Proportional Graphs

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Proportional Graphs · Grade 7 · Worksheet 2

1. Matiu is helping his family install solar panels on their roof. He notices that on a sunny day, the solar panels produce 12,500 watts of power in 5 hours. If the panels continue to produce power at this constant proportional rate, how many watts of power will they produce in 24 hours? Answer: 60,000 Solution: Find the constant rate of power production per hour. Divide the total watts by the number of hours: 12,500 watts / 5 hours = 2,500 watts per hour.
   Full step-by-step solution

   Step 1: Find the constant rate of power production per hour. Divide the total watts by the number of hours: 12,500 watts / 5 hours = 2,500 watts per hour. Step 2: Multiply the rate by the new number of hours to find the total power for 24 hours: 2,500 watts/hour * 24 hours = 60,000 watts. Step 3: Check: The relationship is proportional, so the graph of watts vs. hours is a straight line through (0,0). At 5 hours, the point is (5, 12,500). At 24 hours, the point is (24, 60,000). Both follow y = 2,500x. The answer is 60,000.

2. Moana is training for a race and runs at a constant speed. After 5 minutes, Moana has run 5 miles. If this is a proportional relationship, how many miles does Moana run per minute? Answer: 1 Solution: In a proportional relationship, the constant of proportionality (unit rate) is y ÷ x. Moana runs 5 miles in 5 minutes, so the unit rate is 5 ÷ 5 = 1. This means Moana runs 1 mile(s) per minute.
   Full step-by-step solution

   Step 1: In a proportional relationship, the constant of proportionality (unit rate) is y ÷ x. Step 2: Moana runs 5 miles in 5 minutes, so the unit rate is 5 ÷ 5 = 1. Step 3: This means Moana runs 1 mile(s) per minute. The answer is 1.

3. Matiu is tracking the distance a toy car travels over time on a straight track. The relationship is proportional and is graphed on a coordinate plane where the x-axis represents time in seconds and the y-axis represents distance in centimeters. The line passes through the origin (0,0) and the point (12, 48). If the car continues at this constant speed, what is the distance traveled in centimeters after 30 seconds? Answer: 120 Solution: The relationship is proportional, so the graph is a straight line through (0,0). The constant of proportionality (speed) is distance divided by time. Use the given point (12, 48).
   Full step-by-step solution

   Step 1: The relationship is proportional, so the graph is a straight line through (0,0). The constant of proportionality (speed) is distance divided by time. Step 2: Use the given point (12, 48). Speed = distance / time = 48 / 12 = 4 centimeters per second. Step 3: For a time of 30 seconds, distance = speed * time = 4 * 30 = 120 centimeters. The distance traveled after 30 seconds is 120 centimeters.

4. A city is planning a new bike trail that needs to be paved. The paving company can complete 2.5 kilometers of trail in 8 days working at their current rate. If the city needs 15 kilometers of trail paved for the grand opening in 6 weeks, will the company be able to complete the work on time? Show how many kilometers they can pave in the available time. Answer: 13.125 Solution: Find the daily paving rate: 2.5 km ÷ 8 days = 0.3125 km per day Convert 6 weeks to days: 6 weeks × 7 days/week = 42 days Calculate total distance that can be paved: 0.3125 km/day × 42 days = 13.125 km Compare to required distance: 13.125 km < 15 km The company can pave 13.125 kilometers in the…
   Full step-by-step solution

   Step 1: Find the daily paving rate: 2.5 km ÷ 8 days = 0.3125 km per day Step 2: Convert 6 weeks to days: 6 weeks × 7 days/week = 42 days Step 3: Calculate total distance that can be paved: 0.3125 km/day × 42 days = 13.125 km Step 4: Compare to required distance: 13.125 km < 15 km The company can pave 13.125 kilometers in the available time, which is less than the required 15 kilometers.

5. If y = 3x + 2, what is y when x = 5? Answer: 17 Solution: We are given the equation: y = 3x + 2 We are told x = 5. Substitute x = 5 into the equation. y = 3 * 5 + 2 Perform the multiplication first (order of operations).
   Full step-by-step solution

   We are given the equation: y = 3x + 2 We are told x = 5. Step 1: Substitute x = 5 into the equation. y = 3 * 5 + 2 Step 2: Perform the multiplication first (order of operations). 3 * 5 = 15 So now: y = 15 + 2 Step 3: Perform the addition. 15 + 2 = 17 Step 4: Write the final answer. y = 17 So when x = 5, y equals 17.

6. Isabella is filling orders for her online craft store. She knows that to make 12 beaded bracelets, she needs 84 inches of elastic cord. If she needs to fill an order for 27 bracelets, and she wants to graph the proportional relationship between the number of bracelets and the length of cord needed, what is the constant of proportionality (the slope) that she should use for her graph? Give your answer as a decimal. Answer: 7 Solution: Identify the two quantities: number of bracelets (x) and length of cord in inches (y). The relationship is proportional, so it can be written as y = kx, where k is the constant of proportionality.
   Full step-by-step solution

   Step 1: Identify the two quantities: number of bracelets (x) and length of cord in inches (y). Step 2: The relationship is proportional, so it can be written as y = kx, where k is the constant of proportionality. Step 3: We know that when x = 12 bracelets, y = 84 inches of cord. Step 4: Substitute into the equation: 84 = k * 12. Step 5: Solve for k by dividing both sides by 12: k = 84 / 12. Step 6: Calculate: 84 / 12 = 7. Step 7: The constant of proportionality is 7, meaning each bracelet requires 7 inches of cord. The answer is 7.

7. A construction company needs to transport 48,000 kilograms of materials using trucks that each carry 1,250 kilograms. The company has already used 12 trucks. How many more trucks are needed to transport all the materials? Answer: 27 Solution: Calculate how much material has been transported by the 12 trucks already used. 12 trucks × 1,250 kg/truck = 15,000 kg Calculate how much material still needs to be transported.
   Full step-by-step solution

   Step 1: Calculate how much material has been transported by the 12 trucks already used. 12 trucks × 1,250 kg/truck = 15,000 kg Step 2: Calculate how much material still needs to be transported. 48,000 kg total - 15,000 kg already transported = 33,000 kg remaining Step 3: Calculate how many more trucks are needed for the remaining material. 33,000 kg ÷ 1,250 kg/truck = 26.4 trucks Step 4: Since we can't use a fraction of a truck, we need to round up to the next whole number. 26.4 trucks rounds up to 27 trucks Step 5: Verify the answer. 27 trucks × 1,250 kg/truck = 33,750 kg 15,000 kg (already transported) + 33,750 kg = 48,750 kg This is slightly more than needed, which confirms we need to round up. The answer is 27.