Proportionality Constant

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Proportionality Constant · Grade 7 · Worksheet 3

1. 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.

2. A factory produces electronic components at a constant rate. The quality control team found that for every 125 components produced, 8 are defective. If the factory produces 15,000 components in a day, how many defective components should they expect? Answer: 960 Solution: Identify the ratio of defective components to total components: 8 defective per 125 total Set up a proportion where x represents the number of defective components in 15,000 total: 8/125 = x/15,000 Cross-multiply to solve for x: 8 × 15,000 = 125 × x Calculate: 120,000 = 125x Divide both sides by…
   Full step-by-step solution

   Step 1: Identify the ratio of defective components to total components: 8 defective per 125 total Step 2: Set up a proportion where x represents the number of defective components in 15,000 total: 8/125 = x/15,000 Step 3: Cross-multiply to solve for x: 8 × 15,000 = 125 × x Step 4: Calculate: 120,000 = 125x Step 5: Divide both sides by 125: x = 120,000 ÷ 125 Step 6: Calculate the division: 120,000 ÷ 125 = 960 Step 7: Therefore, they should expect 960 defective components.

3. A construction company needs to determine if their concrete mixing ratio is proportional. For a small test batch, they used 15 kilograms of cement with 45 kilograms of sand. For the actual foundation, the blueprint specifies using 85 kilograms of cement. If the relationship between cement and sand is proportional, how many kilograms of sand should be used for the foundation? Answer: 255 Solution: The mixing ratio of cement to sand should be proportional between the test batch and the foundation batch. Test batch: 15 kg cement, 45 kg sand. Foundation: 85 kg cement, unknown sand (let’s call it S kg).
   Full step-by-step solution

   Step 1: Understand the problem. The mixing ratio of cement to sand should be proportional between the test batch and the foundation batch. Test batch: 15 kg cement, 45 kg sand. Foundation: 85 kg cement, unknown sand (let’s call it S kg). Step 2: Set up the proportion. Since the ratio of cement to sand must be the same, we write: 15 kg cement / 45 kg sand = 85 kg cement / S kg sand. Step 3: Write the proportion in fraction form. 15/45 = 85/S Step 4: Simplify 15/45. 15/45 = 1/3 (because both 15 and 45 can be divided by 15). So 1/3 = 85/S. Step 5: Solve for S. Cross-multiply: 1 * S = 3 * 85 So S = 255. Step 6: Conclusion. For 85 kg of cement, they need 255 kg of sand to keep the same proportional mix. Final answer: 255

4. 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.

5. If y = 3.75x and y = 45, what is x? Answer: 12 Solution: Start with the equation y = 3.75x Substitute the given value y = 45 into the equation: 45 = 3.75x To solve for x, divide both sides by 3.75: x = 45 ÷ 3.75 Calculate 45 ÷ 3.75 = 12 Therefore, x = 12 The answer is 12.
   Full step-by-step solution

   Step 1: Start with the equation y = 3.75x Step 2: Substitute the given value y = 45 into the equation: 45 = 3.75x Step 3: To solve for x, divide both sides by 3.75: x = 45 ÷ 3.75 Step 4: Calculate 45 ÷ 3.75 = 12 Step 5: Therefore, x = 12 The answer is 12.

6. A factory produces 12,800 units of a product in 8 hours of operation. The production rate is constant. What is the constant of proportionality that relates the number of units produced to the number of hours worked? Answer: 1600 Solution: We are told that the factory produces 12,800 units in 8 hours, and the production rate is constant.
   Full step-by-step solution

   We are told that the factory produces 12,800 units in 8 hours, and the production rate is constant. This means the relationship between units produced and hours worked is a direct proportion: units = constant × hours Step 1: Write the proportional relationship. Let k be the constant of proportionality. Then: units = k × hours Step 2: Substitute the given values into the equation. 12,800 = k × 8 Step 3: Solve for k. Divide both sides by 8: k = 12,800 / 8 Step 4: Perform the division. 12,800 ÷ 8 = 1,600 Step 5: Interpret the result. The constant of proportionality is 1,600 units per hour. This means the factory produces 1,600 units every hour. Final answer: 1600

7. If y = 3x and y = 15, what is x? Answer: 5 Solution: 1) y = 3x 2) y = 15 Since both equations are equal to y, we can set them equal to each other: 3x = 15 To solve for x, divide both sides by 3: x = 15 / 3 x = 5 So the value of x is 5.
   Full step-by-step solution

   We are given two equations: 1) y = 3x 2) y = 15 Since both equations are equal to y, we can set them equal to each other: 3x = 15 To solve for x, divide both sides by 3: x = 15 / 3 Now perform the division: x = 5 So the value of x is 5.

8. A research team is studying the relationship between a country's population and the number of hospitals needed. They found that in one region, a population of 2.4 million people requires 60 hospitals. If this proportional relationship holds true, what is the constant of proportionality (hospitals per million people) that relates population to hospitals? Answer: 25 Solution: Identify the two quantities: population (in millions) and number of hospitals. The given data shows 2.4 million people require 60 hospitals.
   Full step-by-step solution

   Step 1: Identify the two quantities: population (in millions) and number of hospitals. Step 2: The given data shows 2.4 million people require 60 hospitals. Step 3: To find the constant of proportionality (hospitals per million people), divide the number of hospitals by the population in millions: 60 ÷ 2.4 Step 4: Calculate 60 ÷ 2.4 = 25 Step 5: This means for every 1 million people, 25 hospitals are needed. The constant of proportionality is 25 hospitals per million people.