Unit Rates

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Unit Rates · Grade 6 · Worksheet 2

1. A rectangular swimming pool is drawn on a coordinate plane with corners at (0, 0), (20, 0), (20, 12), and (0, 12). The pool is divided into sections by drawing lines from (5, 0) to (5, 12), (10, 0) to (10, 12), and (15, 0) to (15, 12). If each section represents a different swimming lane, how many total lanes are created? Answer: 4 Solution: The rectangular pool has corners at (0, 0), (20, 0), (20, 12), and (0, 12), which means it spans from x = 0 to x = 20 and y = 0 to y = 12.
   Full step-by-step solution

   Step 1: The rectangular pool has corners at (0, 0), (20, 0), (20, 12), and (0, 12), which means it spans from x = 0 to x = 20 and y = 0 to y = 12. Step 2: Vertical lines are drawn at x = 5, x = 10, and x = 15 from the bottom to the top of the rectangle. Step 3: Each vertical line divides the rectangle into additional sections. With no lines, there would be 1 section. Step 4: The first line at x = 5 creates 2 sections. Step 5: The second line at x = 10 creates 3 sections. Step 6: The third line at x = 15 creates 4 sections. Step 7: Therefore, the pool is divided into 4 swimming lanes. The answer is 4.

2. Emma is planning a road trip from her hometown to visit her grandparents. Her car's fuel efficiency is 32 miles per gallon on the highway. If the total distance of her trip is 384 miles and gasoline costs $3.75 per gallon, how much will Emma spend on gasoline for the one-way trip? Answer: 45 Solution: Calculate how many gallons of gasoline Emma needs for the trip. Distance = 384 miles Fuel efficiency = 32 miles per gallon Gallons needed = 384 ÷ 32 = 12 gallons Calculate the total cost of gasoline.
   Full step-by-step solution

   Step 1: Calculate how many gallons of gasoline Emma needs for the trip. Distance = 384 miles Fuel efficiency = 32 miles per gallon Gallons needed = 384 ÷ 32 = 12 gallons Step 2: Calculate the total cost of gasoline. Cost per gallon = $3.75 Total cost = 12 × $3.75 = $45 The answer is $45.

3. If a car travels 480 miles on 15 gallons of gas, what is the unit rate in miles per gallon? Answer: 32 Solution: Identify the total distance traveled: 480 miles Identify the total gallons used: 15 gallons Calculate the unit rate by dividing total miles by total gallons: 480 ÷ 15 Perform the division: 480 ÷ 15 = 32 The unit rate is 32 miles per gallon.
   Full step-by-step solution

   Step 1: Identify the total distance traveled: 480 miles Step 2: Identify the total gallons used: 15 gallons Step 3: Calculate the unit rate by dividing total miles by total gallons: 480 ÷ 15 Step 4: Perform the division: 480 ÷ 15 = 32 Step 5: The unit rate is 32 miles per gallon.

4. Liam is mixing paint to create a specific shade of purple. The recipe requires a ratio of 5 parts blue paint to 3 parts red paint. If Liam uses 1.5 liters of red paint, how many liters of blue paint should he use to maintain the correct ratio? Answer: 2.5 Solution: We are told the ratio of blue paint to red paint is 5 parts blue to 3 parts red. That means for every 3 liters of red paint, we need 5 liters of blue paint. Write the ratio as a fraction.
   Full step-by-step solution

   We are told the ratio of blue paint to red paint is 5 parts blue to 3 parts red. That means for every 3 liters of red paint, we need 5 liters of blue paint. Step 1: Write the ratio as a fraction. Blue / Red = 5 / 3 Step 2: Let B be the blue paint needed when red paint is 1.5 liters. So: B / 1.5 = 5 / 3 Step 3: Solve for B by multiplying both sides by 1.5. B = (5 / 3) × 1.5 Step 4: Write 1.5 as a fraction: 1.5 = 3/2. So: B = (5 / 3) × (3 / 2) Step 5: Cancel the 3's: B = 5 × (1 / 2) B = 5 / 2 B = 2.5 So Liam needs 2.5 liters of blue paint.

5. If 2.5 kg of rice costs $8.75, what is the unit rate in dollars per kg? Answer: 3.5 Solution: Identify the total cost and the total amount. Total cost = $8.75, Total amount = 2.5 kg Calculate the unit rate by dividing the total cost by the total amount.
   Full step-by-step solution

   Step 1: Identify the total cost and the total amount. Total cost = $8.75, Total amount = 2.5 kg Step 2: Calculate the unit rate by dividing the total cost by the total amount. Unit rate = Total cost ÷ Total amount Step 3: Perform the division: 8.75 ÷ 2.5 Step 4: 8.75 ÷ 2.5 = 3.5 Step 5: The unit rate is $3.5 per kg.

6. A printing company uses a special ink mixture where the ratio of black ink to color ink is 7:4. If the company needs to prepare 1,650 liters of this ink mixture for a large order, how many liters of black ink should they use? Answer: 1050 Solution: The ratio of black ink to color ink is 7:4, which means for every 7 parts black ink, there are 4 parts color ink.
   Full step-by-step solution

   Step 1: The ratio of black ink to color ink is 7:4, which means for every 7 parts black ink, there are 4 parts color ink. Step 2: Find the total number of parts: 7 + 4 = 11 parts Step 3: The total mixture is 1,650 liters, so each part equals: 1,650 ÷ 11 = 150 liters Step 4: Black ink represents 7 parts, so: 7 × 150 = 1,050 liters Therefore, the company should use 1,050 liters of black ink.

7. A factory produces 2,400 widgets in 8 hours of operation. What is the unit rate of widget production per hour? Answer: 300 Solution: We are told the factory produces 2,400 widgets in 8 hours. We want the unit rate of widget production per hour, which means widgets per 1 hour. Total widgets / Total hours = 2400 widgets / 8 hours.
   Full step-by-step solution

   We are told the factory produces 2,400 widgets in 8 hours. We want the unit rate of widget production per hour, which means widgets per 1 hour. Step 1: Write the rate as a fraction: Total widgets / Total hours = 2400 widgets / 8 hours. Step 2: To find the unit rate (per 1 hour), divide 2400 by 8. Step 3: Perform the division: 2400 ÷ 8 = 300. Step 4: Interpret the result: 300 widgets per hour. So the unit rate is 300 widgets per hour.

8. A rectangular prism is drawn with dimensions: length = 15 cm, width = 10 cm, and height = 8 cm. If you were to paint all the outside faces of this prism, what is the total surface area in square centimeters? Answer: 700 Solution: Calculate the area of the front and back faces (length × height) Area = 15 cm × 8 cm = 120 cm² Since there are 2 such faces: 2 × 120 cm² = 240 cm² Calculate the area of the left and right faces (width × height) Area = 10 cm × 8 cm = 80 cm² Since there are 2 such faces: 2 × 80 cm² = 160 cm²…
   Full step-by-step solution

   Step 1: Calculate the area of the front and back faces (length × height) Area = 15 cm × 8 cm = 120 cm² Since there are 2 such faces: 2 × 120 cm² = 240 cm² Step 2: Calculate the area of the left and right faces (width × height) Area = 10 cm × 8 cm = 80 cm² Since there are 2 such faces: 2 × 80 cm² = 160 cm² Step 3: Calculate the area of the top and bottom faces (length × width) Area = 15 cm × 10 cm = 150 cm² Since there are 2 such faces: 2 × 150 cm² = 300 cm² Step 4: Add all the areas together 240 cm² + 160 cm² + 300 cm² = 700 cm² The total surface area is 700 square centimeters.