Rational Applications

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Answer Key & Explanations

Rational Applications · Grade 7 · Worksheet 2

1. Hana is monitoring the water level in a reservoir. The water level is measured in meters above sea level. On Monday, the level is -35 meters. Over the next three days, the level changes as follows: Tuesday it rises 47 meters, Wednesday it drops 12 meters, and Thursday it rises 28 meters. What is the water level in the reservoir on Thursday? Answer: 28 Solution: Start with Monday's level: -35 meters. Tuesday rises 47 meters: -35 + 47 = 12 meters. Wednesday drops 12 meters: 12 - 12 = 0 meters.
   Full step-by-step solution

   Step 1: Start with Monday's level: -35 meters. Step 2: Tuesday rises 47 meters: -35 + 47 = 12 meters. Step 3: Wednesday drops 12 meters: 12 - 12 = 0 meters. Step 4: Thursday rises 28 meters: 0 + 28 = 28 meters. The water level on Thursday is 28 meters above sea level.

2. (-3/4 + 2/3) × 12 = ? Answer: -1 Solution: Find a common denominator for -3/4 and 2/3. The least common denominator is 12. Convert -3/4 to -9/12 (multiply numerator and denominator by 3).
   Full step-by-step solution

   Step 1: Find a common denominator for -3/4 and 2/3. The least common denominator is 12. Step 2: Convert -3/4 to -9/12 (multiply numerator and denominator by 3). Step 3: Convert 2/3 to 8/12 (multiply numerator and denominator by 4). Step 4: Add the fractions: -9/12 + 8/12 = -1/12. Step 5: Multiply the result by 12: (-1/12) × 12 = -12/12 = -1. The answer is -1.

3. Aroha is tracking the temperature changes in her freezer over a week. On Monday, the temperature was -19°C. On Tuesday, the temperature dropped by 7°C. On Wednesday, it rose by 13°C. On Thursday, it dropped by 11°C. On Friday, it rose by 9°C. What is the final temperature in the freezer on Friday? Answer: -15 Solution: Start at Monday's temperature: -19°C. Tuesday drops by 7°C: -19 - 7 = -26°C. Wednesday rises by 13°C: -26 + 13 = -13°C.
   Full step-by-step solution

   Step 1: Start at Monday's temperature: -19°C. Step 2: Tuesday drops by 7°C: -19 - 7 = -26°C. Step 3: Wednesday rises by 13°C: -26 + 13 = -13°C. Step 4: Thursday drops by 11°C: -13 - 11 = -24°C. Step 5: Friday rises by 9°C: -24 + 9 = -15°C. The final temperature on Friday is -15°C.

4. (-3/4 + 2/3) × (-12) = ? Answer: 1 Solution: Find a common denominator for -3/4 and 2/3. The least common denominator is 12. Convert -3/4 to -9/12 and 2/3 to 8/12.
   Full step-by-step solution

   Step 1: Find a common denominator for -3/4 and 2/3. The least common denominator is 12. Step 2: Convert -3/4 to -9/12 and 2/3 to 8/12. Step 3: Add the fractions: -9/12 + 8/12 = -1/12. Step 4: Multiply the result by -12: (-1/12) × (-12) = 12/12 = 1. The answer is 1.

5. Olivia is tracking the balance of her savings account for a school project. She started with $50. On Monday, she spent $15 on school supplies. On Tuesday, she deposited her birthday money of $25. On Wednesday, she withdrew $30 to buy a gift. On Thursday, her account was charged a $10 fee for using an out-of-network ATM. What is Olivia's final account balance? Answer: 20 Solution: Start with the initial balance: $50 Monday - spent $15 on supplies. New balance: 50 - 15 = $35 Tuesday - deposited $25 birthday money. New balance: 35 + 25 = $60 Wednesday - withdrew $30 for a gift.
   Full step-by-step solution

   Step 1: Start with the initial balance: $50 Step 2: Monday - spent $15 on supplies. New balance: 50 - 15 = $35 Step 3: Tuesday - deposited $25 birthday money. New balance: 35 + 25 = $60 Step 4: Wednesday - withdrew $30 for a gift. New balance: 60 - 30 = $30 Step 5: Thursday - charged $10 ATM fee. New balance: 30 - 10 = $20 Final answer: Olivia's account balance is $20.

6. Emma is scuba diving in the ocean. She starts at the surface, at an elevation of 0 feet. She descends at a constant rate of 15 feet per minute for 4 minutes. Then, she ascends 10 feet per minute for 2 minutes. After that, she descends another 5 feet per minute for 3 minutes. What is Emma's final elevation relative to the surface? (Use negative numbers to represent depths below the surface.) Answer: -55 Solution: Calculate the first descent. She descends 15 feet per minute for 4 minutes. Change in elevation = -15 * 4 = -60 feet.
   Full step-by-step solution

   Step 1: Calculate the first descent. She descends 15 feet per minute for 4 minutes. Change in elevation = -15 * 4 = -60 feet. Her elevation is now -60 feet. Step 2: Calculate the ascent. She ascends 10 feet per minute for 2 minutes. Change in elevation = +10 * 2 = +20 feet. Her new elevation is -60 + 20 = -40 feet. Step 3: Calculate the final descent. She descends 5 feet per minute for 3 minutes. Change in elevation = -5 * 3 = -15 feet. Her final elevation is -40 + (-15) = -55 feet. Final answer: -55 feet.

7. A triangular park is drawn on a coordinate plane with vertices at (0, 0), (12, 0), and (6, 9). A rectangular playground is placed inside the park with corners at (3, 3), (9, 3), (9, 6), and (3, 6). What is the area of the park that is NOT occupied by the playground? Answer: 36 Solution: Find the area of the triangular park. The base of the triangle extends from (0, 0) to (12, 0), so the base length is 12 units.
   Full step-by-step solution

   Step 1: Find the area of the triangular park. The base of the triangle extends from (0, 0) to (12, 0), so the base length is 12 units. The height of the triangle is the vertical distance from the base to the vertex at (6, 9), which is 9 units. Area of triangle = (1/2) × base × height = (1/2) × 12 × 9 = 6 × 9 = 54 square units. Step 2: Find the area of the rectangular playground. The rectangle has corners at (3, 3) and (9, 6). Length of rectangle = 9 - 3 = 6 units Width of rectangle = 6 - 3 = 3 units Area of rectangle = length × width = 6 × 3 = 18 square units. Step 3: Subtract to find the remaining area. Area not occupied by playground = area of triangle - area of rectangle = 54 - 18 = 36 square units. The answer is 36.