Fractional Volume

lessonbunny.com

Answer Key & Explanations

Fractional Volume · Grade 6 · Worksheet 1

1. 7 1/5 × 9 3/8 × 11 1/4 = ? Answer: 759 3/8 Solution: Convert each mixed number to an improper fraction. 7 1/5 = (7 × 5 + 1)/5 = 36/5 9 3/8 = (9 × 8 + 3)/8 = 75/8 11 1/4 = (11 × 4 + 1)/4 = 45/4 Multiply the three fractions.
   Full step-by-step solution

   Step 1: Convert each mixed number to an improper fraction. 7 1/5 = (7 × 5 + 1)/5 = 36/5 9 3/8 = (9 × 8 + 3)/8 = 75/8 11 1/4 = (11 × 4 + 1)/4 = 45/4 Step 2: Multiply the three fractions. (36/5) × (75/8) × (45/4) = (36 × 75 × 45) / (5 × 8 × 4) Step 3: Multiply the numerators. 36 × 75 = 2700 2700 × 45 = 121500 So numerator = 121500 Step 4: Multiply the denominators. 5 × 8 = 40 40 × 4 = 160 So denominator = 160 Step 5: Write the fraction and simplify. 121500/160 Divide numerator and denominator by 20: 6075/8 Step 6: Convert to a mixed number. 6075 ÷ 8 = 759 remainder 3 So 6075/8 = 759 3/8 The answer is 759 3/8.

2. Aroha's rectangular prism has dimensions 9 2/3 cm × 11 1/4 cm × 12 3/5 cm. What is its volume? Answer: 1370 1/4 cm³ Solution: Convert each mixed number to an improper fraction. 9 2/3 = (9 × 3 + 2)/3 = 29/3 11 1/4 = (11 × 4 + 1)/4 = 45/4 12 3/5 = (12 × 5 + 3)/5 = 63/5 Multiply the three fractions.
   Full step-by-step solution

   Step 1: Convert each mixed number to an improper fraction. 9 2/3 = (9 × 3 + 2)/3 = 29/3 11 1/4 = (11 × 4 + 1)/4 = 45/4 12 3/5 = (12 × 5 + 3)/5 = 63/5 Step 2: Multiply the three fractions. (29/3) × (45/4) × (63/5) = (29 × 45 × 63) / (3 × 4 × 5) Step 3: Multiply the numerators. 29 × 45 = 1305 1305 × 63 = 1305 × (60 + 3) = 78300 + 3915 = 82215 So numerator = 82215 Step 4: Multiply the denominators. 3 × 4 = 12 12 × 5 = 60 So denominator = 60 Step 5: Write the fraction and simplify. 82215/60 Divide numerator and denominator by 15: 82215 ÷ 15 = 5481, 60 ÷ 15 = 4 So 5481/4 Step 6: Convert to a mixed number. 5481 ÷ 4 = 1370 remainder 1 So 5481/4 = 1370 1/4 The answer is 1370 1/4 cm³.

3. Olivia is building a rectangular prism-shaped planter box for her garden. The base of the planter measures 3.5 feet long and 1.75 feet wide. The planter has a height of 2.25 feet. What is the volume of the planter box in cubic feet? Answer: 13.78125 Solution: Identify the dimensions: Length = 3.5 feet, Width = 1.75 feet, Height = 2.25 feet. Use the volume formula V = l x w x h. Multiply the length and width: 3.5 x 1.75 = 6.125.
   Full step-by-step solution

   Step 1: Identify the dimensions: Length = 3.5 feet, Width = 1.75 feet, Height = 2.25 feet. Step 2: Use the volume formula V = l x w x h. Step 3: Multiply the length and width: 3.5 x 1.75 = 6.125. Step 4: Multiply the result by the height: 6.125 x 2.25 = 13.78125. Step 5: The volume is 13.78125 cubic feet. The answer is 13.78125.

4. A rectangular prism-shaped fish tank has dimensions 2 1/4 feet long by 1 3/5 feet wide by 1 1/2 feet high. The tank is filled with water to 3/4 of its maximum height. What is the volume of water in the tank in cubic feet? Answer: 4.05 Solution: Length = 2 1/4 = 9/4 feet Width = 1 3/5 = 8/5 feet Height = 1 1/2 = 3/2 feet Full volume = (9/4) × (8/5) × (3/2) = (9 × 8 × 3)/(4 × 5 × 2) = 216/40 = 27/5 = 5.4 cubic feet Water fills 3/4 of the tank's height: Water height = (3/4) × (3/2) = 9/8 = 1.125 feet Water volume = length × width × water…
   Full step-by-step solution

   Step 1: Find the full volume of the tank Convert mixed numbers to improper fractions: Length = 2 1/4 = 9/4 feet Width = 1 3/5 = 8/5 feet Height = 1 1/2 = 3/2 feet Full volume = (9/4) × (8/5) × (3/2) = (9 × 8 × 3)/(4 × 5 × 2) = 216/40 = 27/5 = 5.4 cubic feet Step 2: Find the actual water height Water fills 3/4 of the tank's height: Water height = (3/4) × (3/2) = 9/8 = 1.125 feet Step 3: Calculate the water volume Water volume = length × width × water height Water volume = (9/4) × (8/5) × (9/8) = (9 × 8 × 9)/(4 × 5 × 8) = 648/160 Simplify: 648/160 = 81/20 = 4.05 cubic feet The answer is 4.05 cubic feet.

5. 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 needs to be filled with soil to a height of 0.8 meters. If each bag of soil contains 0.6 cubic meters, how many full bags of soil does Liam need to buy to fill the planter box? Answer: 4 Solution: Find the volume of the planter box. Volume = length × width × height Length = 2.5 m Width = 1.2 m Height of soil = 0.8 m Volume = 2.5 × 1.2 × 0.8 Multiply step by step.
   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 Given: Length = 2.5 m Width = 1.2 m Height of soil = 0.8 m Volume = 2.5 × 1.2 × 0.8 --- **Step 2: Multiply step by step.** First: 2.5 × 1.2 = 3.0 Then: 3.0 × 0.8 = 2.4 So the volume of soil needed = 2.4 cubic meters. --- **Step 3: Determine how many bags of soil are needed.** Each bag contains 0.6 cubic meters. Number of bags = Total volume ÷ Volume per bag Number of bags = 2.4 ÷ 0.6 --- **Step 4: Perform the division.** 2.4 ÷ 0.6 = 24 ÷ 6 = 4 So, 4 bags are needed exactly. --- **Step 5: Check if full bags are enough.** Since 4 is a whole number, Liam needs exactly 4 full bags to fill the planter box. --- **Final Answer:** 4

6. (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