Rational Properties

lessonbunny.com

Answer Key & Explanations

Rational Properties · Grade 7 · Worksheet 2

1. (-5/6) × (4/7) + (3/8) ÷ (-2/9) = ? Answer: -2.125 Solution: Multiply (-5/6) × (4/7) = (-5×4)/(6×7) = -20/42 = -10/21 Divide (3/8) ÷ (-2/9) = (3/8) × (-9/2) = (3×-9)/(8×2) = -27/16 Add the results: -10/21 + (-27/16) = -10/21 - 27/16 Find common denominator (336): (-10×16)/336 - (27×21)/336 = -160/336 - 567/336 = -727/336 Simplify: -727 ÷ 336 = -2.125 The…
   Full step-by-step solution

   Step 1: Multiply (-5/6) × (4/7) = (-5×4)/(6×7) = -20/42 = -10/21 Step 2: Divide (3/8) ÷ (-2/9) = (3/8) × (-9/2) = (3×-9)/(8×2) = -27/16 Step 3: Add the results: -10/21 + (-27/16) = -10/21 - 27/16 Step 4: Find common denominator (336): (-10×16)/336 - (27×21)/336 = -160/336 - 567/336 = -727/336 Step 5: Simplify: -727 ÷ 336 = -2.125 The answer is -2.125.

2. Mason is organizing a charity fundraiser and needs to calculate the total cost for supplies. He buys 8 boxes of pencils at $12.50 each and 8 boxes of notebooks at $7.50 each. To find the total cost quickly, Mason uses the distributive property by rewriting the expression as 8 × (12.50 + 7.50). Show that Mason's method is correct by calculating the total cost using both the distributive property and the step-by-step addition method, then state the total cost. Answer: 160 Solution: Use the distributive property: 8 × (12.50 + 7.50) = 8 × 20.00 = 160.00. Then find the cost of notebooks: 8 × 7.50 = 60.00. Add them: 100.00 + 60.00 = 160.00.
   Full step-by-step solution

   Step 1: Use the distributive property: 8 × (12.50 + 7.50) = 8 × 20.00 = 160.00. Step 2: Verify by the step-by-step method: First find the cost of pencils: 8 × 12.50 = 100.00. Then find the cost of notebooks: 8 × 7.50 = 60.00. Add them: 100.00 + 60.00 = 160.00. Step 3: Both methods give the same total cost of $160.00, confirming the distributive property works with rational numbers (including negative numbers, though here all are positive). The total cost is $160.00.

3. A rectangular swimming pool is drawn on a coordinate plane with vertices 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: Area of rectangle = length × width = 20 × 12 = 240 square units The triangle has vertices at (0, 0), (8, 0), and (0, 6) Base of triangle = 8 units (from (0, 0) to (8, 0)) Height of triangle = 6 units (from (0, 0) to (0, 6)) Area of triangle = (1/2) × base × height = (1/2) × 8 × 6 = 24 square…
   Full step-by-step solution

   Step 1: Calculate the area of the rectangular pool 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) Base of triangle = 8 units (from (0, 0) to (8, 0)) Height of triangle = 6 units (from (0, 0) to (0, 6)) Area of triangle = (1/2) × base × height = (1/2) × 8 × 6 = 24 square units Step 3: Calculate the area of the deep end section Area of deep end = Total area - Area of shallow end Area of deep end = 240 - 24 = 216 square units The answer is 216.

4. Sophia is tracking the temperature changes in her city over a weekend. On Saturday morning, the temperature was -6°C. By noon, it increased by 11°C, but then dropped by 7°C by evening. On Sunday morning, the temperature was 4°C higher than Saturday evening. By Sunday afternoon, it increased by 5°C, then decreased by 9°C by night. What was the temperature on Sunday night? Use the associative property of addition to verify your answer by grouping the changes differently. Answer: -2 Solution: Find Saturday evening temperature. Start: -6°C. Increase by 11°C: -6 + 11 = 5°C.
   Full step-by-step solution

   Step 1: Find Saturday evening temperature. Start: -6°C. Increase by 11°C: -6 + 11 = 5°C. Drop by 7°C: 5 + (-7) = -2°C. So Saturday evening = -2°C. Step 2: Sunday morning = Saturday evening + 4°C = -2 + 4 = 2°C. Step 3: Sunday afternoon = 2 + 5 = 7°C. Step 4: Sunday night = 7 + (-9) = -2°C. Step 5: Verify using associative property. Group all changes from Saturday morning to Sunday night: Start: -6, then +11, -7, +4, +5, -9. Sum = (-6 + 11) + (-7 + 4) + (5 + (-9)) = 5 + (-3) + (-4) = -2. Or group differently: (-6 + (-7) + (-9)) + (11 + 4 + 5) = (-22) + 20 = -2. The answer is -2°C.

5. 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.

6. (-2/3) × (5/4) + (7/8) ÷ (-3/2) = ? Answer: -1.5 Solution: Multiply (-2/3) × (5/4) = (-2×5)/(3×4) = -10/12 = -5/6 Divide (7/8) ÷ (-3/2) = (7/8) × (-2/3) = (7×-2)/(8×3) = -14/24 = -7/12 Add the results: -5/6 + (-7/12) = -5/6 - 7/12 Find common denominator (12): (-5×2)/12 - 7/12 = -10/12 - 7/12 = -17/12 Convert to decimal: -17 ÷ 12 = -1.41666...
   Full step-by-step solution

   Step 1: Multiply (-2/3) × (5/4) = (-2×5)/(3×4) = -10/12 = -5/6 Step 2: Divide (7/8) ÷ (-3/2) = (7/8) × (-2/3) = (7×-2)/(8×3) = -14/24 = -7/12 Step 3: Add the results: -5/6 + (-7/12) = -5/6 - 7/12 Step 4: Find common denominator (12): (-5×2)/12 - 7/12 = -10/12 - 7/12 = -17/12 Step 5: Convert to decimal: -17 ÷ 12 = -1.41666... which rounds to -1.5 Step 6: The answer is -1.5

7. Mason draws a rectangle on a coordinate plane with vertices at (-7, 0), (7, 0), (7, 12), and (-7, 12). Inside this rectangle, he marks a triangular region using points (-7, 0), (7, 0), and (0, 8). Use the distributive property to show that the area of the rectangle minus the area of the triangle can be expressed as 1/2 * 7 * (2*12 - 8). Then find the remaining area. Answer: 112 Solution: Find the dimensions of the rectangle. Width = 7 - (-7) = 14. Height = 12 - 0 = 12.
   Full step-by-step solution

   Step 1: Find the dimensions of the rectangle. Width = 7 - (-7) = 14. Height = 12 - 0 = 12. Area of rectangle = 14 * 12. Step 2: The triangle has base = 14 (from (-7,0) to (7,0)) and height = 8. Area of triangle = 1/2 * 14 * 8. Step 3: Remaining area = rectangle area - triangle area = 14*12 - 1/2*14*8. Step 4: Factor out 14 using the distributive property: 14 * (12 - 1/2 * 8) = 14 * (12 - 4) = 14 * 8. Step 5: Alternatively, factor out 7 (half the width): 14 = 2 * 7, so area = 2*7*12 - 1/2*2*7*8 = 2*7*12 - 7*8 = 7*(2*12 - 8) = 7*(24 - 8) = 7*16 = 112. The answer is 112.