Proportionality Constant

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Answer Key & Explanations

Proportionality Constant · Grade 7 · Worksheet 1

1. Hana is drawing a rectangle on a coordinate plane with vertices at (0,0), (32,0), (32,20), and (0,20). She then draws a straight line from the origin (0,0) through the point (24,15) to the top edge of the rectangle. If this line represents a proportional relationship between the horizontal distance (x) and vertical distance (y) from the origin, what is the constant of proportionality k? Answer: 0.625 Solution: The line passes through the origin and the point (24,15). For a proportional relationship, the constant of proportionality k is found by dividing y by x for any point on the line (except the origin).
   Full step-by-step solution

   Step 1: The line passes through the origin and the point (24,15). For a proportional relationship, the constant of proportionality k is found by dividing y by x for any point on the line (except the origin). Step 2: Use the point (24,15). Here, x = 24 and y = 15. Step 3: Calculate k = y/x = 15/24. Step 4: Simplify the fraction. Both 15 and 24 are divisible by 3: 15 ÷ 3 = 5, 24 ÷ 3 = 8, so 15/24 = 5/8. Step 5: Convert 5/8 to a decimal: 5 ÷ 8 = 0.625. Step 6: The constant of proportionality is 0.625. The answer is 0.625.

2. A factory produces electronic components at a constant rate. The quality control team found that for every 150 components produced, 12 are defective. If the factory produces 2,250 components in one day, how many defective components should they expect based on this proportional relationship? Answer: 180 Solution: Identify the ratio of defective components to total components: 12 defective / 150 total = 12/150 Simplify the ratio: 12/150 = 2/25 Apply this ratio to the new total of 2,250 components: (2/25) × 2,250 Calculate: 2,250 ÷ 25 = 90, then 90 × 2 = 180 Therefore, they should expect 180 defective…
   Full step-by-step solution

   Step 1: Identify the ratio of defective components to total components: 12 defective / 150 total = 12/150 Step 2: Simplify the ratio: 12/150 = 2/25 Step 3: Apply this ratio to the new total of 2,250 components: (2/25) × 2,250 Step 4: Calculate: 2,250 ÷ 25 = 90, then 90 × 2 = 180 Step 5: Therefore, they should expect 180 defective components out of 2,250 total components.

3. 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.

4. Charlotte runs a small bakery. She uses a constant amount of sugar for each batch of cookies she bakes. If she uses 12,500 grams of sugar to make 25 batches of cookies, what is the constant of proportionality that relates the total sugar used (in grams) to the number of batches of cookies? Answer: 500 Solution: Identify the proportional relationship: total sugar = k × number of batches. Step 2: Use the given values: 12,500 grams = k × 25 batches. Step 3: Solve for k by dividing both sides by 25: k = 12,500 ÷ 25.
   Full step-by-step solution

   Step 1: Identify the proportional relationship: total sugar = k × number of batches. Step 2: Use the given values: 12,500 grams = k × 25 batches. Step 3: Solve for k by dividing both sides by 25: k = 12,500 ÷ 25. Step 4: Calculate: 12,500 ÷ 25 = 500. Step 5: The constant of proportionality is 500, meaning Charlotte uses 500 grams of sugar per batch. The answer is 500.

5. If y = 3.75x and y = 67.5, what is x? Answer: 18 Solution: Start with the equation y = 3.75x Substitute the given value: 67.5 = 3.75x To solve for x, divide both sides by 3.75: x = 67.5 ÷ 3.75 Calculate 67.5 ÷ 3.75 = 18 Therefore, x = 18 The answer is 18.
   Full step-by-step solution

   Step 1: Start with the equation y = 3.75x Step 2: Substitute the given value: 67.5 = 3.75x Step 3: To solve for x, divide both sides by 3.75: x = 67.5 ÷ 3.75 Step 4: Calculate 67.5 ÷ 3.75 = 18 Step 5: Therefore, x = 18 The answer is 18.

6. If y = 37x and y = 851, what is x? Answer: 23 Solution: Start with the equation y = 37x. Substitute the given value y = 851 into the equation: 851 = 37x. To solve for x, divide both sides by 37: x = 851 ÷ 37.
   Full step-by-step solution

   Step 1: Start with the equation y = 37x. Step 2: Substitute the given value y = 851 into the equation: 851 = 37x. Step 3: To solve for x, divide both sides by 37: x = 851 ÷ 37. Step 4: Calculate 851 ÷ 37 = 23. The answer is 23.

7. 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

8. If y = 3.8x and y = 45.6, what is x? Answer: 12 Solution: Start with the equation y = 3.8x Substitute the given value: 45.6 = 3.8x To solve for x, divide both sides by 3.8: x = 45.6 ÷ 3.8 Calculate the division: 45.6 ÷ 3.8 = 12 The answer is 12.
   Full step-by-step solution

   Step 1: Start with the equation y = 3.8x Step 2: Substitute the given value: 45.6 = 3.8x Step 3: To solve for x, divide both sides by 3.8: x = 45.6 ÷ 3.8 Step 4: Calculate the division: 45.6 ÷ 3.8 = 12 The answer is 12.