Fractional Volume

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Answer Key & Explanations

Fractional Volume · Grade 6 · Worksheet 2

1. (2/3 × 0.75) ÷ (1/4) = ? Answer: 2 Solution: Simplify 2/3 × 0.75 0.75 is the same as 3/4.
   Full step-by-step solution

   Let's solve step-by-step. Step 1: Simplify 2/3 × 0.75 0.75 is the same as 3/4. So we have: 2/3 × 3/4 Step 2: Multiply the fractions Multiply numerators: 2 × 3 = 6 Multiply denominators: 3 × 4 = 12 So 2/3 × 3/4 = 6/12 Step 3: Simplify 6/12 6/12 = 1/2 Step 4: Now the expression is (1/2) ÷ (1/4) Dividing by 1/4 is the same as multiplying by 4/1. So (1/2) × (4/1) = 4/2 Step 5: Simplify 4/2 4/2 = 2 Final answer: 2

2. (3/4) × (2/3) ÷ (1/2) = ? Answer: 1 Solution: We have: (3/4) × (2/3) ÷ (1/2) Dividing by a fraction is the same as multiplying by its reciprocal. So, ÷ (1/2) is the same as × (2/1).
   Full step-by-step solution

   Let's solve step-by-step. We have: (3/4) × (2/3) ÷ (1/2) --- **Step 1: Understand the division by a fraction** Dividing by a fraction is the same as multiplying by its reciprocal. So, ÷ (1/2) is the same as × (2/1). The expression becomes: (3/4) × (2/3) × (2/1) --- **Step 2: Multiply the fractions** When multiplying fractions, multiply the numerators together and the denominators together. Numerator: 3 × 2 × 2 = 12 Denominator: 4 × 3 × 1 = 12 So we have: 12/12 --- **Step 3: Simplify** 12/12 = 1 --- **Final Answer:** 1

3. (2 1/4 × 3 1/3) ÷ (1 1/2) = ? Answer: 5 Solution: 2 1/4 = (2 × 4 + 1)/4 = 9/4 3 1/3 = (3 × 3 + 1)/3 = 10/3 1 1/2 = (1 × 2 + 1)/2 = 3/2 (9/4) × (10/3) = (9 × 10)/(4 × 3) = 90/12 Simplify 90/12 90 ÷ 6 = 15, 12 ÷ 6 = 2, so 90/12 = 15/2 (15/2) ÷ (3/2) = (15/2) × (2/3) = (15 × 2)/(2 × 3) = 30/6 Simplify 30/6 30 ÷ 6 = 5 The answer is 5.
   Full step-by-step solution

   Step 1: Convert mixed numbers to improper fractions 2 1/4 = (2 × 4 + 1)/4 = 9/4 3 1/3 = (3 × 3 + 1)/3 = 10/3 1 1/2 = (1 × 2 + 1)/2 = 3/2 Step 2: Multiply the first two fractions (9/4) × (10/3) = (9 × 10)/(4 × 3) = 90/12 Step 3: Simplify 90/12 90 ÷ 6 = 15, 12 ÷ 6 = 2, so 90/12 = 15/2 Step 4: Divide by the third fraction (15/2) ÷ (3/2) = (15/2) × (2/3) = (15 × 2)/(2 × 3) = 30/6 Step 5: Simplify 30/6 30 ÷ 6 = 5 The answer is 5.

4. A triangular prism-shaped chocolate bar has a triangular base with sides measuring 3 cm, 4 cm, and 5 cm, and a length of 12 cm. The chocolate bar is divided into smaller triangular prism pieces by making cuts perpendicular to the length. Each piece has a length of 1.5 cm. How many complete pieces can be made from the chocolate bar? Answer: 8 Solution: Identify the total length of the chocolate bar: 12 cm Identify the length of each piece: 1.5 cm Calculate how many pieces can be made by dividing total length by piece length: 12 ÷ 1.5 12 ÷ 1.5 = 120 ÷ 15 = 8 Since we get exactly 8, all pieces are complete.
   Full step-by-step solution

   Step 1: Identify the total length of the chocolate bar: 12 cm Step 2: Identify the length of each piece: 1.5 cm Step 3: Calculate how many pieces can be made by dividing total length by piece length: 12 ÷ 1.5 Step 4: 12 ÷ 1.5 = 120 ÷ 15 = 8 Step 5: Since we get exactly 8, all pieces are complete. The answer is 8 complete pieces.

5. 8 1/3 × 9 1/2 × 10 2/5 = ? Answer: 823 1/3 Solution: Convert each mixed number to an improper fraction. 8 1/3 = (8 × 3 + 1)/3 = 25/3 9 1/2 = (9 × 2 + 1)/2 = 19/2 10 2/5 = (10 × 5 + 2)/5 = 52/5 Multiply the three fractions.
   Full step-by-step solution

   Step 1: Convert each mixed number to an improper fraction. 8 1/3 = (8 × 3 + 1)/3 = 25/3 9 1/2 = (9 × 2 + 1)/2 = 19/2 10 2/5 = (10 × 5 + 2)/5 = 52/5 Step 2: Multiply the three fractions. (25/3) × (19/2) × (52/5) = (25 × 19 × 52) / (3 × 2 × 5) Step 3: Multiply the numerators. 25 × 19 = 475 475 × 52 = 475 × (50 + 2) = 23750 + 950 = 24700 So numerator = 24700 Step 4: Multiply the denominators. 3 × 2 = 6 6 × 5 = 30 So denominator = 30 Step 5: Write the fraction and simplify. 24700/30 Divide numerator and denominator by 10: 2470/3 Step 6: Convert to a mixed number. 2470 ÷ 3 = 823 remainder 1 So 2470/3 = 823 1/3 The answer is 823 1/3.

6. Liam is building a rectangular planter box for his garden. The box has a length of 2.5 meters, a width of 1.2 meters, and a height of 0.8 meters. He needs to fill it with soil, but the soil is sold in bags that each contain 0.5 cubic meters. How many full bags of soil does Liam need to buy to completely fill the planter box? Answer: 5 Solution: Volume = length × width × height Volume = 2.5 m × 1.2 m × 0.8 m First, multiply 2.5 and 1.2: 2.5 × 1.2 = 3.0 Now multiply by 0.8: 3.0 × 0.8 = 2.4 So, the volume of the box is 2.4 cubic meters.
   Full step-by-step solution

   Let's go step by step. --- **Step 1: Find the volume of the planter box** The box is a rectangular prism, so: Volume = length × width × height Volume = 2.5 m × 1.2 m × 0.8 m First, multiply 2.5 and 1.2: 2.5 × 1.2 = 3.0 Now multiply by 0.8: 3.0 × 0.8 = 2.4 So, the volume of the box is **2.4 cubic meters**. --- **Step 2: Determine how many bags are needed** Each bag contains 0.5 cubic meters of soil. Number of bags needed = Volume of box ÷ Volume per bag Number of bags = 2.4 ÷ 0.5 Dividing by 0.5 is the same as multiplying by 2: 2.4 × 2 = 4.8 So, 4.8 bags are needed. --- **Step 3: Decide on full bags** Since Liam can only buy full bags, he must round up to the nearest whole number. 4.8 → next whole number is **5**. --- **Step 4: Conclusion** Liam needs to buy **5 full bags** of soil to fill the planter box completely. --- **Final answer: 5**

7. A rectangular prism has dimensions 3 1/4 cm × 5 1/2 cm × 7 1/3 cm. What is its volume? Answer: 131 1/24 cm³ Solution: 3 1/4 = (3 × 4 + 1)/4 = 13/4 5 1/2 = (5 × 2 + 1)/2 = 11/2 7 1/3 = (7 × 3 + 1)/3 = 22/3 Volume = (13/4) × (11/2) × (22/3) Numerator: 13 × 11 × 22 = 13 × 242 = 3146 Denominator: 4 × 2 × 3 = 24 3146/24 = 1573/12 1573 ÷ 12 = 131 with remainder 1 So 1573/12 = 131 1/12 1/12 cannot be simplified…
   Full step-by-step solution

   Step 1: Convert mixed numbers to improper fractions 3 1/4 = (3 × 4 + 1)/4 = 13/4 5 1/2 = (5 × 2 + 1)/2 = 11/2 7 1/3 = (7 × 3 + 1)/3 = 22/3 Step 2: Multiply the fractions Volume = (13/4) × (11/2) × (22/3) Step 3: Multiply numerators and denominators Numerator: 13 × 11 × 22 = 13 × 242 = 3146 Denominator: 4 × 2 × 3 = 24 Step 4: Simplify the fraction 3146/24 = 1573/12 Step 5: Convert to mixed number 1573 ÷ 12 = 131 with remainder 1 So 1573/12 = 131 1/12 Step 6: Check for further simplification 1/12 cannot be simplified further The volume is 131 1/12 cm³.