Rational Addition/Subtraction

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Answer Key & Explanations

Rational Addition/Subtraction · Grade 7 · Worksheet 1

1. Aisha is tracking the temperature changes in her science experiment. The temperature starts at -5.25°C. In the first phase, it increases by 8.75°C. In the second phase, it decreases by 12.5°C. In the final phase, it increases by 3.25°C. What is the final temperature? Answer: -5.75 Solution: Start with the initial temperature: -5.25°C Add the first phase increase: -5.25 + 8.75 = 3.5°C Subtract the second phase decrease: 3.5 - 12.5 = -9°C Add the final phase increase: -9 + 3.25 = -5.75°C The final temperature is -5.75°C.
   Full step-by-step solution

   Step 1: Start with the initial temperature: -5.25°C Step 2: Add the first phase increase: -5.25 + 8.75 = 3.5°C Step 3: Subtract the second phase decrease: 3.5 - 12.5 = -9°C Step 4: Add the final phase increase: -9 + 3.25 = -5.75°C The final temperature is -5.75°C.

2. -3/4 + 2/3 - 1/6 = ? Answer: -1/4 Solution: We have: -3/4 + 2/3 - 1/6 Find a common denominator for all fractions. The denominators are 4, 3, and 6. The least common multiple of 4, 3, and 6 is 12.
   Full step-by-step solution

   Let's solve step-by-step: We have: -3/4 + 2/3 - 1/6 Step 1: Find a common denominator for all fractions. The denominators are 4, 3, and 6. The least common multiple of 4, 3, and 6 is 12. Step 2: Convert each fraction to have denominator 12. -3/4 = ( -3 × 3 ) / (4 × 3) = -9/12 2/3 = ( 2 × 4 ) / (3 × 4) = 8/12 -1/6 = ( -1 × 2 ) / (6 × 2) = -2/12 Step 3: Rewrite the expression with the common denominator. -9/12 + 8/12 - 2/12 Step 4: Combine the numerators over the common denominator. Numerator: -9 + 8 - 2 First: -9 + 8 = -1 Then: -1 - 2 = -3 So we have: -3/12 Step 5: Simplify the fraction. -3/12 = -1/4 (dividing numerator and denominator by 3) Final answer: -1/4

3. A rectangular prism is drawn on a coordinate plane with vertices at (0,0,0), (8,0,0), (8,5,0), (0,5,0), (0,0,3), (8,0,3), (8,5,3), and (0,5,3). A smaller rectangular prism is removed from the center, creating a hollow box. The inner prism has vertices at (2,1,0), (6,1,0), (6,4,0), (2,4,0), (2,1,3), (6,1,3), (6,4,3), and (2,4,3). What is the volume of the remaining solid? Answer: 84 Solution: Calculate the volume of the large rectangular prism Length = 8 - 0 = 8 units Width = 5 - 0 = 5 units Height = 3 - 0 = 3 units Volume = 8 × 5 × 3 = 120 cubic units Calculate the volume of the smaller rectangular prism that was removed Length = 6 - 2 = 4 units Width = 4 - 1 = 3 units Height = 3 -…
   Full step-by-step solution

   Step 1: Calculate the volume of the large rectangular prism Length = 8 - 0 = 8 units Width = 5 - 0 = 5 units Height = 3 - 0 = 3 units Volume = 8 × 5 × 3 = 120 cubic units Step 2: Calculate the volume of the smaller rectangular prism that was removed Length = 6 - 2 = 4 units Width = 4 - 1 = 3 units Height = 3 - 0 = 3 units Volume = 4 × 3 × 3 = 36 cubic units Step 3: Subtract the smaller volume from the larger volume 120 - 36 = 84 cubic units The answer is 84.

4. Sophia is tracking the altitude of her drone during a flight. The drone starts at an altitude of -18.75 feet (below ground level). It then rises 24.5 feet, then descends 31.25 feet, and finally rises 15.75 feet. What is the final altitude of the drone in feet? Answer: -9.75 Solution: Start with initial altitude: -18.75 feet Add the first rise: -18.75 + 24.5 = 5.75 feet Subtract the descent: 5.75 - 31.25 = -25.5 feet Add the final rise: -25.5 + 15.75 = -9.75 feet The final altitude of the drone is -9.75 feet.
   Full step-by-step solution

   Step 1: Start with initial altitude: -18.75 feet Step 2: Add the first rise: -18.75 + 24.5 = 5.75 feet Step 3: Subtract the descent: 5.75 - 31.25 = -25.5 feet Step 4: Add the final rise: -25.5 + 15.75 = -9.75 feet The final altitude of the drone is -9.75 feet.

5. Charlotte is tracking the depth of a research submarine during an underwater expedition. The submarine starts at -250.75 feet relative to sea level. It then descends (goes deeper) by 185.5 feet to investigate a coral reef. After exploring the reef, it ascends (goes up) by 312.25 feet to avoid an underwater canyon. Finally, it descends again by 98.4 feet to collect samples near the ocean floor. What is the final depth of the submarine relative to sea level? Answer: -222.4 Solution: Start at -250.75 feet. Descend 185.5 feet means subtract 185.5: -250.75 - 185.5 = -436.25 feet. Ascend 312.25 feet means add 312.25: -436.25 + 312.25 = -124 feet.
   Full step-by-step solution

   Step 1: Start at -250.75 feet. Descend 185.5 feet means subtract 185.5: -250.75 - 185.5 = -436.25 feet. Step 2: Ascend 312.25 feet means add 312.25: -436.25 + 312.25 = -124 feet. Step 3: Descend 98.4 feet means subtract 98.4: -124 - 98.4 = -222.4 feet. The final depth is -222.4 feet.

6. A city's water reservoir contained 85,420.75 cubic meters of water. During a drought, the water level decreased by 12,680.5 cubic meters. Then a conservation program helped save 8,945.25 cubic meters through rainwater harvesting. Finally, an emergency water transfer added 15,320.75 cubic meters. What is the current water volume in the reservoir in cubic meters? Answer: 97006.25 Solution: Start with the initial water volume: 85,420.75 cubic meters Subtract the water lost during the drought: 85,420.75 - 12,680.5 = 72,740.25 cubic meters Add the water saved through conservation: 72,740.25 + 8,945.25 = 81,685.5 cubic meters Add the emergency water transfer: 81,685.5 + 15,320.75 =…
   Full step-by-step solution

   Step 1: Start with the initial water volume: 85,420.75 cubic meters Step 2: Subtract the water lost during the drought: 85,420.75 - 12,680.5 = 72,740.25 cubic meters Step 3: Add the water saved through conservation: 72,740.25 + 8,945.25 = 81,685.5 cubic meters Step 4: Add the emergency water transfer: 81,685.5 + 15,320.75 = 97,006.25 cubic meters The answer is 97006.25.