Unit Rate Applications

lessonbunny.com

Answer Key & Explanations

Unit Rate Applications · Grade 6 · Worksheet 1

1. A factory produces 1,800 widgets in 6 hours. The factory operates 8 hours per day. If they need to fill an order for 7,200 widgets, how many full days will it take to complete the order? Answer: 3 Solution: Find the hourly production rate. The factory makes 1,800 widgets in 6 hours, so the rate is 1,800 / 6 = 300 widgets per hour. Find the daily production.
   Full step-by-step solution

   Step 1: Find the hourly production rate. The factory makes 1,800 widgets in 6 hours, so the rate is 1,800 / 6 = 300 widgets per hour. Step 2: Find the daily production. The factory works 8 hours per day, so it makes 300 × 8 = 2,400 widgets per day. Step 3: Find the number of days needed for the order. The order is for 7,200 widgets, so the number of days is 7,200 / 2,400 = 3 days. The answer is 3 full days.

2. Liam is planning a road trip and needs to calculate his fuel costs. His car can travel 315 miles on 15 gallons of gas. If the current price of gas is $3.60 per gallon, how much will it cost him in dollars to drive 420 miles? Answer: 72 Solution: The car travels 315 miles on 15 gallons. Miles per gallon = 315 / 15 315 ÷ 15 = 21 So the car gets 21 miles per gallon.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Find the car's fuel efficiency (miles per gallon)** The car travels 315 miles on 15 gallons. Miles per gallon = 315 / 15 315 ÷ 15 = 21 So the car gets 21 miles per gallon. --- **Step 2: Find gallons needed for 420 miles** Gallons needed = (miles to travel) / (miles per gallon) Gallons needed = 420 / 21 420 ÷ 21 = 20 So Liam needs 20 gallons for 420 miles. --- **Step 3: Calculate cost** Price per gallon = $3.60 Total cost = (gallons needed) × (price per gallon) Total cost = 20 × 3.60 20 × 3.60 = 72.00 --- **Final answer:** 72

3. Sarah is buying oranges at the grocery store. A bag of 7 oranges costs $7. What is the price per orange? Answer: 1 Solution: The total cost is $7 for 7 oranges. To find the cost per orange, divide the total cost by the number of oranges: $7 ÷ 7 = $1.
   Full step-by-step solution

   Step 1: The total cost is $7 for 7 oranges. Step 2: To find the cost per orange, divide the total cost by the number of oranges: $7 ÷ 7 = $1. The answer is $1 per orange.

4. A rectangular prism is drawn with dimensions 12 cm by 8 cm by 5 cm. If you were to paint all the outside surfaces of this prism, what total surface area would you cover in square centimeters? Answer: 392 Solution: Identify the dimensions: length = 12 cm, width = 8 cm, height = 5 cm Calculate area of front and back faces: 2 × (12 × 5) = 2 × 60 = 120 cm² Calculate area of left and right faces: 2 × (8 × 5) = 2 × 40 = 80 cm² Calculate area of top and bottom faces: 2 × (12 × 8) = 2 × 96 = 192 cm² Add all…
   Full step-by-step solution

   Step 1: Identify the dimensions: length = 12 cm, width = 8 cm, height = 5 cm Step 2: Calculate area of front and back faces: 2 × (12 × 5) = 2 × 60 = 120 cm² Step 3: Calculate area of left and right faces: 2 × (8 × 5) = 2 × 40 = 80 cm² Step 4: Calculate area of top and bottom faces: 2 × (12 × 8) = 2 × 96 = 192 cm² Step 5: Add all areas: 120 + 80 + 192 = 392 cm² The answer is 392.

5. A rectangular prism is drawn with dimensions: length = 12 cm, width = 8 cm, and height = 5 cm. If you were to paint all the outside surfaces of this prism, what total surface area would you cover in square centimeters? Answer: 392 Solution: Front/back area = length × height = 12 cm × 5 cm = 60 cm² Since there are 2 of these faces: 60 cm² × 2 = 120 cm² Left/right area = width × height = 8 cm × 5 cm = 40 cm² Since there are 2 of these faces: 40 cm² × 2 = 80 cm² Top/bottom area = length × width = 12 cm × 8 cm = 96 cm² Since there are…
   Full step-by-step solution

   Step 1: Calculate the area of the front and back faces Front/back area = length × height = 12 cm × 5 cm = 60 cm² Since there are 2 of these faces: 60 cm² × 2 = 120 cm² Step 2: Calculate the area of the left and right faces Left/right area = width × height = 8 cm × 5 cm = 40 cm² Since there are 2 of these faces: 40 cm² × 2 = 80 cm² Step 3: Calculate the area of the top and bottom faces Top/bottom area = length × width = 12 cm × 8 cm = 96 cm² Since there are 2 of these faces: 96 cm² × 2 = 192 cm² Step 4: Add all the areas together Total surface area = 120 cm² + 80 cm² + 192 cm² = 392 cm² The answer is 392.

6. Liam is making a large batch of lemonade for a school event. His recipe requires 3.5 cups of lemon juice to make 14 servings. If he needs to prepare enough lemonade for 60 students, how many cups of lemon juice should he use? Answer: 15 Solution: First, find how many cups of lemon juice are needed per serving. The recipe uses 3.5 cups for 14 servings. So, cups per serving = 3.5 cups ÷ 14 servings.
   Full step-by-step solution

   First, find how many cups of lemon juice are needed per serving. The recipe uses 3.5 cups for 14 servings. So, cups per serving = 3.5 cups ÷ 14 servings. 3.5 ÷ 14 = 0.25 cups per serving. Now, Liam needs to make lemonade for 60 students, meaning 60 servings. Multiply the cups per serving by the number of servings needed: 0.25 cups/serving × 60 servings = 15 cups. So, Liam should use 15 cups of lemon juice.

7. A rectangular prism is drawn with dimensions: length = 15 cm, width = 10 cm, and height = 6 cm. The prism is then scaled up proportionally so that its new surface area is exactly 4 times larger than the original surface area. What is the height of the scaled-up prism? Answer: 12 Solution: Calculate the original surface area of the prism. Surface area = 2(length × width + length × height + width × height) Surface area = 2(15 × 10 + 15 × 6 + 10 × 6) Surface area = 2(150 + 90 + 60) Surface area = 2(300) Surface area = 600 cm² The new surface area is 4 times the original.
   Full step-by-step solution

   Step 1: Calculate the original surface area of the prism. Surface area = 2(length × width + length × height + width × height) Surface area = 2(15 × 10 + 15 × 6 + 10 × 6) Surface area = 2(150 + 90 + 60) Surface area = 2(300) Surface area = 600 cm² Step 2: The new surface area is 4 times the original. New surface area = 4 × 600 = 2400 cm² Step 3: When scaling a 3D shape proportionally, surface area increases by the square of the scale factor. Let k be the scale factor. New surface area = k² × original surface area 2400 = k² × 600 k² = 2400 ÷ 600 k² = 4 k = 2 (we take the positive root since it's scaling up) Step 4: Apply the scale factor to find the new height. New height = original height × k New height = 6 × 2 New height = 12 cm The answer is 12 cm.