Proportional Graphs

lessonbunny.com

Answer Key & Explanations

Proportional Graphs · Grade 7 · Worksheet 2

1. Liam is baking cookies for a school fundraiser. His recipe uses 2.5 cups of flour to make 20 cookies. He needs to make 180 cookies for the event. How many cups of flour should Liam use? Answer: 22.5 Solution: First, find out how many cups of flour are needed for 1 cookie. The recipe uses 2.5 cups of flour for 20 cookies. Flour per cookie = 2.5 cups ÷ 20 cookies 2.5 ÷ 20 = 0.125 cups per cookie Now, Liam needs 180 cookies.
   Full step-by-step solution

   First, find out how many cups of flour are needed for 1 cookie. The recipe uses 2.5 cups of flour for 20 cookies. Step 1: Flour per cookie = 2.5 cups ÷ 20 cookies 2.5 ÷ 20 = 0.125 cups per cookie Step 2: Now, Liam needs 180 cookies. Multiply flour per cookie by the number of cookies needed: 0.125 cups/cookie × 180 cookies Step 3: Calculate: 0.125 × 180 = 22.5 cups So, Liam should use 22.5 cups of flour.

2. A proportional relationship is graphed on a coordinate plane, showing the cost of printing photos at a photo lab. The line passes through points (0, 0) and (8, 12). If a customer wants to print 20 photos, what would be the total cost in dollars? Answer: 30 Solution: Identify the two points on the line: (0, 0) and (8, 12) Calculate the unit rate (cost per photo) using the slope formula: (12 - 0)/(8 - 0) = 12/8 = 1.5 This means each photo costs $1.50 Multiply the unit rate by the number of photos: 1.5 × 20 = 30 The total cost for 20 photos is $30
   Full step-by-step solution

   Step 1: Identify the two points on the line: (0, 0) and (8, 12) Step 2: Calculate the unit rate (cost per photo) using the slope formula: (12 - 0)/(8 - 0) = 12/8 = 1.5 Step 3: This means each photo costs $1.50 Step 4: Multiply the unit rate by the number of photos: 1.5 × 20 = 30 Step 5: The total cost for 20 photos is $30

3. If y = 3x and x = 8, then y = ? Answer: 24 Solution: y = 3x x = 8 Substitute the value of x into the first equation. Since y = 3x and x = 8, we replace x with 8: y = 3 * 8 Multiply 3 by 8. 3 * 8 = 24 State the value of y.
   Full step-by-step solution

   We are given two equations: y = 3x and x = 8 Step 1: Substitute the value of x into the first equation. Since y = 3x and x = 8, we replace x with 8: y = 3 * 8 Step 2: Multiply 3 by 8. 3 * 8 = 24 Step 3: State the value of y. Therefore, y = 24. Final answer: 24

4. Noah is designing a proportional garden path. The path requires 6 paving stones for every 1 meter of length. If Noah wants to build a path that is 16 meters long, how many paving stones will he need in total? Answer: 96 Solution: The relationship is proportional, so the number of paving stones (y) is directly proportional to the length in meters (x). The constant of proportionality (k) is 6 stones per meter, so the equation is y = 6x.
   Full step-by-step solution

   Step 1: The relationship is proportional, so the number of paving stones (y) is directly proportional to the length in meters (x). The constant of proportionality (k) is 6 stones per meter, so the equation is y = 6x. Step 2: Substitute x = 16 meters into the equation: y = 6 * 16. Step 3: Calculate: 6 * 16 = 96. Step 4: Therefore, Noah needs 96 paving stones for a 16-meter path. The answer is 96.

5. If y = 2.75x and x = 16, then y = ? Answer: 44 Solution: The equation is y = 2.75x Substitute x = 16 into the equation: y = 2.75 × 16 Multiply 2.75 by 16: 2.75 × 10 = 27.5 and 2.75 × 6 = 16.5 Add the results: 27.5 + 16.5 = 44 Therefore, y = 44 The answer is 44.
   Full step-by-step solution

   Step 1: The equation is y = 2.75x Step 2: Substitute x = 16 into the equation: y = 2.75 × 16 Step 3: Multiply 2.75 by 16: 2.75 × 10 = 27.5 and 2.75 × 6 = 16.5 Step 4: Add the results: 27.5 + 16.5 = 44 Step 5: Therefore, y = 44 The answer is 44.

6. If y = 2.75x + 8 and x = 24, then y = ? Answer: 74 Solution: Start with the equation y = 2.75x + 8 Substitute x = 24 into the equation: y = 2.75 * 24 + 8 Multiply first: 2.75 * 24 = 66 Then add: 66 + 8 = 74 Therefore, y = 74 The answer is 74.
   Full step-by-step solution

   Step 1: Start with the equation y = 2.75x + 8 Step 2: Substitute x = 24 into the equation: y = 2.75 * 24 + 8 Step 3: Multiply first: 2.75 * 24 = 66 Step 4: Then add: 66 + 8 = 74 Step 5: Therefore, y = 74 The answer is 74.

7. A rectangular garden is drawn on a coordinate plane with vertices at (2, 1), (8, 1), (8, 5), and (2, 5). If the gardener wants to create a proportional rectangular flower bed inside this garden where the length is reduced by a scale factor of 0.5 and the width is reduced by the same factor, what would be the coordinates of the vertices of the smaller flower bed if it is centered within the original garden? Answer: (3, 2), (7, 2), (7, 4), (3, 4) Solution: When working with proportional relationships in geometry, scaling an object by a factor affects all dimensions equally.
   Full step-by-step solution

   When working with proportional relationships in geometry, scaling an object by a factor affects all dimensions equally. To center a scaled object within its original, you need to calculate the center point and then adjust the coordinates based on the new dimensions. This concept is important in map reading, model building, and understanding similarity in geometric figures.

8. Liam is mixing paint to create a specific shade of purple. The recipe requires a ratio of 3 parts blue paint to 5 parts red paint. If Liam uses 12,000 milliliters of blue paint, how many liters of red paint does he need to maintain the correct proportion? Answer: 20 Solution: The recipe says 3 parts blue to 5 parts red. Blue : Red = 3 : 5 We know Liam uses 12,000 milliliters of blue paint. Let \( R \) be the milliliters of red paint needed.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the ratio** The recipe says 3 parts blue to 5 parts red. That means: Blue : Red = 3 : 5 --- **Step 2: Set up the proportion** We know Liam uses 12,000 milliliters of blue paint. Let \( R \) be the milliliters of red paint needed. From the ratio: \[ \frac{\text{Blue}}{\text{Red}} = \frac{3}{5} \] So: \[ \frac{12000}{R} = \frac{3}{5} \] --- **Step 3: Solve for \( R \)** Cross-multiply: \[ 12000 \times 5 = 3 \times R \] \[ 60000 = 3R \] \[ R = \frac{60000}{3} \] \[ R = 20000 \ \text{milliliters} \] --- **Step 4: Convert milliliters to liters** We know 1 liter = 1000 milliliters. So: \[ 20000 \ \text{mL} = \frac{20000}{1000} \ \text{liters} \] \[ = 20 \ \text{liters} \] --- **Final Answer:** 20 liters