Proportionality Constant

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Answer Key & Explanations

Proportionality Constant · Grade 7 · Worksheet 2

1. Mason is analyzing a line graph on a coordinate plane that shows a proportional relationship between the number of hours spent studying (x-axis) and the score on a test in points (y-axis). The line passes through the origin and the point (17, 391). What is the constant of proportionality k, which represents the number of points earned per hour of studying? Answer: 23 Solution: For a proportional relationship y = kx, the constant of proportionality k equals y divided by x for any point (x, y) on the line through the origin.
   Full step-by-step solution

   Step 1: For a proportional relationship y = kx, the constant of proportionality k equals y divided by x for any point (x, y) on the line through the origin. Step 2: The line passes through the point (17, 391), so x = 17 and y = 391. Step 3: Calculate k = y/x = 391/17. Step 4: Divide 391 by 17. Since 17 * 20 = 340, and 17 * 3 = 51, then 340 + 51 = 391, so 391/17 = 23. Step 5: The constant of proportionality is 23, meaning Mason earns 23 points per hour of studying.

2. A construction company needs to mix concrete using a specific ratio of cement to sand. For every 8 kilograms of cement, they use 12 kilograms of sand. If they use 240 kilograms of sand for a large project, what is the constant of proportionality that relates the amount of cement used to the amount of sand used? Answer: 2/3 Solution: For every 8 kilograms of cement, they use 12 kilograms of sand.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the ratio given in the problem** The problem says: For every 8 kilograms of cement, they use 12 kilograms of sand. So the ratio of cement to sand is: cement / sand = 8 / 12 --- **Step 2: Simplify the ratio** 8/12 simplifies by dividing numerator and denominator by 4: 8 ÷ 4 = 2 12 ÷ 4 = 3 So cement / sand = 2/3 --- **Step 3: Interpret the constant of proportionality** The problem asks for the constant of proportionality that relates the amount of cement used to the amount of sand used. That means: cement = k × sand From Step 2, we have: cement = (2/3) × sand So k = 2/3. --- **Step 4: Check with the given numbers for the large project** They use 240 kg of sand. Then cement = (2/3) × 240 = 160 kg. Check the original ratio: 160 kg cement / 240 kg sand = 160/240 = 16/24 = 8/12, which matches the original ratio. --- **Step 5: Final answer** The constant of proportionality is 2/3.

3. If y = 24x and y = 168, what is x? Answer: 7 Solution: Start with the equation y = 24x. Substitute the given value y = 168 into the equation: 168 = 24x. To solve for x, divide both sides by 24: x = 168 ÷ 24.
   Full step-by-step solution

   Step 1: Start with the equation y = 24x. Step 2: Substitute the given value y = 168 into the equation: 168 = 24x. Step 3: To solve for x, divide both sides by 24: x = 168 ÷ 24. Step 4: Calculate 168 ÷ 24 = 7. The answer is 7.

4. If y = 2.5x and y = 37.5, what is x? Answer: 15 Solution: The equation is y = 2.5x, and we know y = 37.5 Substitute the known value: 37.5 = 2.5x To solve for x, divide both sides by 2.5: x = 37.5 ÷ 2.5 Calculate the division: 37.5 ÷ 2.5 = 15 Therefore, x = 15
   Full step-by-step solution

   Step 1: The equation is y = 2.5x, and we know y = 37.5 Step 2: Substitute the known value: 37.5 = 2.5x Step 3: To solve for x, divide both sides by 2.5: x = 37.5 ÷ 2.5 Step 4: Calculate the division: 37.5 ÷ 2.5 = 15 Step 5: Therefore, x = 15

5. If y = 4.2x and y = 25.2, what is x? Answer: 6 Solution: Step 1: Start with the equation y = 4.2x Step 2: Substitute the given value: 25.2 = 4.2x Step 3: Divide both sides by 4.2 to solve for x: x = 25.2 ÷ 4.2 Step 4: Calculate the division: 25.2 ÷ 4.2 = 6 Step 5: Verify: 4.2 × 6 = 25.2, which matches the given y value The answer is 6.
   Full step-by-step solution

   Step 1: Start with the equation y = 4.2x Step 2: Substitute the given value: 25.2 = 4.2x Step 3: Divide both sides by 4.2 to solve for x: x = 25.2 ÷ 4.2 Step 4: Calculate the division: 25.2 ÷ 4.2 = 6 Step 5: Verify: 4.2 × 6 = 25.2, which matches the given y value The answer is 6.

6. Olivia is tracking the growth of a sunflower plant over several days. She records that after 3 days, the plant is 21 centimeters tall. After 7 days, it is 49 centimeters tall. Assuming the plant grows at a constant rate, what is the constant of proportionality that relates the plant's height in centimeters to the number of days? Answer: 7 Solution: The relationship is proportional, so height = k × days, where k is the constant of proportionality. Step 2: Use the first pair: 21 = k × 3. Divide both sides by 3: k = 21 ÷ 3 = 7.
   Full step-by-step solution

   Step 1: The relationship is proportional, so height = k × days, where k is the constant of proportionality. Step 2: Use the first pair: 21 = k × 3. Divide both sides by 3: k = 21 ÷ 3 = 7. Step 3: Check with the second pair: 49 = k × 7, so k = 49 ÷ 7 = 7. Both give the same k. Step 4: The constant of proportionality is 7, meaning the plant grows 7 centimeters per day. The answer is 7.

7. If y = 3x, what is the constant of proportionality? Answer: 3 Solution: y = 3x We need to find the constant of proportionality. Recall the definition of direct proportionality. If y is directly proportional to x, then y = k * x, where k is the constant of proportionality.
   Full step-by-step solution

   Step 1: Understand the problem. We are given the equation: y = 3x We need to find the constant of proportionality. Step 2: Recall the definition of direct proportionality. If y is directly proportional to x, then y = k * x, where k is the constant of proportionality. Step 3: Compare the given equation with the direct proportionality form. Given: y = 3x General form: y = k * x We see that the number multiplying x in the given equation is 3. Step 4: Identify the constant of proportionality. From the comparison, k = 3. Step 5: State the final answer. The constant of proportionality is 3.

8. A car rental company charges a fixed fee of $12500 plus $350 per day for renting a luxury vehicle. If the total cost C is directly proportional to the number of days d rented, what is the constant of proportionality? Answer: 350 Solution: The total cost C consists of a fixed fee of $12500 plus $350 per day.
   Full step-by-step solution

   Step 1: The total cost C consists of a fixed fee of $12500 plus $350 per day. Step 2: The relationship can be written as C = 12500 + 350d Step 3: For a relationship to be directly proportional, it must be of the form y = kx, where k is the constant of proportionality. Step 4: In this case, the $350 per day represents the rate at which the cost increases with each additional day rented. Step 5: Therefore, the constant of proportionality is $350 per day. The answer is 350.