Proportional Graphs

lessonbunny.com

Answer Key & Explanations

Proportional Graphs · Grade 7 · Worksheet 1

1. A proportional relationship is graphed on a coordinate plane where the x-axis represents hours worked and the y-axis represents earnings in dollars. The line passes through points (0,0) and (5,87.50). If someone works for 12 hours, how much will they earn? Answer: 210 Solution: Find the constant of proportionality (earnings per hour) using the given points (0,0) and (5,87.50). Calculate the rate: 87.50 ÷ 5 = 17.50 dollars per hour. Apply this rate to 12 hours: 12 × 17.50 = 210 dollars.
   Full step-by-step solution

   Step 1: Find the constant of proportionality (earnings per hour) using the given points (0,0) and (5,87.50). Step 2: Calculate the rate: 87.50 ÷ 5 = 17.50 dollars per hour. Step 3: Apply this rate to 12 hours: 12 × 17.50 = 210 dollars. Step 4: The earnings for 12 hours of work is 210 dollars.

2. Liam is baking cookies for a school fundraiser. His recipe uses 2 cups of flour to make 24 cookies. He needs to make 180 cookies for the event. How many cups of flour should Liam use? Answer: 15 Solution: The recipe uses 2 cups of flour to make 24 cookies. Liam needs 180 cookies. We need to find how many cups of flour are needed for 180 cookies.
   Full step-by-step solution

   Step 1: Understand the problem. The recipe uses 2 cups of flour to make 24 cookies. Liam needs 180 cookies. We need to find how many cups of flour are needed for 180 cookies. Step 2: Find how many cups per cookie. From the recipe: 2 cups / 24 cookies = 2/24 = 1/12 cup of flour per cookie. Step 3: Multiply cups per cookie by the number of cookies needed. Flour needed = (1/12) cup per cookie × 180 cookies = 180 / 12 = 15 cups. Step 4: Conclusion. Liam should use 15 cups of flour to make 180 cookies. Alternative method using proportion: Let x = cups needed for 180 cookies. Set up the proportion: 2 cups / 24 cookies = x cups / 180 cookies Cross-multiply: 2 × 180 = 24 × x 360 = 24x x = 360 / 24 x = 15 cups. Both methods give the same answer: 15 cups.

3. Aroha is helping her uncle design a proportional scale model of a traditional Maori meeting house. The actual meeting house has a height of 9 meters. In their model, every 1.5 meters of actual height is represented by 4 centimeters. If the model's height is proportional to the actual height, how many centimeters tall should the model be? Answer: 24 Solution: Identify the constant of proportionality. For every 1.5 meters actual height, the model uses 4 centimeters. The constant k = model height / actual height = 4 / 1.5 = 8/3 centimeters per meter.
   Full step-by-step solution

   Step 1: Identify the constant of proportionality. For every 1.5 meters actual height, the model uses 4 centimeters. The constant k = model height / actual height = 4 / 1.5 = 8/3 centimeters per meter. Step 2: Use the equation y = kx, where x is the actual height and y is the model height. Here x = 9 meters and k = 8/3. Step 3: y = (8/3) × 9 = 72/3 = 24 centimeters. Step 4: Verify: The ratio 24 : 9 simplifies to 8 : 3, which matches 4 : 1.5 (since 4/1.5 = 8/3). The answer is 24.

4. A city is planning a new bike trail that needs to be paved. The paving crew can complete 3.5 kilometers of trail in 8 hours. If the crew works at the same rate, how many kilometers can they pave in a 40-hour work week? Answer: 17.5 Solution: Identify the rate of work. The crew paves 3.5 km in 8 hours.
   Full step-by-step solution

   Step 1: Identify the rate of work. The crew paves 3.5 km in 8 hours. Step 2: Calculate the paving rate per hour: 3.5 km ÷ 8 hours = 0.4375 km per hour Step 3: Multiply the hourly rate by the total hours in a work week: 0.4375 km/hour × 40 hours = 17.5 km Step 4: The crew can pave 17.5 kilometers in a 40-hour work week.

5. Tane is helping his uncle build a fence around a large garden. They find that 12 fence posts are needed to support 18 meters of fencing. If the garden requires a total of 45 meters of fencing, and the number of fence posts needed is proportional to the length of fencing, how many fence posts will Tane and his uncle need in total? Answer: 30 Solution: Recognize the proportional relationship. For 18 meters of fencing, 12 posts are needed. The relationship is posts = (2/3) * length.
   Full step-by-step solution

   Step 1: Recognize the proportional relationship. For 18 meters of fencing, 12 posts are needed. So the constant of proportionality k is posts per meter: k = 12/18 = 2/3. Step 2: The relationship is posts = (2/3) * length. Step 3: For 45 meters of fencing, calculate posts = (2/3) * 45. Step 4: Multiply: (2/3) * 45 = (2 * 45) / 3 = 90 / 3 = 30. Step 5: Therefore, Tane and his uncle need 30 fence posts. The answer is 30.

6. If y = 2.5x and x = 18, then y = ? Answer: 45 Solution: The equation given is y = 2.5x, and x = 18. Substitute the value of x into the equation: y = 2.5 * 18. Perform the multiplication: 2.5 * 18 = 45.
   Full step-by-step solution

   Step 1: The equation given is y = 2.5x, and x = 18. Step 2: Substitute the value of x into the equation: y = 2.5 * 18. Step 3: Perform the multiplication: 2.5 * 18 = 45. The answer is 45.

7. If y = 2.5x and x = 8, then y = ? Answer: 20 Solution: We are given that y = 2.5x and x = 8. Substitute the value of x into the equation. Since y = 2.5x and x = 8, we replace x with 8: y = 2.5 * 8 Multiply 2.5 by 8.
   Full step-by-step solution

   We are given that y = 2.5x and x = 8. Step 1: Substitute the value of x into the equation. Since y = 2.5x and x = 8, we replace x with 8: y = 2.5 * 8 Step 2: Multiply 2.5 by 8. First, note that 2.5 is the same as 5/2. So 2.5 * 8 = (5/2) * 8 = (5 * 8) / 2 = 40 / 2 = 20. Alternatively, you can multiply directly: 2.5 * 8 = 20. Step 3: Conclusion. Therefore, y = 20. Final answer: 20

8. A city is planning a new bike trail that needs to be paved. The construction crew can pave 2.5 kilometers of trail in 3 days. If the total trail length is 15 kilometers and the crew works at the same rate, how many days will it take to pave the entire trail? Answer: 18 Solution: Identify the rate of work. The crew paves 2.5 km in 3 days. Calculate the daily paving rate: 2.5 km ÷ 3 days = 0.833 km per day.
   Full step-by-step solution

   Step 1: Identify the rate of work. The crew paves 2.5 km in 3 days. Step 2: Calculate the daily paving rate: 2.5 km ÷ 3 days = 0.833 km per day. Step 3: Set up a proportion: If 2.5 km takes 3 days, then 15 km takes x days. Step 4: Write the proportion: 2.5/3 = 15/x Step 5: Cross multiply: 2.5 × x = 3 × 15 Step 6: Calculate: 2.5x = 45 Step 7: Solve for x: x = 45 ÷ 2.5 Step 8: Calculate: 45 ÷ 2.5 = 18 Step 9: The answer is 18 days.