Algebra worksheet featuring word problems on direct and inverse variation with practical applications.
A worksheet titled "Direct and Inverse Variation Word Problems" with algebraic questions involving real-world scenarios like gas consumption, employee productivity, and money earned.
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Step-by-step solution for: Inverse Variation Word Problems: Complete with ease | airSlate SignNow
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Show Answer Key & Explanations
Step-by-step solution for: Inverse Variation Word Problems: Complete with ease | airSlate SignNow
1. The volume V of a gas kept at a constant temperature varies inversely with the pressure P. If the pressure is 25 pounds per square inch, the volume is 13 cubic feet. What will be the volume if the pressure is 15 pounds per square inch?
- Since V varies inversely with P, V = k/P.
- Using the given values: 13 = k/25 → k = 325.
- For P = 15: V = 325/15 ≈ 21.67 cubic feet.
2. The amount of money spent on gas per gallon varies directly with the number of gallons purchased. When 11.5 gallons of gas were purchased they cost $37.32. How much will 8 gallons of gas cost?
- Since cost C varies directly with gallons G, C = kG.
- Using the given values: 37.32 = k × 11.5 → k ≈ 3.2452.
- For G = 8: C = 3.2452 × 8 ≈ $25.96.
3. The time to complete a project varies inversely with the number of employees. If 7 people can complete the project in 5 days, how long will it take 5 people?
- Since time T varies inversely with employees E, T = k/E.
- Using the given values: 5 = k/7 → k = 35.
- For E = 5: T = 35/5 = 7 days.
4. The time needed to travel a certain distance varies inversely with the rate of speed. If it takes 8 hours to travel a certain distance at 60 miles per hour, how long will it take to travel the same distance at 40 miles per hour?
- Since time T varies inversely with speed S, T = k/S.
- Using the given values: 8 = k/60 → k = 480.
- For S = 40: T = 480/40 = 12 hours.
5. The number of centimeters on a linear measurement varies directly with the number of inches in the measurement. Robin’s height is 68 centimeters or 27 inches. What is Michelle’s height in centimeters if she is 64 inches tall?
- Since centimeters C vary directly with inches I, C = kI.
- Using the given values: 68 = k × 27 → k ≈ 2.5185.
- For I = 64: C = 2.5185 × 64 ≈ 161.19 cm.
6. The number of molecules made by an ice-tracing machine varies directly with the radius of the circle. A 12-inch-radius circle makes 100 molecules. How many molecules will a 16-inch-radius circle require to make the same amount?
- Since molecules M vary directly with radius R, M = kR.
- Using the given values: 100 = k × 12 → k ≈ 8.3333.
- For R = 16: M = 8.3333 × 16 ≈ 133.33 molecules.
7. Karen earns $30 for working 6 hours. If the amount she earns varies directly with the number of hours she works, how much will she earn for working 7 hours?
- Since earnings E vary directly with hours H, E = kH.
- Using the given values: 30 = k × 6 → k = 5.
- For H = 7: E = 5 × 7 = $35.
8. A batch of 150 vitamins costs $8.25. If the cost varies directly with the number of vitamins in the bottle, what should a bottle of 250 vitamins cost?
- Since cost C varies directly with vitamins V, C = kV.
- Using the given values: 8.25 = k × 150 → k = 0.055.
- For V = 250: C = 0.055 × 250 = $13.75.
9. For a fixed number of miles, the gas mileage of a car (miles/gallon) varies inversely with the number of gallons used. Suppose each car averaged 25 miles per gallon and used 750 gallons of gas last year. If next year, due to driving more miles, the car uses 900 gallons, how many gallons of gas would be used per gallon, assuming the same total miles?
- Since gas mileage M varies inversely with gallons G, M = k/G.
- Using the given values: 25 = k/750 → k = 18750.
- For G = 900: M = 18750/900 ≈ 20.83 miles per gallon.
10. You invested $975 to earn an interest of $230 in one year. If the interest earned varies directly with the amount invested, how much would you receive if you invested $1500 for one year? How much would you need to invest to earn $2000 in one year?
- Since interest I varies directly with investment A, I = kA.
- Using the given values: 230 = k × 975 → k ≈ 0.2359.
- For A = 1500: I = 0.2359 × 1500 ≈ $353.85.
- To earn I = 2000: 2000 = 0.2359 × A → A ≈ $8478.17.
- Since V varies inversely with P, V = k/P.
- Using the given values: 13 = k/25 → k = 325.
- For P = 15: V = 325/15 ≈ 21.67 cubic feet.
2. The amount of money spent on gas per gallon varies directly with the number of gallons purchased. When 11.5 gallons of gas were purchased they cost $37.32. How much will 8 gallons of gas cost?
- Since cost C varies directly with gallons G, C = kG.
- Using the given values: 37.32 = k × 11.5 → k ≈ 3.2452.
- For G = 8: C = 3.2452 × 8 ≈ $25.96.
3. The time to complete a project varies inversely with the number of employees. If 7 people can complete the project in 5 days, how long will it take 5 people?
- Since time T varies inversely with employees E, T = k/E.
- Using the given values: 5 = k/7 → k = 35.
- For E = 5: T = 35/5 = 7 days.
4. The time needed to travel a certain distance varies inversely with the rate of speed. If it takes 8 hours to travel a certain distance at 60 miles per hour, how long will it take to travel the same distance at 40 miles per hour?
- Since time T varies inversely with speed S, T = k/S.
- Using the given values: 8 = k/60 → k = 480.
- For S = 40: T = 480/40 = 12 hours.
5. The number of centimeters on a linear measurement varies directly with the number of inches in the measurement. Robin’s height is 68 centimeters or 27 inches. What is Michelle’s height in centimeters if she is 64 inches tall?
- Since centimeters C vary directly with inches I, C = kI.
- Using the given values: 68 = k × 27 → k ≈ 2.5185.
- For I = 64: C = 2.5185 × 64 ≈ 161.19 cm.
6. The number of molecules made by an ice-tracing machine varies directly with the radius of the circle. A 12-inch-radius circle makes 100 molecules. How many molecules will a 16-inch-radius circle require to make the same amount?
- Since molecules M vary directly with radius R, M = kR.
- Using the given values: 100 = k × 12 → k ≈ 8.3333.
- For R = 16: M = 8.3333 × 16 ≈ 133.33 molecules.
7. Karen earns $30 for working 6 hours. If the amount she earns varies directly with the number of hours she works, how much will she earn for working 7 hours?
- Since earnings E vary directly with hours H, E = kH.
- Using the given values: 30 = k × 6 → k = 5.
- For H = 7: E = 5 × 7 = $35.
8. A batch of 150 vitamins costs $8.25. If the cost varies directly with the number of vitamins in the bottle, what should a bottle of 250 vitamins cost?
- Since cost C varies directly with vitamins V, C = kV.
- Using the given values: 8.25 = k × 150 → k = 0.055.
- For V = 250: C = 0.055 × 250 = $13.75.
9. For a fixed number of miles, the gas mileage of a car (miles/gallon) varies inversely with the number of gallons used. Suppose each car averaged 25 miles per gallon and used 750 gallons of gas last year. If next year, due to driving more miles, the car uses 900 gallons, how many gallons of gas would be used per gallon, assuming the same total miles?
- Since gas mileage M varies inversely with gallons G, M = k/G.
- Using the given values: 25 = k/750 → k = 18750.
- For G = 900: M = 18750/900 ≈ 20.83 miles per gallon.
10. You invested $975 to earn an interest of $230 in one year. If the interest earned varies directly with the amount invested, how much would you receive if you invested $1500 for one year? How much would you need to invest to earn $2000 in one year?
- Since interest I varies directly with investment A, I = kA.
- Using the given values: 230 = k × 975 → k ≈ 0.2359.
- For A = 1500: I = 0.2359 × 1500 ≈ $353.85.
- To earn I = 2000: 2000 = 0.2359 × A → A ≈ $8478.17.
Parent Tip: Review the logic above to help your child master the concept of direct and inverse variation worksheet with answers.
