Algebra worksheet featuring direct and inverse variation word problems.
A worksheet titled "Direct and Inverse Variation Word Problems" with algebra questions involving real-world scenarios like gas consumption, population growth, and food consumption.
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Step-by-step solution for: Inverse Variation Word Problems Form - Fill Out and Sign Printable ...
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Show Answer Key & Explanations
Step-by-step solution for: Inverse Variation Word Problems Form - Fill Out and Sign Printable ...
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 for the pressure to 40 pounds per square inch?
- Since V ∝ 1/P, we have V = k/P for some constant k.
- Given P = 25 psi, V = 13 ft³: 13 = k/25 → k = 13 × 25 = 325.
- For P = 40 psi: V = 325 / 40 = 8.125 ft³.
2. The amount of money spent on the gas meter varies directly with the number of gallons purchased. When 11.5 gallons of gas were purchased they cost $17.25. How much will 6 gallons of gas cost?
- Since cost C ∝ gallons G, we have C = kG.
- Given G = 11.5, C = 17.25: 17.25 = k × 11.5 → k = 17.25 / 11.5 = 1.5.
- For G = 6: C = 1.5 × 6 = $9.00.
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 ∝ 1/employees E, we have T = k/E.
- Given E = 7, T = 5: 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 ∝ 1/speed R, we have T = k/R.
- Given R = 60 mph, T = 8 hours: 8 = k/60 → k = 480.
- For R = 40 mph: T = 480 / 40 = 12 hours.
5. The number of centimeters in a linear measurement varies directly with the number of inches in the measurement. Robin’s height is 68 inches. How tall is she in cm? (Note: 1 inch = 2.54 cm)
- Since cm ∝ inches, we have cm = k × inches.
- Given 1 inch = 2.54 cm, so k = 2.54.
- For 68 inches: cm = 2.54 × 68 = 172.72 cm.
6. The number of revolutions made by a tire traveling along a fixed distance varies inversely with the radius of the tire. A 12-inch radius tire makes 100 revolutions to travel a certain distance. How many revolutions would a 16-inch radius tire require to travel the same distance?
- Since revolutions N ∝ 1/radius r, we have N = k/r.
- Given r = 12, N = 100: 100 = k/12 → k = 1200.
- For r = 16: N = 1200 / 16 = 75 revolutions.
7. Karen earns $30.50 for working 5 hours. If she earns the same rate each hour, how much will she earn for working 7 hours?
- Since earnings E ∝ hours H, we have E = kH.
- Given H = 5, E = 30.50: 30.50 = k × 5 → k = 6.10.
- For H = 7: E = 6.10 × 7 = $42.70.
8. A batch of 150 vitamins costs $5.25. If the cost varies directly with the number of vitamins in the batch, what should a bottle of 250 vitamins cost?
- Since cost C ∝ vitamins V, we have C = kV.
- Given V = 150, C = 5.25: 5.25 = k × 150 → k = 0.035.
- For V = 250: C = 0.035 × 250 = $8.75.
9. For a fixed number of miles, the gas mileage of a car (miles/gallon) varies inversely with the number of gallons. Suppose you average 25 miles per gallon and used 750 gallons of gas last year. If next year, you drive the same number of miles, but your new car averages 30 miles per gallon, how many gallons of gas will you use?
- Since miles/gallon M ∝ 1/gallons G, we have M = k/G → k = M × G.
- Given M = 25 mpg, G = 750 gal: k = 25 × 750 = 18750 miles.
- For M = 30 mpg: G = k / M = 18750 / 30 = 625 gallons.
10. We invested $975 to earn an interest of $230 for one year. If the interest earned varies directly with the amount invested, how much would we receive if we invest $2000 for one year? (Assume same rate)
- Since interest I ∝ investment P, we have I = kP.
- Given P = 975, I = 230: 230 = k × 975 → k = 230 / 975 ≈ 0.235897.
- For P = 2000: I = 0.235897 × 2000 ≈ $471.79.
- Since V ∝ 1/P, we have V = k/P for some constant k.
- Given P = 25 psi, V = 13 ft³: 13 = k/25 → k = 13 × 25 = 325.
- For P = 40 psi: V = 325 / 40 = 8.125 ft³.
2. The amount of money spent on the gas meter varies directly with the number of gallons purchased. When 11.5 gallons of gas were purchased they cost $17.25. How much will 6 gallons of gas cost?
- Since cost C ∝ gallons G, we have C = kG.
- Given G = 11.5, C = 17.25: 17.25 = k × 11.5 → k = 17.25 / 11.5 = 1.5.
- For G = 6: C = 1.5 × 6 = $9.00.
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 ∝ 1/employees E, we have T = k/E.
- Given E = 7, T = 5: 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 ∝ 1/speed R, we have T = k/R.
- Given R = 60 mph, T = 8 hours: 8 = k/60 → k = 480.
- For R = 40 mph: T = 480 / 40 = 12 hours.
5. The number of centimeters in a linear measurement varies directly with the number of inches in the measurement. Robin’s height is 68 inches. How tall is she in cm? (Note: 1 inch = 2.54 cm)
- Since cm ∝ inches, we have cm = k × inches.
- Given 1 inch = 2.54 cm, so k = 2.54.
- For 68 inches: cm = 2.54 × 68 = 172.72 cm.
6. The number of revolutions made by a tire traveling along a fixed distance varies inversely with the radius of the tire. A 12-inch radius tire makes 100 revolutions to travel a certain distance. How many revolutions would a 16-inch radius tire require to travel the same distance?
- Since revolutions N ∝ 1/radius r, we have N = k/r.
- Given r = 12, N = 100: 100 = k/12 → k = 1200.
- For r = 16: N = 1200 / 16 = 75 revolutions.
7. Karen earns $30.50 for working 5 hours. If she earns the same rate each hour, how much will she earn for working 7 hours?
- Since earnings E ∝ hours H, we have E = kH.
- Given H = 5, E = 30.50: 30.50 = k × 5 → k = 6.10.
- For H = 7: E = 6.10 × 7 = $42.70.
8. A batch of 150 vitamins costs $5.25. If the cost varies directly with the number of vitamins in the batch, what should a bottle of 250 vitamins cost?
- Since cost C ∝ vitamins V, we have C = kV.
- Given V = 150, C = 5.25: 5.25 = k × 150 → k = 0.035.
- For V = 250: C = 0.035 × 250 = $8.75.
9. For a fixed number of miles, the gas mileage of a car (miles/gallon) varies inversely with the number of gallons. Suppose you average 25 miles per gallon and used 750 gallons of gas last year. If next year, you drive the same number of miles, but your new car averages 30 miles per gallon, how many gallons of gas will you use?
- Since miles/gallon M ∝ 1/gallons G, we have M = k/G → k = M × G.
- Given M = 25 mpg, G = 750 gal: k = 25 × 750 = 18750 miles.
- For M = 30 mpg: G = k / M = 18750 / 30 = 625 gallons.
10. We invested $975 to earn an interest of $230 for one year. If the interest earned varies directly with the amount invested, how much would we receive if we invest $2000 for one year? (Assume same rate)
- Since interest I ∝ investment P, we have I = kP.
- Given P = 975, I = 230: 230 = k × 975 → k = 230 / 975 ≈ 0.235897.
- For P = 2000: I = 0.235897 × 2000 ≈ $471.79.
Parent Tip: Review the logic above to help your child master the concept of direct variation worksheet with answer key.
