Unit Rates

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Unit Rates · Grade 6 · Worksheet 1

1. Liam is mixing paint for an art project. He needs to create a specific shade of purple by combining blue and red paint in a ratio of 5:3. If Liam uses 2.5 liters of blue paint, how many liters of red paint should he add to maintain the correct ratio? Answer: 1.5 Solution: The ratio of blue paint to red paint is 5:3. This means for every 5 parts of blue paint, we need 3 parts of red paint. Identify the given quantity.
   Full step-by-step solution

   Let's solve this step by step. Step 1: Understand the ratio. The ratio of blue paint to red paint is 5:3. This means for every 5 parts of blue paint, we need 3 parts of red paint. Step 2: Identify the given quantity. Liam uses 2.5 liters of blue paint. We need to find how many liters of red paint this corresponds to in the ratio. Step 3: Set up the proportion. Let the amount of red paint be R liters. From the ratio: Blue / Red = 5 / 3 So: 2.5 / R = 5 / 3 Step 4: Solve for R. Cross-multiply: 2.5 × 3 = 5 × R 7.5 = 5 × R Divide both sides by 5: R = 7.5 / 5 R = 1.5 Step 5: Conclusion. Liam should add 1.5 liters of red paint to maintain the 5:3 ratio with 2.5 liters of blue paint. Final answer: 1.5

2. If 3.75 kg of rice costs $16.50, what is the unit rate in dollars per kg? Answer: 4.4 Solution: Identify the total cost and the total amount. Total cost = $16.50, Total amount = 3.75 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 = $16.50, Total amount = 3.75 kg Step 2: Calculate the unit rate by dividing the total cost by the total amount. Unit rate = 16.50 ÷ 3.75 Step 3: Perform the division: 16.50 ÷ 3.75 = 4.4 Step 4: The unit rate is $4.4 per kg.

3. Liam is mixing paint for his art project. He needs to create a specific shade of purple by mixing red and blue paint in a ratio of 3:5. If Liam uses 2.4 liters of red paint, how many liters of blue paint should he use to maintain the correct ratio? Answer: 4 Solution: We are told the ratio of red to blue paint is 3:5. This means for every 3 parts of red, we need 5 parts of blue. Liam uses 2.4 liters of red paint.
   Full step-by-step solution

   We are told the ratio of red to blue paint is 3:5. This means for every 3 parts of red, we need 5 parts of blue. Liam uses 2.4 liters of red paint. We can set up a proportion to find the blue paint needed. Step 1: Write the ratio as a fraction: red / blue = 3 / 5 Step 2: Substitute the known red amount: 2.4 / blue = 3 / 5 Step 3: Cross-multiply to solve for blue: 2.4 * 5 = 3 * blue Step 4: Multiply: 12 = 3 * blue Step 5: Divide both sides by 3: blue = 12 / 3 Step 6: Simplify: blue = 4 So, Liam should use 4 liters of blue paint.

4. Emma is planning a hiking trip and needs to calculate how long it will take to complete a 15-kilometer trail. Her hiking app shows that she typically maintains a steady pace of 4 kilometers per hour. If she hikes at this consistent rate, how many hours will it take her to finish the entire trail? Answer: 3.75 Solution: Identify the known values. The total distance is 15 km, and the speed is 4 km per hour. Recall the relationship between time, distance, and speed: time = distance / speed.
   Full step-by-step solution

   Step 1: Identify the known values. The total distance is 15 km, and the speed is 4 km per hour. Step 2: Recall the relationship between time, distance, and speed: time = distance / speed. Step 3: Substitute the known values into the formula: time = 15 km / 4 km per hour. Step 4: Perform the division: 15 ÷ 4 = 3.75. Step 5: The result, 3.75, represents the number of hours. The answer is 3.75 hours.

5. 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.

6. If a car travels 450 km in 5 hours, what is its unit rate in km/h? Answer: 90 Solution: Identify the total distance traveled: 450 km Identify the total time taken: 5 hours Calculate the unit rate by dividing distance by time: 450 km ÷ 5 hours Perform the division: 450 ÷ 5 = 90 The unit rate is 90 km/h.
   Full step-by-step solution

   Step 1: Identify the total distance traveled: 450 km Step 2: Identify the total time taken: 5 hours Step 3: Calculate the unit rate by dividing distance by time: 450 km ÷ 5 hours Step 4: Perform the division: 450 ÷ 5 = 90 Step 5: The unit rate is 90 km/h.

7. 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.

8. Emma is training for a marathon and tracks her running pace. She runs 8 kilometers in 48 minutes. What is Emma's running speed in kilometers per hour? Answer: 10 Solution: Emma runs 8 km in 48 minutes. To find speed in km per hour, we need to find how many km she runs in 60 minutes. First, find her speed in km per minute: 8 km ÷ 48 minutes = 1/6 km per minute.
   Full step-by-step solution

   Step 1: Emma runs 8 km in 48 minutes. Step 2: To find speed in km per hour, we need to find how many km she runs in 60 minutes. Step 3: First, find her speed in km per minute: 8 km ÷ 48 minutes = 1/6 km per minute. Step 4: Multiply by 60 minutes to get km per hour: (1/6) × 60 = 10 km per hour. The answer is 10.

9. If a car travels 528 miles on 16 gallons of gas, what is the unit rate in miles per gallon? Answer: 33 Solution: Identify the total distance traveled: 528 miles Identify the total gallons used: 16 gallons Calculate the unit rate by dividing total miles by total gallons: 528 ÷ 16 Perform the division: 528 ÷ 16 = 33 The unit rate is 33 miles per gallon.
   Full step-by-step solution

   Step 1: Identify the total distance traveled: 528 miles Step 2: Identify the total gallons used: 16 gallons Step 3: Calculate the unit rate by dividing total miles by total gallons: 528 ÷ 16 Step 4: Perform the division: 528 ÷ 16 = 33 Step 5: The unit rate is 33 miles per gallon.