Equivalent Ratio Tables

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Answer Key & Explanations

Equivalent Ratio Tables · Grade 6 · Worksheet 1

1. Complete the ratio table: | 9 | 15 | 21 | 30 | 45 | x | | 12 | 20 | 28 | 40 | 60 | 84 | Answer: 63 Solution: Find the multiplier by looking at a complete column. Use the first column: 9 * m = 12. Solve: m = 12 / 9 = 4/3.
   Full step-by-step solution

   Step 1: Find the multiplier by looking at a complete column. Use the first column: 9 * m = 12. Solve: m = 12 / 9 = 4/3. Step 2: Check with another column: 15 * (4/3) = 60/3 = 20. Correct. Step 3: Apply the multiplier to the column with x: x * (4/3) = 84. Step 4: Solve for x: x = 84 * (3/4) = 252/4 = 63. The answer is 63.

2. Complete the table of equivalent ratios: | Hana's score | 4 | 8 | 12 | 16 | |--------------|---|---|----|----| | Matiu's score| 6 | ? | 18 | ? | Answer: 12 and 24 Solution: Find the ratio from the first column. Hana's score is 4 and Matiu's score is 6. The ratio is 4:6, which simplifies to 2:3.
   Full step-by-step solution

   Step 1: Find the ratio from the first column. Hana's score is 4 and Matiu's score is 6. The ratio is 4:6, which simplifies to 2:3. This means for every 2 points Hana gets, Matiu gets 3 points. Step 2: For the second column, Hana's score is 8. Since 8 = 4 × 2, multiply Matiu's first score by 2: 6 × 2 = 12. So the missing value is 12. Step 3: For the fourth column, Hana's score is 16. Since 16 = 4 × 4, multiply Matiu's first score by 4: 6 × 4 = 24. So the missing value is 24. Step 4: Check the third column: Hana's score is 12, which is 4 × 3, so Matiu's score should be 6 × 3 = 18, which matches. The missing values are 12 and 24.

3. If 12:18 = x:27, then x = ? Answer: 18 Solution: We are given the proportion: 12:18 = x:27. Write the proportion in fraction form. 12:18 means 12/18, and x:27 means x/27.
   Full step-by-step solution

   We are given the proportion: 12:18 = x:27. Step 1: Write the proportion in fraction form. 12:18 means 12/18, and x:27 means x/27. So we have: 12/18 = x/27 Step 2: Simplify 12/18 if possible. 12/18 = (12 ÷ 6)/(18 ÷ 6) = 2/3. So the equation becomes: 2/3 = x/27 Step 3: Solve for x by cross-multiplying. 2/3 = x/27 Multiply both sides by 27: 27 * (2/3) = x Step 4: Perform the multiplication. 27 * 2 = 54 54 / 3 = 18 So x = 18. Step 5: Check the answer. 12/18 = 2/3, and 18/27 = 2/3. Both ratios are equal, so the answer is correct. Final answer: x = 18

4. If 3:7 = x:84, then x = ? Answer: 36 Solution: Write the proportion: 3/7 = x/84 Cross-multiply: 3 * 84 = 7 * x Calculate the left side: 3 * 84 = 252 Write the equation: 252 = 7x Divide both sides by 7: 252 ÷ 7 = x Calculate: 252 ÷ 7 = 36 Therefore, x = 36
   Full step-by-step solution

   Step 1: Write the proportion: 3/7 = x/84 Step 2: Cross-multiply: 3 * 84 = 7 * x Step 3: Calculate the left side: 3 * 84 = 252 Step 4: Write the equation: 252 = 7x Step 5: Divide both sides by 7: 252 ÷ 7 = x Step 6: Calculate: 252 ÷ 7 = 36 Step 7: Therefore, x = 36

5. A fruit juice company uses a special concentrate-to-water ratio of 3:7 to make their premium orange juice. If they want to produce 1500 liters of juice for a large order, how many liters of concentrate should they use? Answer: 450 Solution: The concentrate-to-water ratio is 3:7. This means for every 3 parts of concentrate, there are 7 parts of water. Total parts = 3 (concentrate) + 7 (water) = 10 parts.
   Full step-by-step solution

   Step 1: Understand the ratio The concentrate-to-water ratio is 3:7. This means for every 3 parts of concentrate, there are 7 parts of water. Step 2: Find the total parts Total parts = 3 (concentrate) + 7 (water) = 10 parts. Step 3: Relate parts to actual quantity We know 10 parts correspond to 1500 liters of juice. So, 1 part = 1500 liters ÷ 10 = 150 liters. Step 4: Find concentrate quantity Concentrate is 3 parts, so: Concentrate liters = 3 × 150 liters = 450 liters. Step 5: Conclusion They should use 450 liters of concentrate.

6. If 7:11 = 63:x, then x = ? Answer: 99 Solution: and 63/99 = 0.63636... ✓
   Full step-by-step solution

   Step 1: Write the proportion as fractions: 7/11 = 63/x Step 2: Cross multiply to solve for x: 7 × x = 11 × 63 Step 3: Calculate the right side: 11 × 63 = 693 Step 4: So 7x = 693 Step 5: Divide both sides by 7: x = 693 ÷ 7 Step 6: Calculate: 693 ÷ 7 = 99 Step 7: Therefore, x = 99 Step 8: Verify: 7/11 = 0.63636... and 63/99 = 0.63636... ✓

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 plants flowers in a pattern that creates 3 rows and 4 columns of flower beds within the garden, what is the ratio of the area of one flower bed to the total garden area? Answer: 1:12 Solution: Vertices: (2, 1), (8, 1), (8, 5), (2, 5) Length along x-axis: from x = 2 to x = 8 → length = 8 − 2 = 6 units Length along y-axis: from y = 1 to y = 5 → width = 5 − 1 = 4 units Area of garden = length × width = 6 × 4 = 24 square units.
   Full step-by-step solution

   Let's go step by step. --- **Step 1: Find the dimensions of the garden** Vertices: (2, 1), (8, 1), (8, 5), (2, 5) Length along x-axis: from x = 2 to x = 8 → length = 8 − 2 = 6 units Length along y-axis: from y = 1 to y = 5 → width = 5 − 1 = 4 units Area of garden = length × width = 6 × 4 = 24 square units. --- **Step 2: Understand the flower bed arrangement** The problem says: 3 rows and 4 columns of flower beds. That means the garden is divided into 3 equal parts along the vertical (y) direction and 4 equal parts along the horizontal (x) direction. --- **Step 3: Find the dimensions of one flower bed** Total length (x-direction) = 6 units Divided into 4 columns → width of one flower bed = 6 / 4 = 1.5 units Total width (y-direction) = 4 units Divided into 3 rows → height of one flower bed = 4 / 3 units Area of one flower bed = (1.5) × (4/3) = (3/2) × (4/3) = (3 × 4) / (2 × 3) = 12 / 6 = 2 square units --- **Step 4: Ratio of one flower bed area to total garden area** One flower bed area = 2 Total garden area = 24 Ratio = 2 : 24 Simplify: divide both by 2 → 1 : 12 --- **Final Answer:** 1:12

8. A fruit punch recipe calls for mixing orange juice and pineapple juice in a ratio of 5:3. If Liam wants to make 32 cups of punch for his party, how many cups of orange juice should he use? Answer: 20 Solution: The ratio of orange juice to pineapple juice is 5 : 3. This means for every 5 cups of orange juice, we use 3 cups of pineapple juice. Total parts = 5 + 3 = 8 parts.
   Full step-by-step solution

   Let's go step by step. --- **Step 1: Understand the ratio** The ratio of orange juice to pineapple juice is 5 : 3. This means for every 5 cups of orange juice, we use 3 cups of pineapple juice. --- **Step 2: Find the total parts in the mixture** Total parts = 5 + 3 = 8 parts. --- **Step 3: Relate total parts to total punch** We know the total punch is 32 cups. So, 8 parts = 32 cups. --- **Step 4: Find how many cups are in 1 part** 1 part = 32 cups / 8 = 4 cups. --- **Step 5: Find cups of orange juice** Orange juice is 5 parts. So, orange juice = 5 × 4 cups = 20 cups. --- **Step 6: Conclusion** Liam should use **20 cups** of orange juice. --- **Final answer:** 20

9. If 4:9 = 16:x, then x = ? Answer: 36 Solution: We are given the proportion: 4:9 = 16:x. Write the proportion as fractions. 4:9 means 4/9, and 16:x means 16/x.
   Full step-by-step solution

   We are given the proportion: 4:9 = 16:x. Step 1: Write the proportion as fractions. 4:9 means 4/9, and 16:x means 16/x. So we have: 4/9 = 16/x Step 2: Cross-multiply to eliminate the fractions. 4 * x = 9 * 16 Step 3: Perform the multiplication on the right side. 4 * x = 144 Step 4: Solve for x by dividing both sides by 4. x = 144 / 4 Step 5: Calculate the division. x = 36 So the value of x is 36. We can check: 4/9 = 16/36, because 16/36 simplifies to 4/9. Correct.