Rational Properties

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Answer Key & Explanations

Rational Properties · Grade 7 · Worksheet 3

1. Emma is planning a road trip from her home to her grandmother's house, which is 420 miles away. Her car's fuel efficiency is 28 miles per gallon, and gasoline costs $3.75 per gallon. If she wants to budget for the round trip, how much money should Emma set aside for gasoline? Answer: 112.50 Solution: Calculate the total distance for the round trip. Distance one way = 420 miles Round trip distance = 420 × 2 = 840 miles Calculate how many gallons of gasoline are needed.
   Full step-by-step solution

   Step 1: Calculate the total distance for the round trip. Distance one way = 420 miles Round trip distance = 420 × 2 = 840 miles Step 2: Calculate how many gallons of gasoline are needed. Fuel efficiency = 28 miles per gallon Gallons needed = Total distance ÷ Fuel efficiency = 840 ÷ 28 840 ÷ 28 = 30 gallons Step 3: Calculate the total cost of gasoline. Gasoline price = $3.75 per gallon Total cost = Gallons needed × Price per gallon = 30 × 3.75 30 × 3.75 = 112.50 Emma should set aside $112.50 for gasoline.

2. Noah is managing his online store's inventory and finances. He starts the month with $12,500 in his business account. He then spends $3,200 on new inventory, receives a payment of $8,750 from a customer, pays a shipping fee of $1,500, and finally gets a refund of $600 for a returned item. Noah wants to use the commutative and associative properties of addition to verify his ending balance. He writes the expression: 12500 + (-3200) + 8750 + (-1500) + 600. Show how to use the commutative property to reorder the numbers and the associative property to group them in a way that makes the calculation easier, then find Noah's ending balance. Answer: 17150 Solution: Start with the original expression: 12500 + (-3200) + 8750 + (-1500) + 600. Use the commutative property to reorder the numbers so that all positive numbers come first: 12500 + 8750 + 600 + (-3200) + (-1500).
   Full step-by-step solution

   Step 1: Start with the original expression: 12500 + (-3200) + 8750 + (-1500) + 600. Step 2: Use the commutative property to reorder the numbers so that all positive numbers come first: 12500 + 8750 + 600 + (-3200) + (-1500). Step 3: Use the associative property to group the positive numbers together and the negative numbers together: (12500 + 8750 + 600) + [(-3200) + (-1500)]. Step 4: Calculate the sum of the positive numbers: 12500 + 8750 = 21250, then 21250 + 600 = 21850. Step 5: Calculate the sum of the negative numbers: (-3200) + (-1500) = -4700. Step 6: Add the two results: 21850 + (-4700) = 17150. Noah's ending balance is $17,150.

3. A rectangular swimming pool is drawn on a coordinate plane with corners at (0, 0), (20, 0), (20, 12), and (0, 12). A triangular shallow end is marked off with vertices at (0, 0), (8, 0), and (0, 6). What is the area of the deep end section of the pool? Answer: 216 Solution: Find the area of the entire rectangular pool. The rectangle has length 20 units and width 12 units. Area of rectangle = length × width = 20 × 12 = 240 square units.
   Full step-by-step solution

   Step 1: Find the area of the entire rectangular pool. The rectangle has length 20 units and width 12 units. Area of rectangle = length × width = 20 × 12 = 240 square units. Step 2: Find the area of the triangular shallow end. The triangle has vertices at (0, 0), (8, 0), and (0, 6). This is a right triangle with legs of length 8 and 6 units. Area of triangle = (1/2) × base × height = (1/2) × 8 × 6 = (1/2) × 48 = 24 square units. Step 3: Subtract the triangular area from the rectangular area to find the deep end. Area of deep end = total area - shallow area = 240 - 24 = 216 square units. The answer is 216.

4. Emma is planning a road trip from her home to her grandparents' house, which is 420 kilometers away. Her car's fuel efficiency is 15 kilometers per liter, and the current fuel price is $1.20 per liter. If she needs to make the round trip (there and back), how much will Emma spend on fuel for the entire journey? Answer: $67.20 Solution: Calculate the total distance for the round trip. Distance one way = 420 km Round trip distance = 420 km × 2 = 840 km Calculate the total fuel needed.
   Full step-by-step solution

   Step 1: Calculate the total distance for the round trip. Distance one way = 420 km Round trip distance = 420 km × 2 = 840 km Step 2: Calculate the total fuel needed. Fuel efficiency = 15 km per liter Fuel needed = Total distance ÷ Fuel efficiency = 840 km ÷ 15 km/L = 56 liters Step 3: Calculate the total cost. Fuel price = $1.20 per liter Total cost = Fuel needed × Price per liter = 56 L × $1.20/L = $67.20 The answer is $67.20.

5. Ava is tracking the total distance she hikes over a weekend. On Saturday, she hikes 16 miles. On Sunday morning, she hikes another 21 miles, but then realizes she took a wrong turn and has to hike back 9 miles to get back on the correct trail. Later that afternoon, she hikes an additional 11 miles to reach the campground. Ava wants to use the commutative and associative properties of addition to verify her total distance. She writes the expression: 16 + 21 + (-9) + 11. Show how to reorder and group the numbers using the commutative and associative properties to make the calculation easier, then find Ava's total hiking distance for the weekend. Answer: 39 Solution: Start with the original expression: 16 + 21 + (-9) + 11. Use the commutative property to reorder the numbers so that all positive numbers are together: 16 + 21 + 11 + (-9).
   Full step-by-step solution

   Step 1: Start with the original expression: 16 + 21 + (-9) + 11. Step 2: Use the commutative property to reorder the numbers so that all positive numbers are together: 16 + 21 + 11 + (-9). Step 3: Use the associative property to group the positive numbers: (16 + 21 + 11) + (-9). Step 4: Calculate the sum of the positive numbers: 16 + 21 = 37, then 37 + 11 = 48. Step 5: Add the negative number: 48 + (-9) = 39. Step 6: Verify by adding in the original order: 16 + 21 = 37; 37 + (-9) = 28; 28 + 11 = 39. Ava's total hiking distance is 39 miles.

6. Liam is designing a rectangular garden with a length of 15.75 meters and a width that is 2/3 of the length. He needs to buy fencing material that costs $12.50 per meter to enclose the entire garden. How much will Liam spend on fencing? Answer: 656.25 Solution: Find the width of the garden. The width is 2/3 of the length. Length = 15.75 meters.
   Full step-by-step solution

   Step 1: Find the width of the garden. The width is 2/3 of the length. Length = 15.75 meters. Width = (2/3) * 15.75. First, calculate 15.75 * 2 = 31.5. Then divide by 3: 31.5 / 3 = 10.5. So, width = 10.5 meters. Step 2: Find the perimeter of the garden. Perimeter of a rectangle = 2 * (length + width). Length + width = 15.75 + 10.5 = 26.25 meters. Perimeter = 2 * 26.25 = 52.5 meters. Step 3: Calculate the total cost of fencing. Fencing costs $12.50 per meter. Total cost = perimeter * cost per meter. Total cost = 52.5 * 12.50. First, calculate 52.5 * 12.5. Multiply 52.5 * 10 = 525. Multiply 52.5 * 2.5 = 131.25. Add: 525 + 131.25 = 656.25. So, Liam will spend $656.25 on fencing.