Rational Properties

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Answer Key & Explanations

Rational Properties · Grade 7 · Worksheet 1

1. A rectangular swimming pool is drawn on a coordinate plane with corners at (0, 0), (20, 0), (20, 12), and (0, 12). A triangular safety zone is marked off inside the pool with vertices at (0, 0), (20, 0), and (10, 8). What is the area of the pool that remains available for swimming after excluding the safety zone? 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 vertices at (0, 0), (20, 0), and (10, 8) Base of triangle = distance between (0, 0) and (20, 0) = 20 units Height of triangle = vertical distance from base to…
   Full step-by-step solution

   Step 1: Find 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: Find the area of the triangular safety zone The triangle has vertices at (0, 0), (20, 0), and (10, 8) Base of triangle = distance between (0, 0) and (20, 0) = 20 units Height of triangle = vertical distance from base to point (10, 8) = 8 units Area of triangle = 1/2 × base × height = 1/2 × 20 × 8 = 80 square units Step 3: Find the remaining swimming area Remaining area = area of rectangle - area of triangle Remaining area = 240 - 80 = 160 square units The answer is 160.

2. (-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.

3. Olivia is tracking the temperature changes in her city for a science project. At 6:00 AM, the temperature was -8.5°C. By noon, the temperature had increased by 15.75°C. Then, from noon to 6:00 PM, the temperature decreased by 10.25°C. Olivia wants to use the associative property of addition to verify that she can group the temperature changes differently and still get the same final temperature. What is the final temperature at 6:00 PM? Answer: -3 Solution: Write the temperature changes as an addition expression: -8.5 + 15.75 + (-10.25). Group the first two numbers: (-8.5 + 15.75) + (-10.25). Calculate inside the parentheses: -8.5 + 15.75 = 7.25.
   Full step-by-step solution

   Step 1: Write the temperature changes as an addition expression: -8.5 + 15.75 + (-10.25). Step 2: Group the first two numbers: (-8.5 + 15.75) + (-10.25). Step 3: Calculate inside the parentheses: -8.5 + 15.75 = 7.25. Step 4: Then add the third number: 7.25 + (-10.25) = -3. Step 5: Now group the last two numbers: -8.5 + (15.75 + (-10.25)). Step 6: Calculate inside the parentheses: 15.75 + (-10.25) = 5.5. Step 7: Then add the first number: -8.5 + 5.5 = -3. Both groupings give the same result, verifying the associative property. The final temperature at 6:00 PM is -3°C. The answer is -3.

4. (-12/7) × (21/4) + (27/2) ÷ (-9/4) = ? Answer: -15 Solution: Multiply (-12/7) × (21/4) = (-12 × 21) / (7 × 4) = -252 / 28 = -9 Divide (27/2) ÷ (-9/4) = (27/2) × (-4/9) = (27 × -4) / (2 × 9) = -108 / 18 = -6 Add the results: -9 + (-6) = -9 - 6 = -15 The answer is -15.
   Full step-by-step solution

   Step 1: Multiply (-12/7) × (21/4) = (-12 × 21) / (7 × 4) = -252 / 28 = -9 Step 2: Divide (27/2) ÷ (-9/4) = (27/2) × (-4/9) = (27 × -4) / (2 × 9) = -108 / 18 = -6 Step 3: Add the results: -9 + (-6) = -9 - 6 = -15 The answer is -15.

5. Emma is mixing a special paint color that requires combining red, blue, and yellow paint in a ratio of 3:5:2. If she uses 450 milliliters of blue paint, how many total milliliters of paint will she need to mix the complete color combination? Answer: 900 Solution: The ratio of red:blue:yellow is 3:5:2, so blue represents 5 parts out of the total. We know 5 parts = 450 ml, so 1 part = 450 ÷ 5 = 90 ml. Total parts = 3 + 5 + 2 = 10 parts.
   Full step-by-step solution

   Step 1: The ratio of red:blue:yellow is 3:5:2, so blue represents 5 parts out of the total. Step 2: We know 5 parts = 450 ml, so 1 part = 450 ÷ 5 = 90 ml. Step 3: Total parts = 3 + 5 + 2 = 10 parts. Step 4: Total paint = 10 parts × 90 ml per part = 900 ml. The answer is 900 milliliters.

6. Mere draws a rectangular vegetable garden on a coordinate grid. The corners are at (-12, 0), (16, 0), (16, 11), and (-12, 11). Inside the garden, she marks a triangular compost area with vertices at (-12, 0), (16, 0), and (2, 11). Use the distributive property to calculate the area of the garden that is NOT the compost area. Answer: 154 Solution: Find the area of the rectangular garden. Length = 16 - (-12) = 28 units. Width = 11 units.
   Full step-by-step solution

   Step 1: Find the area of the rectangular garden. Length = 16 - (-12) = 28 units. Width = 11 units. Area of rectangle = 28 * 11 = 308 square units. Step 2: Find the area of the triangular compost area. The base of the triangle is from (-12, 0) to (16, 0), so base = 28 units. The height is the vertical distance from the base to the top vertex (2, 11), so height = 11 units. Area of triangle = 1/2 * base * height = 1/2 * 28 * 11 = 154 square units. Step 3: Subtract the triangle area from the rectangle area to find the remaining garden area. Using the distributive property: 308 - 154 = 28*11 - 1/2*28*11 = 28*11*(1 - 1/2) = 28*11*1/2 = 154 square units. The answer is 154.

7. Emma is mixing a special fruit punch for a school event. She combines 2/3 of a liter of orange juice with 1.5 liters of apple juice and 0.75 liters of cranberry juice. Then she pours the mixture into cups that each hold 0.25 liters. How many full cups of fruit punch can Emma serve? Answer: 11 Solution: Orange juice: 2/3 liter ≈ 0.6667 liters Apple juice: 1.5 liters Cranberry juice: 0.75 liters Total = 2/3 + 1.5 + 0.75 = 0.6667 + 1.5 + 0.75 = 2.9167 liters Convert 2/3 to decimal for easier calculation 2/3 = 0.6667 Total = 0.6667 + 1.5 + 0.75 = 2.9167 liters 2.9167 ÷ 0.25 = 11.6668 Since we can…
   Full step-by-step solution

   Step 1: Find the total volume of fruit punch Orange juice: 2/3 liter ≈ 0.6667 liters Apple juice: 1.5 liters Cranberry juice: 0.75 liters Total = 2/3 + 1.5 + 0.75 = 0.6667 + 1.5 + 0.75 = 2.9167 liters Step 2: Convert 2/3 to decimal for easier calculation 2/3 = 0.6667 Total = 0.6667 + 1.5 + 0.75 = 2.9167 liters Step 3: Divide total volume by cup size 2.9167 ÷ 0.25 = 11.6668 Step 4: Determine number of full cups Since we can only serve full cups, we take the whole number part: 11 cups Emma can serve 11 full cups of fruit punch.