Rational Applications

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Answer Key & Explanations

Rational Applications · Grade 7 · Worksheet 2

1. A factory produces 15,000 units of a product in 8 hours. Due to a machine upgrade, their production rate increases by 25%. How many units will they produce in 12 hours at the new rate? Answer: 28125 Solution: Find the original production rate per hour. The factory produces 15,000 units in 8 hours. So, original rate = 15,000 ÷ 8 = 1,875 units per hour.
   Full step-by-step solution

   Step 1: Find the original production rate per hour. The factory produces 15,000 units in 8 hours. So, original rate = 15,000 ÷ 8 = 1,875 units per hour. Step 2: Calculate the new production rate after a 25% increase. A 25% increase means the new rate is 100% + 25% = 125% of the original rate. 125% as a decimal is 1.25. New rate = 1.25 × 1,875 = 2,343.75 units per hour. Step 3: Find the total units produced in 12 hours at the new rate. Units = new rate × time = 2,343.75 × 12. First, 2,343 × 12 = 28,116. Then, 0.75 × 12 = 9. So total = 28,116 + 9 = 28,125 units. Final answer: 28,125

2. 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.

3. Kaia's bank account balance was -$137. She deposited $213. What is her new balance? Answer: 76 Solution: Start with the initial balance: -$137. Add the deposit: -137 + 213. Since 213 is larger than 137, the result will be positive.
   Full step-by-step solution

   Step 1: Start with the initial balance: -$137. Step 2: Add the deposit: -137 + 213. Step 3: Since 213 is larger than 137, the result will be positive. Subtract 137 from 213: 213 - 137 = 76. Step 4: The new balance is $76. The answer is 76.

4. (-3/4 + 2/3) × (-24) = ? Answer: 2 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 -24: (-1/12) × (-24) = 24/12. Step 6: Simplify 24/12 = 2. The answer is 2.

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. Nikau has 97 dollars in their bank account. They write a check for 45 dollars to pay for a school field trip. What is the new balance in their account? Answer: 52 Solution: Start with the initial balance: $97. Subtract the amount of the check: $97 - $45 = $52. The new balance is $52.
   Full step-by-step solution

   Step 1: Start with the initial balance: $97. Step 2: Subtract the amount of the check: $97 - $45 = $52. The new balance is $52.

7. A recipe requires 3/4 cup of flour per batch. If you have 5 1/2 cups of flour, how many complete batches can you make? Answer: 7 Solution: Convert 5 1/2 cups to an improper fraction: 5 1/2 = 11/2 Divide total flour by flour per batch: (11/2) ÷ (3/4) When dividing fractions, multiply by the reciprocal: (11/2) × (4/3) = 44/6 Simplify 44/6 = 22/3 = 7 1/3 Since we can only make complete batches, we take the whole number part: 7 You can…
   Full step-by-step solution

   Step 1: Convert 5 1/2 cups to an improper fraction: 5 1/2 = 11/2 Step 2: Divide total flour by flour per batch: (11/2) ÷ (3/4) Step 3: When dividing fractions, multiply by the reciprocal: (11/2) × (4/3) = 44/6 Step 4: Simplify 44/6 = 22/3 = 7 1/3 Step 5: Since we can only make complete batches, we take the whole number part: 7 You can make 7 complete batches of the recipe.

8. Aroha is managing the inventory for a warehouse. On Monday, the warehouse had 12,500 kilograms of grain. On Tuesday, they received a shipment of 3,800 kilograms. On Wednesday, they shipped out 5,200 kilograms to a bakery. On Thursday, a leak caused a loss of 1,100 kilograms. On Friday, they received another shipment of 2,700 kilograms. How many kilograms of grain are in the warehouse now? Answer: 12700 Solution: Start with Monday's amount: 12,500 kg. Tuesday's shipment adds grain: 12,500 + 3,800 = 16,300 kg. Wednesday's shipment removes grain: 16,300 - 5,200 = 11,100 kg.
   Full step-by-step solution

   Step 1: Start with Monday's amount: 12,500 kg. Step 2: Tuesday's shipment adds grain: 12,500 + 3,800 = 16,300 kg. Step 3: Wednesday's shipment removes grain: 16,300 - 5,200 = 11,100 kg. Step 4: Thursday's leak removes grain: 11,100 - 1,100 = 10,000 kg. Step 5: Friday's shipment adds grain: 10,000 + 2,700 = 12,700 kg. The warehouse now has 12,700 kilograms of grain.