Rational Properties

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Answer Key & Explanations

Rational Properties · Grade 7 · Worksheet 3

1. Aroha draws a rectangular garden on a coordinate grid with vertices at (-9, 0), (5, 0), (5, 7), and (-9, 7). Inside the garden, she marks a triangular flower bed with vertices at (-9, 0), (5, 0), and (-2, 7). Use the distributive property to show that the area of the remaining garden (excluding the flower bed) can be expressed as (1/2) * 14 * 7. What is the area of the remaining garden? Answer: 49 Solution: Find the area of the rectangle. Length = 5 - (-9) = 14 units. Width = 7 - 0 = 7 units.
   Full step-by-step solution

   Step 1: Find the area of the rectangle. Length = 5 - (-9) = 14 units. Width = 7 - 0 = 7 units. Area_rectangle = 14 * 7 = 98 square units. Step 2: Find the area of the triangle. Base = 5 - (-9) = 14 units. Height = vertical distance from base (y=0) to the third vertex at y=7, so height = 7 units. Area_triangle = (1/2) * base * height = (1/2) * 14 * 7 = 49 square units. Step 3: The remaining area = Area_rectangle - Area_triangle = 98 - 49. Use the distributive property: 98 - 49 = 14*7 - (1/2)*14*7 = 14*7 * (1 - 1/2) = 14*7 * (1/2) = (1/2) * 14 * 7. Step 4: Calculate (1/2) * 14 * 7 = 7 * 7 = 49 square units. The answer is 49.

2. Emma is planning a road trip from her home to her grandmother's 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, how much will Emma spend on fuel for the entire journey? Answer: $67.20 Solution: Calculate the total distance for the round trip: 420 km × 2 = 840 km Calculate the fuel needed: 840 km ÷ 15 km/L = 56 liters Calculate the total cost: 56 liters × $1.20/L = $67.20 The answer is $67.20.
   Full step-by-step solution

   Step 1: Calculate the total distance for the round trip: 420 km × 2 = 840 km Step 2: Calculate the fuel needed: 840 km ÷ 15 km/L = 56 liters Step 3: Calculate the total cost: 56 liters × $1.20/L = $67.20 The answer is $67.20.

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 by connecting points (0, 0), (8, 0), and (0, 6). What is the area of the deep end section of the pool? Answer: 216 Solution: Rectangle length = 20 units Rectangle width = 12 units Area of rectangle = length × width = 20 × 12 = 240 square units The triangle has vertices at (0, 0), (8, 0), and (0, 6) This is a right triangle with base = 8 units and height = 6 units Area of triangle = (1/2) × base × height = (1/2) × 8 ×…
   Full step-by-step solution

   Step 1: Calculate the area of the rectangular pool Rectangle length = 20 units Rectangle width = 12 units Area of rectangle = length × width = 20 × 12 = 240 square units Step 2: Calculate 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 base = 8 units and height = 6 units Area of triangle = (1/2) × base × height = (1/2) × 8 × 6 = 24 square units Step 3: Calculate the area of the deep end section Deep end area = Total pool area - Shallow end area Deep end area = 240 - 24 = 216 square units The answer is 216.

4. 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.

5. (-3/4) × (2/3) + (5/6) ÷ (1/4) = ? Answer: 3 Solution: First, multiply the fractions: (-3/4) × (2/3) = (-3×2)/(4×3) = -6/12 = -1/2 Next, divide the fractions: (5/6) ÷ (1/4) = (5/6) × (4/1) = (5×4)/(6×1) = 20/6 = 10/3 Now add the results: -1/2 + 10/3 Find a common denominator (6): -3/6 + 20/6 = 17/6 Convert to mixed number: 17/6 = 2 5/6 Simplify: 2…
   Full step-by-step solution

   Step 1: First, multiply the fractions: (-3/4) × (2/3) = (-3×2)/(4×3) = -6/12 = -1/2 Step 2: Next, divide the fractions: (5/6) ÷ (1/4) = (5/6) × (4/1) = (5×4)/(6×1) = 20/6 = 10/3 Step 3: Now add the results: -1/2 + 10/3 Step 4: Find a common denominator (6): -3/6 + 20/6 = 17/6 Step 5: Convert to mixed number: 17/6 = 2 5/6 Step 6: Simplify: 2 5/6 = 3 The answer is 3.

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 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 the width is 10.5 meters. Step 2: Find the perimeter of the rectangular garden. Perimeter formula for a rectangle: P = 2 * (length + width). P = 2 * (15.75 + 10.5). First, add length and width: 15.75 + 10.5 = 26.25. Then multiply by 2: 2 * 26.25 = 52.5. So the perimeter is 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. Break it down: 52.5 * 10 = 525. 52.5 * 2.5 = 52.5 * (2 + 0.5) = 52.5 * 2 = 105, plus 52.5 * 0.5 = 26.25, so 105 + 26.25 = 131.25. Now add: 525 + 131.25 = 656.25. So the total cost is $656.25. Final Answer: 656.25

7. (-3/4) × (2/3) + (5/6) ÷ (2/5) = ? Answer: 1.5 Solution: First, multiply the fractions: (-3/4) × (2/3) = (-3×2)/(4×3) = -6/12 = -1/2 Next, divide the fractions: (5/6) ÷ (2/5) = (5/6) × (5/2) = (5×5)/(6×2) = 25/12 Now add the results: -1/2 + 25/12 Find a common denominator (12): -6/12 + 25/12 = 19/12 Convert to decimal: 19 ÷ 12 = 1.5833...
   Full step-by-step solution

   Step 1: First, multiply the fractions: (-3/4) × (2/3) = (-3×2)/(4×3) = -6/12 = -1/2 Step 2: Next, divide the fractions: (5/6) ÷ (2/5) = (5/6) × (5/2) = (5×5)/(6×2) = 25/12 Step 3: Now add the results: -1/2 + 25/12 Step 4: Find a common denominator (12): -6/12 + 25/12 = 19/12 Step 5: Convert to decimal: 19 ÷ 12 = 1.5833... which rounds to 1.5 Step 6: The answer is 1.5

8. Emma is mixing a special cleaning solution that requires a 3:5 ratio of vinegar to water. She needs to make 12 liters of the solution for her science project. How many liters of vinegar and water should she use? Answer: 4.5 liters of vinegar and 7.5 liters of water Solution: The ratio is 3 parts vinegar to 5 parts water, so total parts = 3 + 5 = 8 parts Each part represents 12 liters ÷ 8 = 1.5 liters Vinegar amount = 3 parts × 1.5 liters = 4.5 liters Water amount = 5 parts × 1.5 liters = 7.5 liters Check: 4.5 + 7.5 = 12 liters total Emma should use 4.5 liters of…
   Full step-by-step solution

   Step 1: The ratio is 3 parts vinegar to 5 parts water, so total parts = 3 + 5 = 8 parts Step 2: Each part represents 12 liters ÷ 8 = 1.5 liters Step 3: Vinegar amount = 3 parts × 1.5 liters = 4.5 liters Step 4: Water amount = 5 parts × 1.5 liters = 7.5 liters Step 5: Check: 4.5 + 7.5 = 12 liters total Emma should use 4.5 liters of vinegar and 7.5 liters of water.