Unit Rates

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Unit Rates · Grade 6 · Worksheet 3

1. 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.

2. Liam is mixing paint for an 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 blue paint, how many liters of red paint should he use to maintain the correct ratio? Answer: 1.44 Solution: We are told that the ratio of red to blue paint is 3:5. That means for every 3 parts red, there are 5 parts blue. Liam uses 2.4 liters of blue paint.
   Full step-by-step solution

   We are told that the ratio of red to blue paint is 3:5. That means for every 3 parts red, there are 5 parts blue. Liam uses 2.4 liters of blue paint. We need to find how many liters of red paint (let's call it R) will keep the ratio 3:5. Step 1: Set up the proportion Red / Blue = 3 / 5 So R / 2.4 = 3 / 5 Step 2: Solve for R Multiply both sides by 2.4: R = (3 / 5) × 2.4 Step 3: Calculate First, 3 / 5 = 0.6 Then 0.6 × 2.4 = 1.44 Step 4: Conclusion Liam should use 1.44 liters of red paint. ANSWER: 1.44

3. Noah is planning a hiking trip and needs to calculate his walking pace. On his last practice hike, he walked 12.5 kilometers in 2.5 hours. What was Noah's average walking speed in kilometers per hour? Answer: 5 Solution: Identify the distance walked: 12.5 kilometers Identify the time taken: 2.5 hours To find speed in kilometers per hour, divide distance by time: 12.5 ÷ 2.5 Calculate the division: 12.5 ÷ 2.5 = 5 Noah's average walking speed was 5 kilometers per hour.
   Full step-by-step solution

   Step 1: Identify the distance walked: 12.5 kilometers Step 2: Identify the time taken: 2.5 hours Step 3: To find speed in kilometers per hour, divide distance by time: 12.5 ÷ 2.5 Step 4: Calculate the division: 12.5 ÷ 2.5 = 5 Noah's average walking speed was 5 kilometers per hour.

4. 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.

5. If 2.5 kilograms of flour costs $8.75, what is the unit rate in dollars per kilogram? 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: 8.75 ÷ 2.5 Perform the division: 8.75 ÷ 2.5 = 3.5 The unit rate is $3.5 per kilogram.
   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: 8.75 ÷ 2.5 Step 3: Perform the division: 8.75 ÷ 2.5 = 3.5 Step 4: The unit rate is $3.5 per kilogram.

6. If 4.5 liters of paint cover 15 square meters, what is the unit rate in liters per square meter? Answer: 0.3 Solution: We are given that 4.5 liters of paint cover 15 square meters. We want the unit rate in liters per square meter. Understand what "unit rate in liters per square meter" means.
   Full step-by-step solution

   We are given that 4.5 liters of paint cover 15 square meters. We want the unit rate in liters per square meter. Step 1: Understand what "unit rate in liters per square meter" means. It means: how many liters are used for 1 square meter. So we divide liters by square meters. Step 2: Write the division: 4.5 liters ÷ 15 square meters Step 3: Perform the division: 4.5 ÷ 15 = ? First, 15 × 0.3 = 4.5, so 4.5 ÷ 15 = 0.3. Alternatively, you can write it as a fraction: 4.5 / 15 = 45 / 150 = 9 / 30 = 3 / 10 = 0.3 Step 4: Interpret the result: 0.3 liters per square meter. So the unit rate is 0.3 liters per square meter. Final answer: 0.3

7. 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.

8. Liam is making a large batch of lemonade for a school fair. His recipe requires 8 cups of lemon juice for every 12 cups of water to make 20 cups of lemonade. If he wants to make 75 cups of lemonade, how many cups of lemon juice will he need? Answer: 30 Solution: 8 cups lemon juice + 12 cups water = 20 cups lemonade. 8 cups lemon juice / 20 cups lemonade. Ratio = 8/20 = 2/5.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the original recipe** The recipe says: 8 cups lemon juice + 12 cups water = 20 cups lemonade. So the ratio of lemon juice to total lemonade is: 8 cups lemon juice / 20 cups lemonade. --- **Step 2: Find the ratio of lemon juice to total lemonade** Ratio = 8/20 = 2/5. This means 2/5 of the total lemonade is lemon juice. --- **Step 3: Apply the ratio to the new total** Liam wants 75 cups of lemonade. Lemon juice needed = (2/5) × 75. --- **Step 4: Calculate** (2/5) × 75 = (2 × 75) / 5 = 150 / 5 = 30. --- **Step 5: Conclusion** Liam will need **30 cups of lemon juice** to make 75 cups of lemonade. --- **Final answer:** 30