Proportionality Constant

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Proportionality Constant · Grade 7 · Worksheet 3

1. 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 150 kilograms of cement for a large project, how many kilograms of sand should they use to maintain the same proportional mixture? Answer: 225 Solution: The problem says: for every 8 kg of cement, they use 12 kg of sand.
   Full step-by-step solution

   Let's go step-by-step. Step 1: Understand the ratio The problem says: for every 8 kg of cement, they use 12 kg of sand. So the ratio of cement to sand is 8 : 12. Step 2: Simplify the ratio (optional, but helps understanding) 8 : 12 can be simplified by dividing both numbers by 4: 8 ÷ 4 = 2 12 ÷ 4 = 3 So the ratio is 2 : 3. This means for every 2 kg of cement, you need 3 kg of sand. Step 3: Set up the proportion We know they use 150 kg of cement. Let S be the required sand in kg. From the ratio 8 : 12, we have: 8 kg cement / 12 kg sand = 150 kg cement / S kg sand So: 8 / 12 = 150 / S Step 4: Solve for S Cross-multiply: 8 × S = 12 × 150 8S = 1800 S = 1800 / 8 S = 225 Step 5: Conclusion They need 225 kg of sand to maintain the same proportional mixture. Final answer: 225

2. y = 37x, 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.

3. A factory produces electronic components at a constant rate. The production manager notices that in 3.5 hours, the factory produces 1,750 components. If the factory operates for 12 hours, how many components will it produce? Answer: 6000 Solution: Find the production rate per hour by dividing total components by hours: 1,750 ÷ 3.5 = 500 components per hour Multiply the hourly rate by the total operating time: 500 × 12 = 6,000 components The factory will produce 6,000 components in 12 hours.
   Full step-by-step solution

   Step 1: Find the production rate per hour by dividing total components by hours: 1,750 ÷ 3.5 = 500 components per hour Step 2: Multiply the hourly rate by the total operating time: 500 × 12 = 6,000 components Step 3: The factory will produce 6,000 components in 12 hours.

4. If y = 23x and y = 529, what is x? Answer: 23 Solution: Start with the equation y = 23x. Here, k = 23. Substitute the given value y = 529 into the equation: 529 = 23x.
   Full step-by-step solution

   Step 1: Start with the equation y = 23x. Here, k = 23. Step 2: Substitute the given value y = 529 into the equation: 529 = 23x. Step 3: To solve for x, divide both sides by 23: x = 529 ÷ 23. Step 4: Calculate 529 ÷ 23 = 23. The answer is 23.

5. A factory produces 450 units of a product in 6 hours. If the production rate remains constant, how many units will be produced in 10 hours? Answer: 750 Solution: The factory produces 450 units in 6 hours. We need to find how many units are produced in 10 hours at the same rate. Find the production rate per hour.
   Full step-by-step solution

   Step 1: Understand the problem. The factory produces 450 units in 6 hours. We need to find how many units are produced in 10 hours at the same rate. Step 2: Find the production rate per hour. Units produced per hour = Total units / Total hours Units per hour = 450 / 6 450 ÷ 6 = 75 So, the factory produces 75 units per hour. Step 3: Calculate units produced in 10 hours. Units in 10 hours = Units per hour × 10 Units in 10 hours = 75 × 10 75 × 10 = 750 Step 4: Conclusion. At the same constant rate, the factory will produce 750 units in 10 hours. ANSWER: 750

6. 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

7. y = 42x, y = 3024 Answer: 72 Solution: Identify the equation y = 42x. Here, k = 42. Substitute y = 3024 into the equation: 3024 = 42x.
   Full step-by-step solution

   Step 1: Identify the equation y = 42x. Here, k = 42. Step 2: Substitute y = 3024 into the equation: 3024 = 42x. Step 3: Solve for x by dividing both sides by 42: x = 3024 ÷ 42. Step 4: Calculate 3024 ÷ 42 = 72. The answer is 72.

8. 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.

9. A proportional relationship is represented on a coordinate plane by a straight line passing through the origin and the point (15, 24). What is the constant of proportionality for this relationship? Answer: 1.6 Solution: Identify the coordinates of the point the line passes through: (15, 24). In a proportional relationship, the constant of proportionality (k) is the ratio of y to x for any point on the line (k = y/x).
   Full step-by-step solution

   Step 1: Identify the coordinates of the point the line passes through: (15, 24). Step 2: In a proportional relationship, the constant of proportionality (k) is the ratio of y to x for any point on the line (k = y/x). Step 3: Calculate the ratio: k = 24 / 15. Step 4: Simplify the fraction: 24 ÷ 15 = 1.6. Step 5: The constant of proportionality is 1.6.