Proportional Reasoning: – Solving problems involving direct proportion.

PROPORTIONAL REASONING Solving problems involving direct proportion. Grade 7 Mathematics: Proportional Reasoning – Solving problems involving direct proportion Subtopic Navigator Introduction Concept of Direct Proportion Using Ratios in Direct Proportion Formulas for Direct Proportion Word Problems on Direct Proportion Applications and Mixed Problems Cumulative Exercises Conclusion Lesson Objectives Understand the meaning of direct proportion. Use ratios to solve direct proportion problems. Apply the formula of direct proportion in calculations. Solve real-life word problems involving direct proportion. Lesson Introduction Direct proportion is a relationship between two quantities in which, as one increases, the other increases at the same rate. For example, the cost of buying oranges is directly proportional to the number of oranges purchased: more oranges mean more cost. If two variables [latex]x[/latex] and [latex]y[/latex] are in direct proportion, then [latex]frac{y}{x}[/latex] is constant, or equivalently, [latex]y = kx[/latex] where [latex]k[/latex] is the constant of proportionality. Concept of Direct Proportion When two values increase (or decrease) together in such a way that their ratio remains constant, they are said to be in direct proportion. Example 1: If 5 pencils cost ₦100, find the cost of 8 pencils (assuming the cost is directly proportional to the number of pencils). Solution: Ratio of cost to pencils is constant. [latex]frac{100}{5} = frac{C}{8}[/latex] [latex]C = frac{100}{5} times 8 = 160[/latex] The cost of 8 pencils is ₦160. Example 2: If 3 kg of rice costs ₦450, find the cost of 7 kg of rice. Solution: [latex]frac{450}{3} = frac{C}{7}[/latex] [latex]C = frac{450}{3} times 7 = 1050[/latex] The cost of 7 kg of rice is ₦1050. Exercises (Concept of Direct Proportion) If 2 litres of milk cost ₦800, what is the cost of 5 litres? If 7 pens cost ₦1400, how much will 10 pens cost? Using Ratios in Direct Proportion Ratios help us compare proportional relationships. If [latex]a : b = c : d[/latex], then we can solve for missing terms by cross multiplication. Example 3: If 4 mangoes cost ₦200, what is the cost of 9 mangoes? Solution: Ratio: [latex]4 : 200 = 9 : C[/latex]. Cross multiply: [latex]4C = 200 times 9[/latex]. [latex]C = frac{1800}{4} = 450[/latex]. The cost of 9 mangoes is ₦450. Example 4: A car travels 150 km using 12 litres of fuel. How much fuel will it need to travel 250 km? Solution: [latex]150 : 12 = 250 : F[/latex]. Cross multiply: [latex]150F = 12 times 250 = 3000[/latex]. [latex]F = frac{3000}{150} = 20[/latex]. The car will need 20 litres of fuel. Exercises (Using Ratios) If 12 apples cost ₦600, find the cost of 20 apples. A machine produces 45 items in 3 hours. How many items will it produce in 8 hours? Formulas for Direct Proportion The general formula for direct proportion is: [latex]y = kx[/latex], where [latex]k[/latex] is the constant of proportionality. Example 5: If [latex]y[/latex] is directly proportional to [latex]x[/latex], and [latex]y = 12[/latex] when [latex]x = 4[/latex], find [latex]k[/latex] and [latex]y[/latex] when [latex]x = 9[/latex]. Solution: [latex]y = kx[/latex]. When [latex]y = 12, x = 4[/latex]. [latex]12 = k times 4 implies k = 3[/latex]. When [latex]x = 9[/latex], [latex]y = 3 times 9 = 27[/latex]. Exercises (Formulas) If [latex]y[/latex] is directly proportional to [latex]x[/latex], and [latex]y = 20[/latex] when [latex]x = 5[/latex], find [latex]y[/latex] when [latex]x = 15[/latex]. If [latex]y[/latex] is directly proportional to [latex]x[/latex], and [latex]y = 48[/latex] when [latex]x = 8[/latex], find [latex]y[/latex] when [latex]x = 12[/latex]. Word Problems on Direct Proportion Direct proportion is very common in real-life situations like speed, distance, cost, recipes, and scaling. Example 6: If a worker is paid ₦15,000 for 10 days of work, how much will he be paid for 24 days? Solution: [latex]10 : 15000 = 24 : P[/latex]. Cross multiply: [latex]10P = 15000 times 24[/latex]. [latex]P = 36000[/latex]. He will be paid ₦36,000. Example 7: A recipe for 4 people requires 600 g of flour. How much flour is needed for 10 people? Solution: [latex]frac{600}{4} = frac{F}{10}[/latex]. [latex]F = frac{600}{4} times 10 = 1500[/latex]. 1500 g of flour is needed. Exercises (Word Problems) If 8 workers can build a wall in 12 days, how many days will 2 workers take (assuming direct proportion)? A bus covers 180 km in 3 hours. How far will it cover in 7 hours? Applications and Mixed Problems Example 8:If 15 books cost ₦9000, find the cost of 27 books.Solution: [latex]frac{9000}{15} = frac{C}{27} implies C = 16200[/latex]. Cost is ₦16,200. Example 9:A car moves 120 km with 8 litres of petrol. How much distance will it cover with 20 litres?Solution: [latex]frac{120}{8} = frac{D}{20} implies D = 300[/latex]. Distance = 300 km. Example 10:A worker earns ₦2500 for 5 hours. How much will he earn for 14 hours?Solution: [latex]frac{2500}{5} = frac{E}{14} implies E = 7000[/latex]. Exercises (Applications) A man walks 9 km in 2 hours. How far will he walk in 5 hours? ₦8400 is shared equally among 12 children. How much will 20 children receive if the same amount is to be shared equally? Cumulative Exercises If 6 notebooks cost ₦2400, find the cost of 15 notebooks. 5 workers can complete a task in 12 days. How many days will 15 workers take? If ₦1500 is paid for 3 kg of beans, how much will 12 kg cost? A train covers 90 km in 1.5 hours. How far will it cover in 4 hours? If 8 litres of paint covers 56 m², how many litres are needed to cover 140 m²? ₦5000 is paid for 20 litres of petrol. Find the cost of 32 litres. If 4 men finish digging in 6 days, how many days will 12 men take? A printer produces 200 pages in 10 minutes. How many pages will it produce in 45 minutes? If 7 kg of sugar costs ₦3500, what is the cost of 20 kg? A car covers 300 km in 5 hours. How long will it take to cover 720 km? Conclusion/Recap In this lesson, we have studied direct proportion and its applications. We saw that if two quantities are directly proportional, their ratio is constant, and we can solve problems using ratios or the formula [latex]y = kx[/latex]. Direct proportion is widely used in real-life contexts like costs, recipes, fuel consumption, and wages. Clip It! Share your ANSWER in the Chat. Indicate TITLE e.g Linear Equation 1. .....2. e.t.c