Rational Addition/Subtraction

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Answer Key & Explanations

Rational Addition/Subtraction · Grade 7 · Worksheet 1

1. A rectangular garden is drawn on a coordinate plane with vertices at (-3, 2), (5, 2), (5, -4), and (-3, -4). A diagonal path is drawn from (-3, 2) to (5, -4). What is the length of this diagonal path? Round your answer to the nearest tenth. Answer: 10.0 Solution: Identify the coordinates of the two endpoints of the diagonal path. The problem says the diagonal goes from (-3, 2) to (5, -4). Let (x1, y1) = (-3, 2) and (x2, y2) = (5, -4).
   Full step-by-step solution

   Step 1: Identify the coordinates of the two endpoints of the diagonal path. The problem says the diagonal goes from (-3, 2) to (5, -4). Let (x1, y1) = (-3, 2) and (x2, y2) = (5, -4). Step 2: Recall the distance formula for two points in a coordinate plane. Distance = sqrt( (x2 - x1)^2 + (y2 - y1)^2 ) Step 3: Calculate the horizontal difference (x2 - x1). x2 - x1 = 5 - (-3) = 5 + 3 = 8 Step 4: Calculate the vertical difference (y2 - y1). y2 - y1 = -4 - 2 = -6 Step 5: Square each difference. (8)^2 = 64 (-6)^2 = 36 Step 6: Add the squares. 64 + 36 = 100 Step 7: Take the square root. sqrt(100) = 10.0 Step 8: Round to the nearest tenth. 10.0 is already to the nearest tenth. Final answer: The length of the diagonal path is 10.0.

2. 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.

3. Kaia is tracking the altitude of a hot air balloon. The balloon starts at an altitude of -135 feet (below sea level). It then ascends 287 feet. After that, it descends 419 feet. Finally, it ascends 173 feet. What is the final altitude of the balloon? Answer: -94 Solution: Start at -135 feet. Ascend 287 feet: -135 + 287 = 152 feet. Descend 419 feet: 152 - 419 = -267 feet.
   Full step-by-step solution

   Step 1: Start at -135 feet. Step 2: Ascend 287 feet: -135 + 287 = 152 feet. Step 3: Descend 419 feet: 152 - 419 = -267 feet. Step 4: Ascend 173 feet: -267 + 173 = -94 feet. The final altitude is -94 feet.

4. A scientist is tracking the temperature change in a chemical reaction. The reaction started at 15.75°C. In the first hour, the temperature increased by 7.25°C. In the second hour, it decreased by 12.50°C due to cooling. In the third hour, it increased again by 5.80°C. What was the final temperature of the reaction after all three hours? Answer: 16.3 Solution: Start with the initial temperature: 15.75°C Add the first hour's increase: 15.75 + 7.25 = 23.00°C Subtract the second hour's decrease: 23.00 - 12.50 = 10.50°C Add the third hour's increase: 10.50 + 5.80 = 16.30°C The final temperature is 16.3°C.
   Full step-by-step solution

   Step 1: Start with the initial temperature: 15.75°C Step 2: Add the first hour's increase: 15.75 + 7.25 = 23.00°C Step 3: Subtract the second hour's decrease: 23.00 - 12.50 = 10.50°C Step 4: Add the third hour's increase: 10.50 + 5.80 = 16.30°C The final temperature is 16.3°C.

5. Mason is keeping track of the stock price of a company over four days. On Monday, the stock price started at $72.50. On Tuesday, it dropped by $15.75. On Wednesday, it gained $9.40. On Thursday, it fell by $12.80. What is the final stock price after all these changes? Answer: 53.35 Solution: Start with Monday's price: $72.50 Subtract Tuesday's drop: 72.50 - 15.75 = 56.75 Add Wednesday's gain: 56.75 + 9.40 = 66.15 Subtract Thursday's fall: 66.15 - 12.80 = 53.35 The final stock price is $53.35.
   Full step-by-step solution

   Step 1: Start with Monday's price: $72.50 Step 2: Subtract Tuesday's drop: 72.50 - 15.75 = 56.75 Step 3: Add Wednesday's gain: 56.75 + 9.40 = 66.15 Step 4: Subtract Thursday's fall: 66.15 - 12.80 = 53.35 The final stock price is $53.35.

6. Isabella is tracking the value of a rare collectible coin over four months. In January, the coin's value was -$12.50 below its initial purchase price. In February, its value increased by $18.75. In March, its value decreased by $9.40. In April, its value increased again by $7.30. What is the final value of the coin relative to its initial purchase price? Answer: 4.15 Solution: Start with the initial value relative to purchase price: -12.50 dollars. Step 2: Add the February increase: -12.50 + 18.75 = 6.25 dollars. Step 3: Subtract the March decrease: 6.25 - 9.40 = -3.15 dollars.
   Full step-by-step solution

   Step 1: Start with the initial value relative to purchase price: -12.50 dollars. Step 2: Add the February increase: -12.50 + 18.75 = 6.25 dollars. Step 3: Subtract the March decrease: 6.25 - 9.40 = -3.15 dollars. Step 4: Add the April increase: -3.15 + 7.30 = 4.15 dollars. The final value is $4.15 above the initial purchase price.

7. Mason is tracking the value of a rare trading card. The card's value started at -$7.25 relative to its original purchase price. Last month, the value increased by $12.75. This week, the value decreased by $5.50. Then, a new trend caused the value to increase by $8.25. What is the current value of the trading card relative to its original purchase price? Answer: 8.25 Solution: Start with the initial value: -7.25 Add the first increase: -7.25 + 12.75 = 5.50 Subtract the decrease: 5.50 - 5.50 = 0.00 Add the final increase: 0.00 + 8.25 = 8.25 The final value is $8.25 above the original purchase price.
   Full step-by-step solution

   Step 1: Start with the initial value: -7.25 Step 2: Add the first increase: -7.25 + 12.75 = 5.50 Step 3: Subtract the decrease: 5.50 - 5.50 = 0.00 Step 4: Add the final increase: 0.00 + 8.25 = 8.25 The final value is $8.25 above the original purchase price.