Rational Applications

lessonbunny.com

Answer Key & Explanations

Rational Applications · Grade 7 · Worksheet 3

1. (-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.

2. Emma's bank account balance was -$85. She deposited $142. What is her new balance? Answer: 57 Solution: Start with the initial balance: -85 dollars. Add the deposit: -85 + 142. Think of this as 142 - 85 = 57.
   Full step-by-step solution

   Step 1: Start with the initial balance: -85 dollars. Step 2: Add the deposit: -85 + 142. Step 3: Think of this as 142 - 85 = 57. Step 4: Since 142 is larger than 85, the result is positive: 57. The answer is 57.

3. Olivia has 81 dollars in their bank account. They write a check for 37 dollars to pay for a school field trip. What is the new balance in their account? Answer: 44 Solution: Start with the initial balance: $81. Subtract the amount of the check: $81 - $37 = $44. The new balance is $44.
   Full step-by-step solution

   Step 1: Start with the initial balance: $81. Step 2: Subtract the amount of the check: $81 - $37 = $44. The new balance is $44.

4. Emma is planning a community mural project and needs to mix paint colors. She wants to create a specific shade of purple by mixing red and blue paint in a ratio of 3:5. If she needs a total of 4.8 liters of purple paint for the mural, how many liters of blue paint should she use? Answer: 3 Solution: The ratio of red to blue paint is 3:5, which means for every 3 parts red, there are 5 parts blue. The total number of parts is 3 + 5 = 8 parts. Blue paint represents 5 out of the 8 total parts, or 5/8 of the mixture.
   Full step-by-step solution

   Step 1: The ratio of red to blue paint is 3:5, which means for every 3 parts red, there are 5 parts blue. Step 2: The total number of parts is 3 + 5 = 8 parts. Step 3: Blue paint represents 5 out of the 8 total parts, or 5/8 of the mixture. Step 4: Multiply the total amount of paint (4.8 liters) by the fraction representing blue paint: 4.8 × (5/8) Step 5: Calculate: 4.8 × 5 = 24, then 24 ÷ 8 = 3 Step 6: Emma needs 3 liters of blue paint.

5. Liam is creating a scale model of his city for a school project. The actual height of the city's tallest building is 480 meters. If Liam uses a scale where 2 centimeters on the model represents 15 meters in real life, how many centimeters tall should the model building be? Answer: 64 Solution: The scale says 2 centimeters on the model represents 15 meters in real life. Model length / Real length = 2 cm / 15 m. Identify the real height of the building.
   Full step-by-step solution

   Step 1: Understand the scale. The scale says 2 centimeters on the model represents 15 meters in real life. So the ratio is: Model length / Real length = 2 cm / 15 m. Step 2: Identify the real height of the building. The real building height is 480 meters. Step 3: Let the model height be \( x \) centimeters. Using the scale ratio: \( x \) cm / 480 m = 2 cm / 15 m. Step 4: Solve for \( x \). Multiply both sides by 480 m: \( x \) = (2 cm / 15 m) × 480 m. Step 5: Simplify. The "meters" units cancel: \( x \) = (2 × 480) / 15 cm. Step 6: Calculate 2 × 480 = 960. So \( x \) = 960 / 15. Step 7: Divide 960 by 15. 15 × 64 = 960, so 960 / 15 = 64. Step 8: Conclusion. The model building should be 64 centimeters tall.

6. Mere's bank account balance was -$48. She deposited $62. What is her new balance? Answer: 14 Solution: Start with the initial balance: -$48. Add the deposit of $62: -48 + 62. To add a positive to a negative, find the difference: 62 - 48 = 14.
   Full step-by-step solution

   Step 1: Start with the initial balance: -$48. Step 2: Add the deposit of $62: -48 + 62. Step 3: To add a positive to a negative, find the difference: 62 - 48 = 14. Step 4: Since the positive number is larger, the result is positive: 14. The answer is $14.

7. A rectangular garden is drawn on a coordinate plane with corners at (2, 1), (18, 1), (18, 13), and (2, 13). A circular fountain with radius 3 units is placed at the center of the garden. What is the area of the garden that is NOT covered by the fountain? (Use π = 3.14) Answer: 163.74 Solution: Find the area of the rectangular garden. The corners are (2, 1), (18, 1), (18, 13), and (2, 13). Length in x-direction = 18 - 2 = 16 units.
   Full step-by-step solution

   Step 1: Find the area of the rectangular garden. The corners are (2, 1), (18, 1), (18, 13), and (2, 13). Length in x-direction = 18 - 2 = 16 units. Length in y-direction = 13 - 1 = 12 units. Area of rectangle = length × width = 16 × 12 = 192 square units. Step 2: Find the center of the garden. The center of the rectangle is halfway between x = 2 and x = 18, and halfway between y = 1 and y = 13. Center x = (2 + 18)/2 = 20/2 = 10. Center y = (1 + 13)/2 = 14/2 = 7. So center is at (10, 7). Step 3: Find the area of the circular fountain. Radius r = 3 units. Area of circle = π × r^2 = 3.14 × (3^2) = 3.14 × 9 = 28.26 square units. Step 4: Find the area NOT covered by the fountain. Area not covered = Area of rectangle − Area of fountain = 192 − 28.26 = 163.74 square units. Final answer: 163.74

8. Sophia is managing the inventory for a small grocery store. The store's freezer temperature was recorded as -6°C at 6:00 AM. By 10:00 AM, the temperature had risen by 11°C due to a malfunction. The repair team fixed the issue, and the temperature then dropped by 16°C by 2:00 PM. What was the final temperature in the freezer at 2:00 PM? Answer: -11 Solution: Start with the initial temperature at 6:00 AM: -6°C. The temperature rises by 11°C by 10:00 AM. Add: -6 + 11 = 5°C.
   Full step-by-step solution

   Step 1: Start with the initial temperature at 6:00 AM: -6°C. Step 2: The temperature rises by 11°C by 10:00 AM. Add: -6 + 11 = 5°C. Step 3: The temperature then drops by 16°C by 2:00 PM. Subtract: 5 - 16 = -11°C. The final temperature at 2:00 PM is -11°C.