Certainly! Here's a worksheet on the topic of Direct and Inverse Variation for 8th-grade students following the ICSE board curriculum. The problems are categorized into easy, medium, and difficult levels.
**Direct and Inverse Variation Worksheet**
**Instructions:**
1. Solve all the problems.
2. Show your work and calculations where necessary.
3. Circle or underline your final answers.
**Easy Problems:**
1. If y varies directly with x, and y = 12 when x = 4, find the constant of variation (k).
2. If y varies inversely with x, and y = 10 when x = 5, find the constant of variation (k).
3. If y varies directly with x, and y = 25 when x = 5, find y when x = 8.
4. If y varies inversely with x, and y = 6 when x = 9, find y when x = 12.
5. If y varies directly with x, and y = 15 when x = 3, find x when y = 30.
**Medium Problems:**
6. The cost (C) of printing flyers is directly proportional to the number of flyers (n). If it costs $40 to print 200 flyers, find the cost to print 600 flyers.
7. The time (t) it takes to complete a task is inversely proportional to the number of workers (w). If it takes 8 hours for 6 workers to complete the task, how long will it take for 12 workers to finish the same task?
8. A car travels at a constant speed. If it covers 60 miles in 2 hours, how long will it take to cover 150 miles at the same speed?
9. The force (F) of attraction between two objects is directly proportional to the product of their masses (m1 and m2) and inversely proportional to the square of the distance (d) between them. If F = 12 when m1 = 4, m2 = 6, and d = 3, find F when m1 = 8, m2 = 9, and d = 5.
10. The pressure (P) in a closed container is inversely proportional to its volume (V). If P = 48 kPa when V = 4 liters, find the pressure when V = 10 liters.
**Difficult Problems:**
11. A car's fuel efficiency (miles per gallon) varies inversely with its speed (in miles per hour). If the car gets 30 miles per gallon at 60 mph, find the fuel efficiency at 70 mph.
12. The force (F) of gravity between two objects varies directly with the product of their masses (m1 and m2) and inversely with the square of the distance (d) between them. If F = 9.8 N when m1 = 5 kg, m2 = 10 kg, and d = 1 m, find F when m1 = 3 kg, m2 = 8 kg, and d = 2 m.
13. The time (t) it takes for a pendulum to complete one full swing varies directly with the square root of its length (L). If a pendulum takes 2 seconds to complete one swing when L = 9 meters, find the time it takes when L = 16 meters.
14. The resistance (R) in an electrical circuit is inversely proportional to the square of the current (I). If R = 25 ohms when I = 5 amperes, find R when I = 10 amperes.
15. The force (F) required to lift an object with a pulley system varies directly with the weight (W) of the object and inversely with the number (n) of supporting ropes. If F = 120 N when W = 600 N and n = 4, find F when W = 800 N and n = 6.
**Answers:**
**Easy Problems:**
1. k = 3
2. k = 50
3. y = 40
4. y = 4
5. x = 6
**Medium Problems:**
6. $150
7. 4 hours
8. 5 hours
9. F = 5.76
10. P = 19.2 kPa
**Difficult Problems:**
11. Fuel efficiency at 70 mph = 25 mpg
12. F = 4.35 N
13. Time = 3 seconds
14. R = 6.25 ohms
15. F = 160 N
