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Proportionality Constant

Grade 7 · Ratios · Worksheet 1

  1. If y = 14x and y = 168, what is x? Answer: ______________
  2. A proportional relationship is represented on a coordinate plane by a straight line that passes through the origin and the point (15, 21). What is the constant of proportionality for this relationship? Answer: ______________
  3. A factory produces 3,600 widgets in 8 hours of operation. The production rate remains constant throughout the day. If the factory operates for 15 hours, how many widgets will it produce? Answer: ______________
  4. If y = 2.5x, what is the constant of proportionality? Answer: ______________
  5. A factory produces electronic components at a constant rate. The quality control team found that for every 8 hours of production, 1,200 components pass inspection. If the factory operates for 42 hours in a week, how many components will pass inspection, assuming the production rate remains constant? Answer: ______________
  6. Emma runs a small organic farm. She notices that the total weight of carrots harvested is directly proportional to the number of rows planted. When she plants 9 rows, she harvests 153 kilograms of carrots. What is the constant of proportionality that relates the total weight of carrots (in kilograms) to the number of rows planted? Answer: ______________
  7. A factory produces 4,800 units of a product in 6 hours. The production rate remains constant throughout the day. If the factory operates for 15 hours, how many units will it produce? Answer: ______________
  8. A proportional relationship is represented on a coordinate plane by a straight line passing through the origin and the point (12, 18). What is the constant of proportionality for this relationship? Answer: ______________
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Answer Key & Explanations

Proportionality Constant · Grade 7 · Worksheet 1

  1. If y = 14x and y = 168, what is x? Answer: 12 Solution: Start with the equation y = 14x. Substitute the given value y = 168: 168 = 14x. To solve for x, divide both sides by 14: x = 168 ÷ 14.
    Full step-by-step solution

    Step 1: Start with the equation y = 14x. Step 2: Substitute the given value y = 168: 168 = 14x. Step 3: To solve for x, divide both sides by 14: x = 168 ÷ 14. Step 4: Calculate 168 ÷ 14 = 12. The answer is 12.

  2. A proportional relationship is represented on a coordinate plane by a straight line that passes through the origin and the point (15, 21). What is the constant of proportionality for this relationship? Answer: 1.4 Solution: Identify the coordinates of the point on the line: (15, 21) In a proportional relationship, the constant of proportionality k equals y/x for any point (x,y) on the line Calculate k = 21/15 Simplify the fraction: 21 ÷ 3 = 7 and 15 ÷ 3 = 5, so 21/15 = 7/5 Convert to decimal: 7/5 = 1.4 The constant…
    Full step-by-step solution

    Step 1: Identify the coordinates of the point on the line: (15, 21) Step 2: In a proportional relationship, the constant of proportionality k equals y/x for any point (x,y) on the line Step 3: Calculate k = 21/15 Step 4: Simplify the fraction: 21 ÷ 3 = 7 and 15 ÷ 3 = 5, so 21/15 = 7/5 Step 5: Convert to decimal: 7/5 = 1.4 Step 6: The constant of proportionality is 1.4

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

    Step 1: Find the production rate per hour. We know the factory produces 3,600 widgets in 8 hours. So, widgets per hour = 3600 ÷ 8. 3600 ÷ 8 = 450 widgets per hour. Step 2: Multiply the hourly production by the new number of hours. The factory now operates for 15 hours. Total widgets = 450 × 15. Step 3: Calculate 450 × 15. First, 450 × 10 = 4500. Then, 450 × 5 = 2250. Add them: 4500 + 2250 = 6750. Step 4: State the final answer. The factory will produce 6750 widgets in 15 hours.

  4. If y = 2.5x, what is the constant of proportionality? Answer: 2.5 Solution: The equation is y = 2.5x In the form y = kx, k represents the constant of proportionality Comparing y = 2.5x with y = kx, we see that k = 2.5 Therefore, the constant of proportionality is 2.5
    Full step-by-step solution

    Step 1: The equation is y = 2.5x Step 2: In the form y = kx, k represents the constant of proportionality Step 3: Comparing y = 2.5x with y = kx, we see that k = 2.5 Step 4: Therefore, the constant of proportionality is 2.5

  5. A factory produces electronic components at a constant rate. The quality control team found that for every 8 hours of production, 1,200 components pass inspection. If the factory operates for 42 hours in a week, how many components will pass inspection, assuming the production rate remains constant? Answer: 6300 Solution: 1,200 components ÷ 8 hours = 150 components per hour Calculate total components for 42 hours 150 components/hour × 42 hours = 6,300 components The answer is 6,300 components.
    Full step-by-step solution

    Step 1: Find the production rate per hour 1,200 components ÷ 8 hours = 150 components per hour Step 2: Calculate total components for 42 hours 150 components/hour × 42 hours = 6,300 components The answer is 6,300 components.

  6. Emma runs a small organic farm. She notices that the total weight of carrots harvested is directly proportional to the number of rows planted. When she plants 9 rows, she harvests 153 kilograms of carrots. What is the constant of proportionality that relates the total weight of carrots (in kilograms) to the number of rows planted? Answer: 17 Solution: Identify the proportional relationship: total weight (in kg) = k × number of rows. Substitute the given values: 153 = k × 9. Solve for k by dividing both sides by 9: k = 153 ÷ 9.
    Full step-by-step solution

    Step 1: Identify the proportional relationship: total weight (in kg) = k × number of rows. Step 2: Substitute the given values: 153 = k × 9. Step 3: Solve for k by dividing both sides by 9: k = 153 ÷ 9. Step 4: Perform the division: 153 ÷ 9 = 17. Step 5: The constant of proportionality is 17, meaning each row produces 17 kilograms of carrots. The answer is 17.

  7. A factory produces 4,800 units of a product in 6 hours. The production rate remains constant throughout the day. If the factory operates for 15 hours, how many units will it produce? Answer: 12000 Solution: Find the production rate per hour. The factory produces 4800 units in 6 hours. So, production rate = total units ÷ total hours = 4800 ÷ 6.
    Full step-by-step solution

    Step 1: Find the production rate per hour. The factory produces 4800 units in 6 hours. So, production rate = total units ÷ total hours = 4800 ÷ 6. 4800 ÷ 6 = 800 units per hour. Step 2: Use the rate to find production for 15 hours. If the factory makes 800 units each hour, then in 15 hours: Units produced = rate × hours = 800 × 15. Step 3: Calculate 800 × 15. 800 × 15 = 800 × (10 + 5) = 800 × 10 + 800 × 5 = 8000 + 4000 = 12000. Final answer: The factory will produce 12000 units in 15 hours.

  8. A proportional relationship is represented on a coordinate plane by a straight line passing through the origin and the point (12, 18). What is the constant of proportionality for this relationship? Answer: 1.5 Solution: A proportional relationship means the graph is a straight line through the origin (0,0) and another point (12, 18). In such a case, the equation is of the form y = k * x, where k is the constant of proportionality.
    Full step-by-step solution

    Step 1: Understand the problem. A proportional relationship means the graph is a straight line through the origin (0,0) and another point (12, 18). In such a case, the equation is of the form y = k * x, where k is the constant of proportionality. Step 2: Identify the coordinates. The line passes through (0,0) and (12, 18). So for x = 12, y = 18. Step 3: Use the formula for the constant of proportionality. k = y / x Substitute the given point (but not the origin, since 0/0 is undefined): k = 18 / 12 Step 4: Simplify the fraction. 18 divided by 12 is the same as (18 ÷ 6) / (12 ÷ 6) = 3/2. 3/2 = 1.5 Step 5: Conclusion. The constant of proportionality is 1.5. Final answer: 1.5