Proportional Tables

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Proportional Tables · Grade 7 · Worksheet 2

1. A construction company needs to mix concrete using a proportional relationship between cement and sand. The table shows that 8 bags of cement are mixed with 24 bags of sand for a small project, and 12 bags of cement are mixed with 36 bags of sand for a medium project. If they need to use 30 bags of cement for a large project, how many bags of sand should they use to maintain the same mixture ratio? Answer: 90 Solution: Identify the ratio from the table. For the small project: 8 cement : 24 sand. For the medium project: 12 cement : 36 sand.
   Full step-by-step solution

   Step 1: Identify the ratio from the table. For the small project: 8 cement : 24 sand. For the medium project: 12 cement : 36 sand. Step 2: Check if the ratios are equivalent. 8/24 = 1/3 and 12/36 = 1/3. Both simplify to the same ratio of 1:3 (cement to sand). Step 3: Apply the ratio to 30 bags of cement. Since cement:sand = 1:3, multiply 30 by 3. Step 4: Calculate: 30 × 3 = 90. The construction company needs 90 bags of sand.

2. Sophia is organizing a school fundraiser where she sells homemade candles. She notices that the cost of wax is proportional to the number of candles she makes. The table below shows the relationship between the number of candles and the cost of wax. If Sophia wants to make 50 candles, how much will the wax cost? | Number of Candles | Cost of Wax (dollars) | |-------------------|----------------------| | 8 | 44 | | 12 | 66 | | 20 | 110 | Answer: 275 Solution: Find the constant of proportionality (cost per candle) using the first row: 44 dollars ÷ 8 candles = 5.5 dollars per candle.
   Full step-by-step solution

   Step 1: Find the constant of proportionality (cost per candle) using the first row: 44 dollars ÷ 8 candles = 5.5 dollars per candle. Step 2: Verify with the second row: 66 ÷ 12 = 5.5 dollars per candle. Step 3: Verify with the third row: 110 ÷ 20 = 5.5 dollars per candle. Step 4: Use the constant to find the cost for 50 candles: 50 candles × 5.5 dollars per candle = 275 dollars. The wax will cost $275 for 50 candles.

3. x | y 1 | 16 3 | 48 5 | 80 7 | ? 9 | 144 Complete the table for the proportional relationship. Find the missing value of y when x = 7. Answer: 112 Solution: For x = 1, y = 16: y/x = 16/1 = 16 For x = 3, y = 48: y/x = 48/3 = 16 For x = 5, y = 80: y/x = 80/5 = 16 For x = 9, y = 144: y/x = 144/9 = 16 All ratios equal 16, so the constant of proportionality k = 16.
   Full step-by-step solution

   Step 1: Check that the relationship is proportional by verifying y/x is constant for the given rows. For x = 1, y = 16: y/x = 16/1 = 16 For x = 3, y = 48: y/x = 48/3 = 16 For x = 5, y = 80: y/x = 80/5 = 16 For x = 9, y = 144: y/x = 144/9 = 16 All ratios equal 16, so the constant of proportionality k = 16. Step 2: The equation is y = 16x. Step 3: Substitute x = 7 into the equation: y = 16 × 7 = 112. The missing value is 112.

4. Liam is mixing a special cleaning solution that requires a specific ratio of concentrate to water. The table shows the proportional relationship between the amount of concentrate and the total solution. If the table indicates that 4 liters of concentrate make 18 liters of total solution, how many liters of total solution will Liam make if he uses 10 liters of concentrate? Answer: 45 Solution: The table shows a proportional relationship between concentrate and total solution. 4 liters of concentrate → 18 liters of total solution. Total solution = concentrate + water.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the relationship** The table shows a proportional relationship between concentrate and total solution. We are told: 4 liters of concentrate → 18 liters of total solution. That means: Total solution = concentrate + water. So the water in this case is 18 − 4 = 14 liters. But the ratio given is between concentrate and **total solution**, not concentrate and water. --- **Step 2: Set up the proportion** Let \( k \) be the ratio of concentrate to total solution. From the given data: \( k = \frac{\text{concentrate}}{\text{total solution}} = \frac{4}{18} \). Simplify: \( k = \frac{2}{9} \). This means: for every 2 liters of concentrate, you get 9 liters of total solution. --- **Step 3: Apply to 10 liters of concentrate** Let \( T \) = total solution when concentrate = 10 liters. From the proportion: \( \frac{\text{concentrate}}{\text{total solution}} = \frac{2}{9} \). So: \( \frac{10}{T} = \frac{2}{9} \). --- **Step 4: Solve for \( T \)** Cross-multiply: \( 10 \times 9 = 2 \times T \) \( 90 = 2T \) \( T = 45 \). --- **Step 5: Conclusion** If Liam uses 10 liters of concentrate, he will make 45 liters of total solution. --- **Final answer:** 45

5. x | y 2 | 34 4 | 68 6 | 102 8 | 136 10 | ? Answer: 170 Solution: Check if the relationship is proportional by finding y/x for each given pair. For x=2, y=34: y/x = 34/2 = 17 For x=4, y=68: y/x = 68/4 = 17 For x=6, y=102: y/x = 102/6 = 17 For x=8, y=136: y/x = 136/8 = 17 Since y/x = 17 for all pairs, the constant of proportionality k = 17.
   Full step-by-step solution

   Step 1: Check if the relationship is proportional by finding y/x for each given pair. For x=2, y=34: y/x = 34/2 = 17 For x=4, y=68: y/x = 68/4 = 17 For x=6, y=102: y/x = 102/6 = 17 For x=8, y=136: y/x = 136/8 = 17 Since y/x = 17 for all pairs, the constant of proportionality k = 17. Step 2: Use the constant to find y when x=10: y = k * x = 17 * 10 = 170. The answer is 170.

6. x | y 3 | 57 7 | 133 11 | 209 15 | ? 19 | 361 Answer: 285 Solution: Check if the relationship is proportional by calculating y/x for known pairs. For (3, 57): 57 ÷ 3 = 19 For (7, 133): 133 ÷ 7 = 19 For (11, 209): 209 ÷ 11 = 19 For (19, 361): 361 ÷ 19 = 19 Since y/x = 19 for all known pairs, the constant of proportionality k = 19.
   Full step-by-step solution

   Step 1: Check if the relationship is proportional by calculating y/x for known pairs. For (3, 57): 57 ÷ 3 = 19 For (7, 133): 133 ÷ 7 = 19 For (11, 209): 209 ÷ 11 = 19 For (19, 361): 361 ÷ 19 = 19 Step 2: Since y/x = 19 for all known pairs, the constant of proportionality k = 19. Step 3: Use the equation y = 19x to find the missing value when x = 15. y = 19 × 15 = 285 The answer is 285.

7. Isabella is mixing a special fertilizer for her garden. The table below shows the proportional relationship between the amount of fertilizer concentrate (in cups) and the amount of water (in gallons) needed to maintain the correct strength. If the table shows that 2 cups of concentrate require 7 gallons of water, and 4 cups of concentrate require 14 gallons of water, how many gallons of water are needed for 12 cups of fertilizer concentrate? Answer: 42 Solution: Identify the ratio from the table. For 2 cups of concentrate to 7 gallons of water, the ratio is 2/7. For 4 cups to 14 gallons, the ratio is 4/14 = 2/7.
   Full step-by-step solution

   Step 1: Identify the ratio from the table. For 2 cups of concentrate to 7 gallons of water, the ratio is 2/7. For 4 cups to 14 gallons, the ratio is 4/14 = 2/7. The constant of proportionality k = 2/7 (or 0.2857 cups per gallon). Alternatively, find gallons per cup: 7/2 = 3.5 gallons per cup. Step 2: Use the proportion: 2/7 = 12/x. Step 3: Cross-multiply: 2 * x = 7 * 12. Step 4: 2x = 84. Step 5: Divide both sides by 2: x = 42. Isabella needs 42 gallons of water for 12 cups of fertilizer concentrate.