Rational Properties

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Rational Properties · Grade 7 · Worksheet 2

1. Tane is tracking the total profit from his online store over four weeks. In week 1, he makes a profit of $315. In week 2, he has a loss of $189 (shown as -189). In week 3, he makes a profit of $427. In week 4, he has a loss of $273 (shown as -273). Tane writes the expression 315 + (-189) + 427 + (-273). He wants to use the commutative and associative properties of addition to reorder and group the numbers to make the calculation easier. Show how Tane can use these properties to simplify the expression, then find his total profit for the four weeks. Answer: 280 Solution: Start with the original expression: 315 + (-189) + 427 + (-273). Use the commutative property to reorder the numbers so all positive numbers come first: 315 + 427 + (-189) + (-273).
   Full step-by-step solution

   Step 1: Start with the original expression: 315 + (-189) + 427 + (-273). Step 2: Use the commutative property to reorder the numbers so all positive numbers come first: 315 + 427 + (-189) + (-273). Step 3: Use the associative property to group the positive numbers together and the negative numbers together: (315 + 427) + [(-189) + (-273)]. Step 4: Add the positive numbers: 315 + 427 = 742. Step 5: Add the negative numbers: (-189) + (-273) = -462. Step 6: Add the two results: 742 + (-462) = 280. Tane's total profit for the four weeks is $280.

2. A rectangular swimming pool is drawn on a coordinate plane with vertices at (0, 0), (20, 0), (20, 12), and (0, 12). A triangular diving area is marked off inside the pool with vertices at (0, 0), (20, 0), and (10, 8). What is the area of the remaining pool space that is not part of the diving area? Answer: 160 Solution: Length = 20 - 0 = 20 units Width = 12 - 0 = 12 units Area of rectangle = length × width = 20 × 12 = 240 square units The triangle has a base along the bottom of the rectangle from (0,0) to (20,0), so base = 20 units The height of the triangle is the vertical distance from the base to point…
   Full step-by-step solution

   Step 1: Calculate the area of the rectangular pool Length = 20 - 0 = 20 units Width = 12 - 0 = 12 units Area of rectangle = length × width = 20 × 12 = 240 square units Step 2: Calculate the area of the triangular diving area The triangle has a base along the bottom of the rectangle from (0,0) to (20,0), so base = 20 units The height of the triangle is the vertical distance from the base to point (10,8), so height = 8 units Area of triangle = (1/2) × base × height = (1/2) × 20 × 8 = 80 square units Step 3: Calculate the remaining pool area Remaining area = rectangle area - triangle area = 240 - 80 = 160 square units The answer is 160.

3. (-14) × [(-28) + 42] = ? Answer: -196 Solution: Use the distributive property: a × (b + c) = a × b + a × c Here, a = -14, b = -28, c = 42 Compute (-14) × (-28) = 392 (negative × negative = positive) Compute (-14) × 42 = -588 (negative × positive = negative) Add the results: 392 + (-588) = 392 - 588 = -196 The answer is -196.
   Full step-by-step solution

   Step 1: Use the distributive property: a × (b + c) = a × b + a × c Here, a = -14, b = -28, c = 42 Step 2: Compute (-14) × (-28) = 392 (negative × negative = positive) Step 3: Compute (-14) × 42 = -588 (negative × positive = negative) Step 4: Add the results: 392 + (-588) = 392 - 588 = -196 The answer is -196.

4. Emma is planning a road trip from her home to her grandmother's house, which is 450 kilometers away. Her car's fuel efficiency is 12 kilometers per liter, and the current gas price is $1.25 per liter. If she also needs to pay a $15 toll for using the highway, how much will the entire trip cost Emma? Answer: 61.875 Solution: Distance = 450 km Fuel efficiency = 12 km per liter Fuel needed = 450 ÷ 12 = 37.5 liters Gas price = $1.25 per liter Fuel cost = 37.5 × 1.25 = $46.875 Toll = $15 Total cost = Fuel cost + Toll = 46.875 + 15 = $61.875 The answer is 61.875.
   Full step-by-step solution

   Step 1: Calculate the fuel needed for the trip Distance = 450 km Fuel efficiency = 12 km per liter Fuel needed = 450 ÷ 12 = 37.5 liters Step 2: Calculate the fuel cost Gas price = $1.25 per liter Fuel cost = 37.5 × 1.25 = $46.875 Step 3: Add the toll cost Toll = $15 Total cost = Fuel cost + Toll = 46.875 + 15 = $61.875 The answer is 61.875.

5. Mason is designing a rectangular mural on a coordinate grid. The mural has corners at (-12, 0), (12, 0), (12, 17), and (-12, 17). He plans to paint a triangular section with vertices at (-12, 0), (12, 0), and (0, 17) in one color, and the remaining rectangular area in another color. Using the distributive property of rational numbers, show that the area of the remaining rectangular part can be expressed as 2 * 12 * (17 - 17/2), and then calculate this area. Answer: 204 Solution: Find the area of the full rectangular mural. Length = distance from x = -12 to x = 12 = 12 - (-12) = 24 units. Width = 17 units.
   Full step-by-step solution

   Step 1: Find the area of the full rectangular mural. Length = distance from x = -12 to x = 12 = 12 - (-12) = 24 units. Width = 17 units. Area = 24 * 17 = 408 square units. Step 2: Find the area of the triangular section. Base = 24 units (from -12 to 12 along the x-axis). Height = 17 units (vertical distance from y = 0 to y = 17). Area of triangle = 1/2 * base * height = 1/2 * 24 * 17 = 204 square units. Step 3: Subtract to find the remaining area. Remaining area = 408 - 204 = 204 square units. Step 4: Show using the distributive property. The remaining area can be expressed as: 24 * 17 - 1/2 * 24 * 17. Factor out the common factor 24: 24 * (17 - 1/2 * 17) = 24 * (17 - 17/2) = 2 * 12 * (17 - 17/2). Step 5: Calculate: 2 * 12 * (17 - 17/2) = 2 * 12 * (34/2 - 17/2) = 2 * 12 * (17/2) = 2 * 12 * 8.5 = 2 * 102 = 204. The answer is 204.

6. Noah is designing a rectangular solar panel array on a coordinate grid. The corners of the array are at (-8, 0), (19, 0), (19, 14), and (-8, 14). A triangular section for wiring is marked with vertices at (-8, 0), (19, 0), and (5, 14). Using the distributive property of rational numbers, calculate the area of the solar panel array that remains after removing the triangular wiring section. Answer: 189 Solution: Find the area of the full rectangular solar panel array. Length = distance from x = -8 to x = 19 = 19 - (-8) = 27 units. Width = 14 units.
   Full step-by-step solution

   Step 1: Find the area of the full rectangular solar panel array. Length = distance from x = -8 to x = 19 = 19 - (-8) = 27 units. Width = 14 units. Area = 27 * 14 = 378 square units. Step 2: Find the area of the triangular wiring section. Base = 27 units (from -8 to 19 along the x-axis). Height = 14 units (vertical distance from y = 0 to y = 14). Area of triangle = 1/2 * base * height = 1/2 * 27 * 14 = 189 square units. Step 3: Use the distributive property to find the remaining area. Remaining area = area of rectangle - area of triangle = 27 * 14 - 1/2 * 27 * 14. Factor out the common factor 27: 27 * (14 - 1/2 * 14) = 27 * (14 - 7) = 27 * 7 = 189. The answer is 189.

7. (-24/5) × (15/8) + (35/6) ÷ (-7/3) = ? Answer: -11.5 Solution: Multiply (-24/5) × (15/8) = (-24 × 15) / (5 × 8) = -360 / 40 = -9 Divide (35/6) ÷ (-7/3) = (35/6) × (-3/7) = (35 × -3) / (6 × 7) = -105 / 42 = -2.5 Add the results: -9 + (-2.5) = -9 - 2.5 = -11.5 The answer is -11.5.
   Full step-by-step solution

   Step 1: Multiply (-24/5) × (15/8) = (-24 × 15) / (5 × 8) = -360 / 40 = -9 Step 2: Divide (35/6) ÷ (-7/3) = (35/6) × (-3/7) = (35 × -3) / (6 × 7) = -105 / 42 = -2.5 Step 3: Add the results: -9 + (-2.5) = -9 - 2.5 = -11.5 The answer is -11.5.