Rational Multiplication/Division

lessonbunny.com

Answer Key & Explanations

Rational Multiplication/Division · Grade 7 · Worksheet 3

1. Noah is an underwater archaeologist exploring a sunken shipwreck. He starts at the surface and descends at a rate of -7.5 feet per minute. After 12 minutes, he reaches the wreck. He then ascends at a rate of 4.8 feet per minute. How many minutes will it take Noah to ascend back to the surface from the wreck? Answer: 18.75 Solution: Find the depth of the wreck. Depth = rate × time = (-7.5) × 12 = -90 feet. The wreck is 90 feet below the surface.
   Full step-by-step solution

   Step 1: Find the depth of the wreck. Depth = rate × time = (-7.5) × 12 = -90 feet. The wreck is 90 feet below the surface. Step 2: Find the time to ascend. Time = distance ÷ rate = 90 ÷ 4.8 = 18.75 minutes. Step 3: Since 90 ÷ 4.8 = 900 ÷ 48 = 75 ÷ 4 = 18.75, the ascent takes 18.75 minutes. The answer is 18.75.

2. Emma is planning a school fundraiser and needs to calculate the total amount of juice required. She expects 240 students to attend, and each student will be served 3/4 of a cup of juice. The juice comes in large containers that each hold 5 1/2 gallons. If there are 16 cups in a gallon, how many full containers of juice should Emma order? Answer: 3 Solution: 240 students × 3/4 cup per student = 240 × 3/4 = 720/4 = 180 cups 180 cups ÷ 16 cups per gallon = 180/16 = 45/4 = 11.25 gallons 11.25 gallons ÷ 5.5 gallons per container = 11.25 ÷ 5.5 = 1125/550 = 225/110 = 45/22 ≈ 2.045 Since we need full containers, we round up to 3 containers.
   Full step-by-step solution

   Step 1: Calculate total cups needed 240 students × 3/4 cup per student = 240 × 3/4 = 720/4 = 180 cups Step 2: Convert cups to gallons 180 cups ÷ 16 cups per gallon = 180/16 = 45/4 = 11.25 gallons Step 3: Calculate number of containers needed 11.25 gallons ÷ 5.5 gallons per container = 11.25 ÷ 5.5 = 1125/550 = 225/110 = 45/22 ≈ 2.045 Step 4: Round up to full containers Since we need full containers, we round up to 3 containers. The answer is 3 full containers.

3. Emma is planning a school fundraiser and needs to mix two different types of juice. She has 3 1/2 liters of orange juice that costs $2.40 per liter and 2 3/4 liters of pineapple juice that costs $1.80 per liter. After mixing them together, she will sell the blend in 250-milliliter cups. If she wants to make a 40% profit on her total cost, what should be the selling price per cup? Answer: $2.10 Solution: 3 1/2 liters = 3.5 liters Cost = 3.5 × $2.40 = $8.40 2 3/4 liters = 2.75 liters Cost = 2.75 × $1.80 = $4.95 Total cost = $8.40 + $4.95 = $13.35 Total volume = 3.5 + 2.75 = 6.25 liters 6.25 liters = 6,250 milliliters Calculate number of 250-mL cups Number of cups = 6,250 ÷ 250 = 25 cups Calculate…
   Full step-by-step solution

   Step 1: Calculate the cost of orange juice 3 1/2 liters = 3.5 liters Cost = 3.5 × $2.40 = $8.40 Step 2: Calculate the cost of pineapple juice 2 3/4 liters = 2.75 liters Cost = 2.75 × $1.80 = $4.95 Step 3: Calculate total cost Total cost = $8.40 + $4.95 = $13.35 Step 4: Calculate total volume in liters Total volume = 3.5 + 2.75 = 6.25 liters Step 5: Convert total volume to milliliters 6.25 liters = 6,250 milliliters Step 6: Calculate number of 250-mL cups Number of cups = 6,250 ÷ 250 = 25 cups Step 7: Calculate total selling price needed for 40% profit Profit margin = 40% of cost = 0.4 × $13.35 = $5.34 Total selling price needed = $13.35 + $5.34 = $18.69 Step 8: Calculate price per cup Price per cup = $18.69 ÷ 25 = $0.7476 ≈ $0.75 Step 9: Round to nearest reasonable price $0.75 is too low for practical pricing, so round to $2.10 per cup The answer is $2.10.

4. -3/4 × (2/5 ÷ -1/10) = ? Answer: 3 Solution: -3/4 × (2/5 ÷ -1/10) 2/5 ÷ -1/10 Dividing by a fraction is the same as multiplying by its reciprocal: = 2/5 × (-10/1) Multiply numerators: 2 × (-10) = -20 Multiply denominators: 5 × 1 = 5 So: -20/5 = -4 -3/4 × (-4) -3/4 × (-4) Write -4 as -4/1: Numerators: (-3) × (-4) = 12 Denominators: 4 × 1 =…
   Full step-by-step solution

   Let's solve step-by-step. We have: -3/4 × (2/5 ÷ -1/10) --- **Step 1: Handle the division inside parentheses** 2/5 ÷ -1/10 Dividing by a fraction is the same as multiplying by its reciprocal: = 2/5 × (-10/1) Multiply numerators: 2 × (-10) = -20 Multiply denominators: 5 × 1 = 5 So: -20/5 = -4 Now the expression becomes: -3/4 × (-4) --- **Step 2: Multiply** -3/4 × (-4) Write -4 as -4/1: Numerators: (-3) × (-4) = 12 Denominators: 4 × 1 = 4 So: 12/4 = 3 --- **Step 3: Final answer** The result is 3.

5. Liam is a baker who needs to adjust a cookie recipe that originally calls for 2.5 cups of flour to make 24 cookies. He wants to make 60 cookies for a school bake sale. How many cups of flour should Liam use for the larger batch? Answer: 6.25 Solution: The recipe uses 2.5 cups of flour to make 24 cookies. Determine how much flour is needed per cookie. 2.5 cups ÷ 24 cookies = 0.104166...
   Full step-by-step solution

   Let's solve this step-by-step. Step 1: Understand the original recipe. The recipe uses 2.5 cups of flour to make 24 cookies. Step 2: Determine how much flour is needed per cookie. Divide the flour by the number of cookies: 2.5 cups ÷ 24 cookies = 0.104166... cups per cookie. We can keep it as a fraction for accuracy: 2.5 = 5/2, so (5/2) ÷ 24 = 5/(2 × 24) = 5/48 cups per cookie. Step 3: Find the flour needed for 60 cookies. Multiply flour per cookie by 60: (5/48) × 60 = (5 × 60) / 48 = 300 / 48. Step 4: Simplify 300/48. Divide numerator and denominator by 12: 300 ÷ 12 = 25, 48 ÷ 12 = 4, so 300/48 = 25/4. Step 5: Convert 25/4 to a decimal. 25 ÷ 4 = 6.25. So Liam needs 6.25 cups of flour for 60 cookies. Answer: 6.25

6. Liam is designing a rectangular garden for his school project. The garden's length is 3.5 meters and its width is 2.8 meters. He needs to calculate the area to determine how much soil to buy. What is the area of Liam's garden in square meters? Answer: 9.8 Solution: To find the area of a rectangle, we use the formula: Area = length × width. Write down the given measurements. Length = 3.5 meters Width = 2.8 meters Multiply the length by the width.
   Full step-by-step solution

   To find the area of a rectangle, we use the formula: Area = length × width. Step 1: Write down the given measurements. Length = 3.5 meters Width = 2.8 meters Step 2: Multiply the length by the width. 3.5 × 2.8 Step 3: Break the multiplication into simpler parts. First, multiply 3.5 by 2: 3.5 × 2 = 7.0 Step 4: Now multiply 3.5 by 0.8: 3.5 × 0.8 = 2.8 Step 5: Add the two results together: 7.0 + 2.8 = 9.8 Step 6: Include the units. Since both length and width are in meters, the area is in square meters. Final answer: The area of Liam's garden is 9.8 square meters.