Practice Problems

Test your understanding of direct and inverse proportions with these word problems.

Problem 1: The Bakery (Direct)

A bakery uses 12 cups of flour to make 30 large cookies. How many cups of flour are needed to make 75 cookies?

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Step 1: Identify the type. More cookies require more flour, so this is Direct Proportion. $$y = kx$$

Step 2: Find the constant ($k$). Let $y$ be flour and $x$ be cookies. $$k = \frac{y}{x} = \frac{12}{30} = 0.4 \text{ cups per cookie}$$

Step 3: Solve for 75 cookies. $$y = 0.4 \times 75 = 30$$ Answer: 30 cups of flour.

It takes 6 workers 10 days to build a small shed. If the owner wants the shed finished in only 4 days, how many workers are needed in total?

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Step 1: Identify the type. To finish in less time, you need more workers. This is Inverse Proportion. $$xy = k$$

Step 2: Find the constant ($k$). Let $x$ be workers and $y$ be days. $$k = 6 \times 10 = 60 \text{ (total man-days required)}$$

Step 3: Solve for 4 days. $$x \times 4 = 60$$ $$x = \frac{60}{4} = 15$$ Answer: 15 workers are needed.

A car travels 150 miles on 5 gallons of gas. How far can it travel on a full tank of 14 gallons?

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Step 1: Find the constant. $$k = \frac{150}{5} = 30 \text{ miles per gallon}$$

Step 2: Calculate for 14 gallons. $$d = 30 \times 14 = 420$$ Answer: 420 miles.

Three identical pumps can empty a water tank in 8 hours. How long would it take if 2 more pumps were added?

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Step 1: Note the total pumps. Starting pumps = 3. New pumps = $3 + 2 = 5$.

Step 2: Find the constant ($k$). $$k = 3 \text{ pumps} \times 8 \text{ hours} = 24$$

Step 3: Solve for 5 pumps. $$5 \times t = 24$$ $$t = \frac{24}{5} = 4.8 \text{ hours}$$ Answer: 4.8 hours (or 4 hours and 48 minutes).