Proportional Tables

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Proportional Tables · Grade 7 · Worksheet 1

1. x | y 2 | 18 4 | 36 6 | 54 8 | ? 10 | 90 Is this table proportional? If yes, find the constant of proportionality k and the missing y-value. Answer: k = 9, y = 72 Solution: For x=2, y=18: 18/2 = 9 For x=4, y=36: 36/4 = 9 For x=6, y=54: 54/6 = 9 For x=10, y=90: 90/10 = 9 All ratios equal 9, so the table is proportional. The constant of proportionality k = 9.
   Full step-by-step solution

   Step 1: Check the ratio y/x for each complete pair: For x=2, y=18: 18/2 = 9 For x=4, y=36: 36/4 = 9 For x=6, y=54: 54/6 = 9 For x=10, y=90: 90/10 = 9 All ratios equal 9, so the table is proportional. Step 2: The constant of proportionality k = 9. Step 3: Use k to find the missing y when x=8: y = k * x = 9 * 8 = 72. The answer is k = 9, y = 72.

2. x | y 5 | 25 10 | 50 15 | 75 20 | 100 25 | ? Answer: 125 Solution: Check if y/x is constant for each given pair. For (5, 25): 25/5 = 5 For (10, 50): 50/10 = 5 For (15, 75): 75/15 = 5 For (20, 100): 100/20 = 5 The constant of proportionality k = 5. The relationship is y = 5x.
   Full step-by-step solution

   Step 1: Check if y/x is constant for each given pair. For (5, 25): 25/5 = 5 For (10, 50): 50/10 = 5 For (15, 75): 75/15 = 5 For (20, 100): 100/20 = 5 The constant of proportionality k = 5. Step 2: The relationship is y = 5x. Step 3: Substitute x = 25 into the equation: y = 5 * 25 = 125. The answer is 125.

3. If y = 3x, what is y when x = 7? Answer: 21 Solution: We are given the equation: y = 3x We are also told: x = 7 Substitute the value of x into the equation. Since y = 3x and x = 7, we replace x with 7: y = 3 * 7 Multiply 3 by 7. 3 * 7 = 21 State the value of y.
   Full step-by-step solution

   We are given the equation: y = 3x We are also told: x = 7 Step 1: Substitute the value of x into the equation. Since y = 3x and x = 7, we replace x with 7: y = 3 * 7 Step 2: Multiply 3 by 7. 3 * 7 = 21 Step 3: State the value of y. Therefore, y = 21 Final answer: 21

4. Liam is mixing paint to create a specific shade of purple. The recipe requires mixing red and blue paint in a ratio of 3:5. If Liam uses 2.4 liters of blue paint, how many liters of red paint does he need to maintain the correct proportion? Answer: 1.44 Solution: We are told the ratio of red to blue paint is 3:5. That means for every 5 liters of blue paint, we need 3 liters of red paint. Write the ratio as a fraction.
   Full step-by-step solution

   We are told the ratio of red to blue paint is 3:5. That means for every 5 liters of blue paint, we need 3 liters of red paint. Step 1: Write the ratio as a fraction. Red / Blue = 3 / 5 Step 2: We know blue paint used is 2.4 liters. Let R = liters of red paint needed. So: R / 2.4 = 3 / 5 Step 3: Solve for R by multiplying both sides by 2.4: R = (3 / 5) × 2.4 Step 4: Calculate (3 / 5) first: 3 / 5 = 0.6 Step 5: Multiply 0.6 by 2.4: 0.6 × 2.4 = 1.44 Step 6: Conclusion: Liam needs 1.44 liters of red paint. Final answer: 1.44

5. A rectangular garden is drawn on a coordinate plane with vertices at (2, 1), (8, 1), (8, 5), and (2, 5). The gardener creates a proportional scale drawing where all dimensions are multiplied by 2.5. What are the coordinates of the vertices in the new scale drawing? Answer: (5, 2.5), (20, 2.5), (20, 12.5), (5, 12.5) Solution: A = (2, 1) B = (8, 1) C = (8, 5) D = (2, 5) The gardener makes a scale drawing where all dimensions are multiplied by 2.5. That means we multiply both the x-coordinates and y-coordinates by 2.5.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the problem** We have a rectangle with vertices: A = (2, 1) B = (8, 1) C = (8, 5) D = (2, 5) The gardener makes a scale drawing where **all dimensions are multiplied by 2.5**. That means we multiply both the x-coordinates and y-coordinates by 2.5. --- **Step 2: Multiply each coordinate by 2.5** For point A = (2, 1): New x = 2 × 2.5 = 5 New y = 1 × 2.5 = 2.5 So A' = (5, 2.5) For point B = (8, 1): New x = 8 × 2.5 = 20 New y = 1 × 2.5 = 2.5 So B' = (20, 2.5) For point C = (8, 5): New x = 8 × 2.5 = 20 New y = 5 × 2.5 = 12.5 So C' = (20, 12.5) For point D = (2, 5): New x = 2 × 2.5 = 5 New y = 5 × 2.5 = 12.5 So D' = (5, 12.5) --- **Step 3: Write the new vertices in order** The new vertices are: (5, 2.5), (20, 2.5), (20, 12.5), (5, 12.5) --- **Step 4: Check** Original width = 8 - 2 = 6 New width = 20 - 5 = 15 15 / 6 = 2.5 ✓ Original height = 5 - 1 = 4 New height = 12.5 - 2.5 = 10 10 / 4 = 2.5 ✓ --- **Final Answer:** (5, 2.5), (20, 2.5), (20, 12.5), (5, 12.5)

6. Mason is a quality control inspector at a factory that produces electronic components. He records the number of defective components found in different batch sizes. The table shows a proportional relationship between the batch size and the number of defective components found. | Batch Size | Defective Components | |------------|----------------------| | 27 | 12 | | 54 | 24 | | 81 | 36 | | 108 | 48 | If the factory produces a batch of 162 components, how many defective components should Mason expect to find, assuming the same proportional relationship holds? Answer: 72 Solution: Check that the relationship is proportional by finding the ratio of defective components to batch size for each row. Row 1: 12 / 27 = 12/27 = 4/9 Row 2: 24 / 54 = 24/54 = 4/9 Row 3: 36 / 81 = 36/81 = 4/9 Row 4: 48 / 108 = 48/108 = 4/9 All ratios are equal to 4/9, so the relationship is proportional.
   Full step-by-step solution

   Step 1: Check that the relationship is proportional by finding the ratio of defective components to batch size for each row. Row 1: 12 / 27 = 12/27 = 4/9 Row 2: 24 / 54 = 24/54 = 4/9 Row 3: 36 / 81 = 36/81 = 4/9 Row 4: 48 / 108 = 48/108 = 4/9 All ratios are equal to 4/9, so the relationship is proportional. The constant of proportionality is 4/9. Step 2: Use the constant of proportionality to find the number of defective components for a batch of 162. Let d be the number of defective components. d / 162 = 4/9 Step 3: Solve for d by multiplying both sides by 162. d = (4/9) * 162 Step 4: Simplify. First, divide 162 by 9: 162 / 9 = 18 Then, multiply 4 by 18: 4 * 18 = 72 So, d = 72. The answer is 72.

7. Isabella runs a small bakery. She uses a proportional relationship between the amount of flour (in cups) and the number of cookies she bakes. The table below shows some of her baking data. If Isabella uses 36 cups of flour, how many cookies will she bake? | Flour (cups) | Cookies baked | |--------------|---------------| | 8 | 184 | | 14 | 322 | | 22 | 506 | Answer: 828 Solution: Check if the relationship is proportional by finding the ratio of cookies to flour for each row. For 8 cups: 184 / 8 = 23 cookies per cup For 14 cups: 322 / 14 = 23 cookies per cup For 22 cups: 506 / 22 = 23 cookies per cup The constant ratio is 23 cookies per cup.
   Full step-by-step solution

   Step 1: Check if the relationship is proportional by finding the ratio of cookies to flour for each row. For 8 cups: 184 / 8 = 23 cookies per cup For 14 cups: 322 / 14 = 23 cookies per cup For 22 cups: 506 / 22 = 23 cookies per cup The constant ratio is 23 cookies per cup. Step 2: Use the constant ratio to find cookies for 36 cups. 36 cups * 23 cookies per cup = 828 cookies Isabella will bake 828 cookies using 36 cups of flour.