Products should be equal Unit Rate Check: Unit rates should match Estimate Check: Answer should be reasonable Alternative Method: Solve different way Example 1: Checking with Ratios Solve $frac{3}{7} = frac{12}{x}$, get x = 28. Check your answer. Your answer: x = 28 Check: Original: $frac{3}{7} = frac{12}{28}$ Simplify 3/7 = 3/7 (already simplified) Simplify 12/28 = 3/7 (divide by 4) ✓ Both equal 3/7 Estimate: 3/7 ≈ 0.43, 12/28 ≈ 0.43 ✓ Example 2: Multiple Method Check 5 books cost ₦27.50. How much for 8 books? Your answer: ₦44 Check: Method 1 (unit rate): ₦27.50 ÷ 5 = ₦5.50/book 8 × ₦5.50 = ₦44 ✓ Method 2 (proportion): $frac{5}{27.50} = frac{8}{x}$ 5x = 27.50 × 8 5x = 220 x = ₦44 ✓ Method 3 (estimation): About ₦5.50 each, 8 cost about ₦44 ✓ Example 3: Units Check Car: 240 km in 3 hours. How far in 5.5 hours? Your answer: 440 km Check: Calculation: Speed = 240 ÷ 3 = 80 km/h Distance = 80 × 5.5 = 440 km ✓ Units: km/h × h = km (hours cancel) ✓ Estimate: 80 × 5 = 400, 80 × 6 = 480, 440 is between ✓ Visual: If 3 hours → 240 km, 5.5 hours should be less than double ✓ Common Checking Mistakes: Incorrect: Only checking with same method used to solve Correct: Use different method to check Incorrect: Not checking if answer makes sense Correct: Always ask "Is this reasonable?" Problem Type Setup Solution Method Check Method Shopping (price) $frac{text{quantity}_1}{text{price}_1} = frac{text{quantity}_2}{text{price}_2}$ Cross-multiplication Unit rate comparison Speed/Distance $frac{text{distance}_1}{text{time}_1} = frac{text{distance}_2}{text{time}_2}$ Find speed first Estimate reasonable Recipes $frac{text{ingredient}_1}{text{servings}_1} = frac{text{ingredient}_2}{text{servings}_2}$ Unit rate Ratio simplification Scale Models $frac{text{model}}{text{real}} = text{scale}$ Cross-multiplication Visual check Skills Practice Solve $frac{4}{9} = frac{x}{27}$ and check with simplification. 6 pens cost ₦4.80. Find cost of 11 pens and check with unit rate. Car: 180 km in 2.5 hours. Find distance in 4 hours. Check with estimation. Check if 3:5 = 9:15 is true using cross products. Solve and check: $frac{2.4}{3.6} = frac{4}{x}$ Cumulative Exercises If 5 oranges cost ₦3.25, how much for 12 oranges? A car travels 315 km in 4.5 hours. How far in 7 hours? The ratio of boys to girls is 3:4. If there are 21 boys, how many girls? On a map, 2 cm = 15 km. How many cm for 45 km? 4 workers build a fence in 9 days. How long for 6 workers? (Is this direct proportion?) If y = 42 when x = 6, find y when x = 11. A recipe uses 250g flour for 8 pancakes. How much for 20 pancakes? Which is better: 500ml for ₦1.80 or 750ml for ₦2.50? Solve: $frac{5}{8} = frac{x}{24}$ Solve: $frac{3}{x} = frac{12}{28}$ On scale 1:50, model boat is 12cm long. Real length? If 2.5 inches = 6.35cm, how many inches in 20cm? A printer prints 96 pages in 4 minutes. How many in 15 minutes? If 7:9 = x:45, find x. Water flows at 18 liters per minute. How much in 2.5 hours? Check if 4:7 = 12:21 is a true proportion. A shadow 4m long when person 1.6m tall. How tall if shadow 6m? Exchange rate: ₦1 = €0.92. How many Naira for €115? Paint: 3L covers 24m². How much for 40m²? If k = 3.5 and x = 8, find y. Show/Hide Answers Exercise 1: ₦7.80 (Unit rate: ₦0.65 per orange) Exercise 2: 490 km (Speed: 70 km/h) Exercise 3: 28 girls (3:4 = 21:28) Exercise 4: 6 cm (2:15 = x:45) Exercise 5: 6 days (NOT direct - more workers = less time: inverse proportion) Exercise 6: y = 77 (k = 7, y = 7×11) Exercise 7: 625g (250/8 = x/20) Exercise 8: 500ml: ₦0.0036/ml, 750ml: ₦0.0033/ml - 750ml better Exercise 9: x = 15 (5/8 = x/24) Exercise 10: x = 7 (3/x = 12/28) Exercise 11: 600cm or 6m (1:50 = 12:x) Exercise 12: 7.87 inches (2.5/6.35 = x/20) Exercise 13: 360 pages (96/4 = x/15) Exercise 14: x = 35 (7/9 = x/45) Exercise 15: 2700 liters (18 × 150 minutes) Exercise 16: Yes (4×21 = 7×12 = 84) Exercise 17: 2.4m (1.6/4 = x/6) Exercise 18: ₦125 (1/0.92 = x/115) Exercise 19: 5L (3/24 = x/40) Exercise 20: y = 28 (y = 3.5×8) Conclusion/Recap Excellent work! You've now mastered Proportional Reasoning: Solving Direct Proportion Problems. You've learned how to identify when two quantities are directly proportional, set up proportion equations, and solve them using multiple methods. Key Concepts to Remember: 1. Direct Proportion: When two quantities change at the same rate (y = kx) 2. Constant of Proportionality (k): The constant ratio y/x 3. Proportion Equation: $frac{a}{b} = frac{c}{d}$ means a:b = c:d 4. Cross-Multiplication: If $frac{a}{b} = frac{c}{d}$, then a×d = b×c 5. Unit Rate Method: Find value for one unit first 6. Setting Up: Keep same items in same positions in ratios 7. Checking: Always verify with alternative method Common Mistakes to Avoid: • Setting up proportion with items in wrong positions • Assuming all increasing relationships are proportional • Not checking if answer is reasonable • Forgetting to include units in final answer • Confusing direct and inverse proportion • Not simplifying ratios before solving Proportional Reasoning is used in shopping (price comparisons), cooking (recipe adjustments), traveling (speed and distance), maps (scale drawings), and currency exchange. Every time you compare prices per unit, adjust a recipe, or read a map, you're using proportional reasoning. Keep practicing by looking for proportions in your textbooks, in grocery stores, on road trips, and in recipes. Clip It! Share your ANSWER in the Chat. Indicate TITLE e.g Linear Equation 1. .....2. e.t.c
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