Proportionality Constant

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Proportionality Constant · Grade 7 · Worksheet 2

1. Liam is mixing paint to create a specific shade of purple. The recipe requires mixing blue and red paint in a constant ratio. When Liam uses 12 ounces of blue paint, he needs 8 ounces of red paint to get the perfect color. If Liam uses 27 ounces of blue paint, how many ounces of red paint should he use to maintain the same proportional relationship? Answer: 18 Solution: We are told that the ratio of blue paint to red paint must stay constant. First, find the ratio from the given situation. Blue = 12 ounces, Red = 8 ounces.
   Full step-by-step solution

   We are told that the ratio of blue paint to red paint must stay constant. First, find the ratio from the given situation. Step 1: Write the ratio of blue to red from the first case. Blue = 12 ounces, Red = 8 ounces. So the ratio of blue to red is 12 : 8. Step 2: Simplify the ratio. 12 : 8 = 12/8 = 3/2. So the ratio is 3 parts blue to 2 parts red. Step 3: Apply the same ratio to the new amount of blue paint. New blue = 27 ounces. Let R = ounces of red needed. We set up the proportion: Blue / Red = 3 / 2 27 / R = 3 / 2 Step 4: Solve for R. Cross-multiply: 27 * 2 = 3 * R 54 = 3 * R R = 54 / 3 R = 18 Step 5: Conclusion. Liam needs 18 ounces of red paint for 27 ounces of blue paint to keep the same shade of purple. ANSWER: 18

2. A factory produces electronic components at a constant rate. The quality control team found that for every 3 hours of production, the factory makes 2,400 components. If the factory operates continuously for 18 hours, how many components will it produce? Answer: 14400 Solution: Identify the constant of proportionality (production rate per hour) The factory makes 2,400 components in 3 hours, so the rate is 2,400 ÷ 3 = 800 components per hour For 18 hours of production: 800 × 18 = 14,400 components 2,400 components in 3 hours means the ratio is 2,400:3 For 18 hours: 18 ÷…
   Full step-by-step solution

   Step 1: Identify the constant of proportionality (production rate per hour) The factory makes 2,400 components in 3 hours, so the rate is 2,400 ÷ 3 = 800 components per hour Step 2: Apply this rate to the new time period For 18 hours of production: 800 × 18 = 14,400 components Step 3: Verify the proportional relationship 2,400 components in 3 hours means the ratio is 2,400:3 For 18 hours: 18 ÷ 3 = 6 times longer 2,400 × 6 = 14,400 components Both methods give the same result: 14,400 components.

3. If y = 12.5x and y = 150, what is x? Answer: 12 Solution: Start with the equation y = 12.5x Substitute the given value y = 150 into the equation: 150 = 12.5x To solve for x, divide both sides by 12.5: x = 150 ÷ 12.5 Calculate 150 ÷ 12.5 = 12 Therefore, x = 12 The answer is 12.
   Full step-by-step solution

   Step 1: Start with the equation y = 12.5x Step 2: Substitute the given value y = 150 into the equation: 150 = 12.5x Step 3: To solve for x, divide both sides by 12.5: x = 150 ÷ 12.5 Step 4: Calculate 150 ÷ 12.5 = 12 Step 5: Therefore, x = 12 The answer is 12.

4. A factory produces 3,600 units of a product in 8 hours of operation. The production rate remains constant throughout the day. If the factory operates for 15 hours, how many units will it produce? Answer: 6750 Solution: Find the production rate per hour. The factory produces 3600 units in 8 hours. So, production rate = Total units / Total hours = 3600 / 8.
   Full step-by-step solution

   Step 1: Find the production rate per hour. The factory produces 3600 units in 8 hours. So, production rate = Total units / Total hours = 3600 / 8. 3600 ÷ 8 = 450 units per hour. Step 2: Calculate units produced in 15 hours. If the factory runs for 15 hours at 450 units per hour, then total units = Rate × Time = 450 × 15. Step 3: Multiply. 450 × 15 = 450 × (10 + 5) = 4500 + 2250 = 6750. Step 4: Conclusion. The factory will produce 6750 units in 15 hours.

5. A proportional relationship is represented on a coordinate plane by a straight line that passes through the origin and the point (15, 24). What is the constant of proportionality for this relationship? Answer: 1.6 Solution: In a proportional relationship represented by a line through the origin, the constant of proportionality is the ratio of y to x for any point on the line. The line passes through the point (15, 24), so we can use these coordinates to find the ratio.
   Full step-by-step solution

   Step 1: In a proportional relationship represented by a line through the origin, the constant of proportionality is the ratio of y to x for any point on the line. Step 2: The line passes through the point (15, 24), so we can use these coordinates to find the ratio. Step 3: Calculate the ratio: y/x = 24/15 Step 4: Simplify the fraction: 24 ÷ 15 = 1.6 Step 5: Therefore, the constant of proportionality is 1.6.

6. Emma is working at a community garden where the number of tomato plants is proportional to the number of garden beds. In one section, 7 garden beds contain 133 tomato plants. What is the constant of proportionality that relates the number of tomato plants to the number of garden beds? Answer: 19 Solution: Identify the proportional relationship: tomato plants = k × garden beds Use the given values: 133 tomato plants = k × 7 garden beds Solve for k by dividing both sides by 7: k = 133 ÷ 7 Calculate: 133 ÷ 7 = 19 The constant of proportionality is 19, meaning each garden bed has 19 tomato plants.
   Full step-by-step solution

   Step 1: Identify the proportional relationship: tomato plants = k × garden beds Step 2: Use the given values: 133 tomato plants = k × 7 garden beds Step 3: Solve for k by dividing both sides by 7: k = 133 ÷ 7 Step 4: Calculate: 133 ÷ 7 = 19 Step 5: The constant of proportionality is 19, meaning each garden bed has 19 tomato plants. The answer is 19.

7. Kaia is a beekeeper who sells honey at a local market. The amount of honey produced is proportional to the number of beehives she has. Last season, Kaia had 15 beehives and harvested 270 kilograms of honey. This season, she expanded her apiary to 24 beehives. What is the constant of proportionality that relates the kilograms of honey produced to the number of beehives? Answer: 18 Solution: The relationship is proportional, so it follows the form y = kx, where y is the kilograms of honey, x is the number of beehives, and k is the constant of proportionality.
   Full step-by-step solution

   Step 1: The relationship is proportional, so it follows the form y = kx, where y is the kilograms of honey, x is the number of beehives, and k is the constant of proportionality. Step 2: From last season, we know that 270 kilograms of honey come from 15 beehives. So, 270 = k * 15. Step 3: To find k, divide both sides of the equation by 15: k = 270 / 15. Step 4: Calculate 270 divided by 15. Since 15 * 18 = 270, then 270 / 15 = 18. Step 5: The constant of proportionality is 18. This means each beehive produces 18 kilograms of honey. The answer is 18.

8. A factory produces electronic components at a constant rate. The production manager notices that in 3.5 hours, the factory produces 1,890 components. If the factory operates for 12 hours, how many components will it produce? Answer: 6480 Solution: Find the production rate per hour by dividing total components by hours: 1,890 ÷ 3.5 1,890 ÷ 3.5 = 540 components per hour Multiply the hourly rate by the total hours: 540 × 12 540 × 12 = 6,480 The factory will produce 6,480 components in 12 hours.
   Full step-by-step solution

   Step 1: Find the production rate per hour by dividing total components by hours: 1,890 ÷ 3.5 Step 2: 1,890 ÷ 3.5 = 540 components per hour Step 3: Multiply the hourly rate by the total hours: 540 × 12 Step 4: 540 × 12 = 6,480 The factory will produce 6,480 components in 12 hours.