Rational Properties

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Answer Key & Explanations

Rational Properties · Grade 7 · Worksheet 1

1. Emma is planning a road trip from her home to visit her grandparents. The total distance is 420 miles. Her car's fuel efficiency is 28 miles per gallon, and gasoline costs $3.75 per gallon. If she wants to budget for a round trip, how much money should Emma set aside for gasoline? Answer: 112.50 Solution: Calculate the total distance for the round trip. Round trip distance = 420 miles × 2 = 840 miles Calculate how many gallons of gasoline are needed for the trip.
   Full step-by-step solution

   Step 1: Calculate the total distance for the round trip. Round trip distance = 420 miles × 2 = 840 miles Step 2: Calculate how many gallons of gasoline are needed for the trip. Gallons needed = Total distance ÷ Fuel efficiency Gallons needed = 840 miles ÷ 28 miles/gallon = 30 gallons Step 3: Calculate the total cost for gasoline. Total cost = Gallons needed × Cost per gallon Total cost = 30 gallons × $3.75/gallon = $112.50 Emma should set aside $112.50 for gasoline.

2. (-3/4) × (8/9) + (7/12) ÷ (-1/6) = ? Answer: -4 Solution: First, multiply (-3/4) × (8/9) = (-3 × 8)/(4 × 9) = (-24)/36 = -2/3 Next, divide (7/12) ÷ (-1/6) = (7/12) × (-6/1) = (7 × -6)/(12 × 1) = (-42)/12 = -7/2 Now add the results: -2/3 + (-7/2) = -2/3 - 7/2 Find a common denominator (6): -4/6 - 21/6 = -25/6 Convert to mixed number: -25/6 = -4 1/6 The…
   Full step-by-step solution

   Step 1: First, multiply (-3/4) × (8/9) = (-3 × 8)/(4 × 9) = (-24)/36 = -2/3 Step 2: Next, divide (7/12) ÷ (-1/6) = (7/12) × (-6/1) = (7 × -6)/(12 × 1) = (-42)/12 = -7/2 Step 3: Now add the results: -2/3 + (-7/2) = -2/3 - 7/2 Step 4: Find a common denominator (6): -4/6 - 21/6 = -25/6 Step 5: Convert to mixed number: -25/6 = -4 1/6 The answer is -4 1/6.

3. Hana is tracking her weekly savings to buy a new bicycle. On Monday, she deposits $12.75 into her savings account. On Tuesday, she withdraws $8.50 to buy a gift for her friend. On Wednesday, she deposits another $15.30 from her allowance. On Thursday, she withdraws $5.20 for a school lunch. Hana wants to verify that the total amount in her account after these transactions can be found using the commutative property of addition. Write an expression showing how Hana can rearrange the deposits and withdrawals (using the commutative property) to group positive amounts together and negative amounts together, then calculate the final balance. Answer: $14.35 Solution: Represent deposits as positive numbers and withdrawals as negative numbers: +12.75, -8.50, +15.30, -5.20. The commutative property of addition states that a + b = b + a.
   Full step-by-step solution

   Step 1: Represent deposits as positive numbers and withdrawals as negative numbers: +12.75, -8.50, +15.30, -5.20. Step 2: The commutative property of addition states that a + b = b + a. So we can rearrange the order of addition: 12.75 + (-8.50) + 15.30 + (-5.20) = 12.75 + 15.30 + (-8.50) + (-5.20). Step 3: Group the positive numbers together and the negative numbers together: (12.75 + 15.30) + (-8.50 + (-5.20)). Step 4: Add the positive numbers: 12.75 + 15.30 = 28.05. Step 5: Add the negative numbers: -8.50 + (-5.20) = -13.70. Step 6: Add the results: 28.05 + (-13.70) = 14.35. The final balance in Hana's account is $14.35.

4. Olivia is organizing a charity event and needs to calculate the total donation amount. She receives three donations: -$15 (a refund), $45, and -$20 (another refund). She wants to use the associative property of addition to group the negative donations together first. Show that (-15 + 45) + (-20) equals -15 + (45 + (-20)) and then find the total donation amount. Answer: 10 Solution: Calculate (-15 + 45) + (-20). First, add -15 and 45: -15 + 45 = 30. Then add -20: 30 + (-20) = 10.
   Full step-by-step solution

   Step 1: Calculate (-15 + 45) + (-20). First, add -15 and 45: -15 + 45 = 30. Then add -20: 30 + (-20) = 10. Step 2: Calculate -15 + (45 + (-20)). First, add 45 and -20: 45 + (-20) = 25. Then add -15: -15 + 25 = 10. Step 3: Both expressions equal 10, showing the associative property holds. The total donation amount is $10. The answer is 10.

5. (-3/4) × (2/5) ÷ (-1/10) = ? Answer: 3 Solution: We have: (-3/4) × (2/5) ÷ (-1/10) Dividing by a fraction is the same as multiplying by its reciprocal.
   Full step-by-step solution

   Let's solve step-by-step. We have: (-3/4) × (2/5) ÷ (-1/10) --- **Step 1: Understand the division** Dividing by a fraction is the same as multiplying by its reciprocal. So: ÷ (-1/10) = × (-10/1) Now the expression becomes: (-3/4) × (2/5) × (-10/1) --- **Step 2: Multiply the first two fractions** (-3/4) × (2/5) = (-3 × 2) / (4 × 5) = -6 / 20 Simplify -6/20 by dividing numerator and denominator by 2: -6/20 = -3/10 So now we have: (-3/10) × (-10/1) --- **Step 3: Multiply the result by the third fraction** (-3/10) × (-10/1) = (-3 × -10) / (10 × 1) = 30 / 10 --- **Step 4: Simplify** 30 / 10 = 3 --- **Step 5: Check sign reasoning** Negative × positive = negative Negative × negative = positive So overall sign is positive, which matches our result 3. --- **Final Answer:** 3

6. A rectangular swimming pool is drawn on a coordinate plane with corners at (0, 0), (20, 0), (20, 12), and (0, 12). A triangular shallow end is marked off with vertices at (0, 0), (8, 0), and (0, 6). What is the area of the deep end section of the pool? Answer: 216 Solution: Find the area of the entire rectangular pool. The rectangle has length 20 units and width 12 units. Area of rectangle = length × width = 20 × 12 = 240 square units.
   Full step-by-step solution

   Step 1: Find the area of the entire rectangular pool. The rectangle has length 20 units and width 12 units. Area of rectangle = length × width = 20 × 12 = 240 square units. Step 2: Find the area of the triangular shallow end. The triangle has vertices at (0, 0), (8, 0), and (0, 6). This is a right triangle with legs of length 8 and 6 units. Area of triangle = (1/2) × base × height = (1/2) × 8 × 6 = (1/2) × 48 = 24 square units. Step 3: Subtract the triangular area from the rectangular area to find the deep end. Area of deep end = total area - shallow area = 240 - 24 = 216 square units. The answer is 216.

7. (-3/4) × (2/3) ÷ (-1/2) = ? Answer: 1 Solution: Write the expression clearly. (-3/4) × (2/3) ÷ (-1/2) Understand that division by a fraction is the same as multiplication by its reciprocal. So, ÷ (-1/2) = × (-2/1).
   Full step-by-step solution

   Let's solve step-by-step: Step 1: Write the expression clearly. (-3/4) × (2/3) ÷ (-1/2) Step 2: Understand that division by a fraction is the same as multiplication by its reciprocal. So, ÷ (-1/2) = × (-2/1). Now the expression becomes: (-3/4) × (2/3) × (-2/1) Step 3: Multiply the numerators together and the denominators together. Numerators: (-3) × 2 × (-2) = (-6) × (-2) = 12 Denominators: 4 × 3 × 1 = 12 So we have 12/12. Step 4: Simplify 12/12 = 1. Step 5: Check sign rules: Negative × positive = negative Negative × negative = positive So overall sign is positive. Final answer: 1