Rational Applications

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Answer Key & Explanations

Rational Applications · Grade 7 · Worksheet 3

1. Emma is planning a road trip from her home in Chicago to visit her grandparents in St. Louis, a distance of 300 miles. Her car gets 25 miles per gallon on the highway. If gasoline costs $3.80 per gallon and she also needs to pay $12 in tolls for the entire trip, how much will Emma spend in total for gasoline and tolls? Answer: 57.60 Solution: Calculate the gallons of gasoline needed for the trip. Distance = 300 miles Fuel efficiency = 25 miles per gallon Gallons needed = 300 ÷ 25 = 12 gallons Calculate the cost of gasoline.
   Full step-by-step solution

   Step 1: Calculate the gallons of gasoline needed for the trip. Distance = 300 miles Fuel efficiency = 25 miles per gallon Gallons needed = 300 ÷ 25 = 12 gallons Step 2: Calculate the cost of gasoline. Gasoline cost = 12 gallons × $3.80 per gallon 12 × 3.80 = 45.60 Gasoline cost = $45.60 Step 3: Add the toll cost to the gasoline cost. Toll cost = $12.00 Total cost = $45.60 + $12.00 = $57.60 Emma will spend $57.60 in total for gasoline and tolls.

2. A square-shaped farm is drawn on a coordinate grid with corners at (-7, -7), (7, -7), (7, 7), and (-7, 7). A rectangular pond is placed inside the farm with corners at (-5, -3), (5, -3), (5, 3), and (-5, 3). What is the area of the farm that is NOT covered by the pond? Answer: 136 Solution: Find the side length of the square farm. The x-coordinates of the left and right corners are -7 and 7, so the side length is 7 - (-7) = 14 units. Find the area of the square farm.
   Full step-by-step solution

   Step 1: Find the side length of the square farm. The x-coordinates of the left and right corners are -7 and 7, so the side length is 7 - (-7) = 14 units. Step 2: Find the area of the square farm. Area = side × side = 14 × 14 = 196 square units. Step 3: Find the length and width of the rectangular pond. The x-coordinates of the left and right corners are -5 and 5, so the length is 5 - (-5) = 10 units. The y-coordinates of the bottom and top corners are -3 and 3, so the width is 3 - (-3) = 6 units. Step 4: Find the area of the rectangular pond. Area = length × width = 10 × 6 = 60 square units. Step 5: Subtract the pond area from the farm area. Area not covered = 196 - 60 = 136 square units. The answer is 136.

3. Olivia is tracking her scuba diving depth. She starts at sea level (0 meters) and descends at a rate of 3 meters per minute for 7 minutes. Then she ascends 11 meters. Later, she descends another 5 meters. What is her final depth relative to sea level? Answer: -15 Solution: Olivia descends 3 meters per minute for 7 minutes. Total descent = 3 × 7 = 21 meters. Since descending is below sea level, her depth is -21 meters.
   Full step-by-step solution

   Step 1: Olivia descends 3 meters per minute for 7 minutes. Total descent = 3 × 7 = 21 meters. Since descending is below sea level, her depth is -21 meters. Step 2: She ascends 11 meters. Ascending moves toward the surface, so add 11: -21 + 11 = -10 meters. Step 3: She descends another 5 meters. Descending means going further down, so subtract 5: -10 - 5 = -15 meters. Olivia's final depth is 15 meters below sea level, or -15 meters.

4. Mason is monitoring the water level in a reservoir. The water level is currently at 27 feet below the normal level. After a heavy rainfall, the water level rises by 12 feet. Then, due to a drought, the water level drops by 7 feet. What is the final water level relative to the normal level? Answer: -22 Solution: The current water level is 27 feet below normal. Represent this as -27. The water level rises by 12 feet.
   Full step-by-step solution

   Step 1: The current water level is 27 feet below normal. Represent this as -27. Step 2: The water level rises by 12 feet. This is an increase, so add +12: -27 + 12 = -15. Step 3: Then the water level drops by 7 feet. This is a decrease, so subtract 7 or add -7: -15 + (-7) = -22. Step 4: The final water level is 22 feet below the normal level. The answer is -22.

5. Emma's bank account balance was -$45. She deposited $80 and then wrote a check for $25. What is her new balance? Answer: $10 Solution: Start with the initial balance: -$45. Add the deposit of $80: -45 + 80 = 35. Subtract the check amount of $25: 35 - 25 = 10.
   Full step-by-step solution

   Step 1: Start with the initial balance: -$45. Step 2: Add the deposit of $80: -45 + 80 = 35. Step 3: Subtract the check amount of $25: 35 - 25 = 10. The new balance is $10.

6. Matiu is tracking the water level in a local river for a science project. On Monday, the water level was 24 centimeters above the normal level. On Tuesday, it dropped by 18 centimeters. On Wednesday, it rose by 12 centimeters. On Thursday, it dropped by 26 centimeters. What is the water level on Thursday relative to the normal level? Answer: -8 centimeters Solution: Start at the normal level, which is 0 centimeters. Monday's level: 0 + 24 = 24 cm above normal. Tuesday's change: 24 - 18 = 6 cm above normal.
   Full step-by-step solution

   Step 1: Start at the normal level, which is 0 centimeters. Step 2: Monday's level: 0 + 24 = 24 cm above normal. Step 3: Tuesday's change: 24 - 18 = 6 cm above normal. Step 4: Wednesday's change: 6 + 12 = 18 cm above normal. Step 5: Thursday's change: 18 - 26 = -8 cm. The water level on Thursday is 8 centimeters below the normal level, or -8 cm.

7. Aroha's bank account balance was -$137. She deposited $95 and then withdrew $213. What is her new balance? Answer: -$255 Solution: Start with the initial balance: -$137. Add the deposit of $95: -137 + 95 = -42. Subtract the withdrawal of $213: -42 - 213 = -255.
   Full step-by-step solution

   Step 1: Start with the initial balance: -$137. Step 2: Add the deposit of $95: -137 + 95 = -42. Step 3: Subtract the withdrawal of $213: -42 - 213 = -255. The final balance is -$255.

8. Emma's bank account balance was -$137. She deposited $95 and then withdrew $213. What is her new balance? Answer: -255 Solution: Start with the initial balance: -$137 Deposit $95 means adding 95: -137 + 95 = -42 Withdraw $213 means subtracting 213: -42 - 213 = -255 The final balance is -$255.
   Full step-by-step solution

   Step 1: Start with the initial balance: -$137 Step 2: Deposit $95 means adding 95: -137 + 95 = -42 Step 3: Withdraw $213 means subtracting 213: -42 - 213 = -255 The final balance is -$255.