Direct and Inverse Proportions – Class 8
Hi everyone! This chapter is about Direct and Inverse Proportions. These concepts help us understand how quantities relate to each other.
1. Direct Proportion
Two quantities are said to be in direct proportion if an increase in one quantity leads to a corresponding increase in the other quantity, and a decrease in one leads to a decrease in the other, such that the ratio of the two quantities remains the same.
Example: The cost of apples is directly proportional to the number of apples you buy. If 2 apples cost Rs. 50, then 4 apples will cost Rs. 100 (double the apples, double the cost). The ratio of cost to apples (50/2 = 100/4 = 25) remains constant.
If x and y are in direct proportion, then x/y = k (where k is a constant).
Example: If a car travels 100 km in 2 hours, how far will it travel in 5 hours (assuming a constant speed)?
Solution: Distance and time are in direct proportion. 100 km / 2 hours = x km / 5 hours. Solving for x gives x = 250 km.
2. Inverse Proportion
Two quantities are said to be in inverse proportion if an increase in one quantity leads to a corresponding decrease in the other quantity, and vice-versa, such that the product of the two quantities remains the same.
Example: The number of workers and the time taken to complete a job are inversely proportional. If 2 workers can complete a job in 6 days, then 4 workers will complete the same job in 3 days (double the workers, half the time). The product of workers and days (2 * 6 = 4 * 3 = 12) remains constant.
If x and y are in inverse proportion, then x * y = k (where k is a constant).
Example: 5 taps can fill a tank in 10 hours. How long will it take 8 taps to fill the same tank?
Solution: Number of taps and time are inversely proportional. 5 taps * 10 hours = 8 taps * x hours. Solving for x gives x = 6.25 hours.
Applications of Direct and Inverse Proportions
1. Unitary Method:
Used to calculate the value of a certain number of units when the value of a different number of units is given. This often involves direct proportion.
2. Time and Work Problems:
Relating the number of workers, the time taken to complete a task, and the amount of work done. This can involve both direct and inverse proportions.
3. Speed, Distance, and Time:
Calculating the distance travelled given the speed and time, or vice versa. This often involves direct proportion (for constant speed).
4. Scaling:
Used in maps and models where dimensions are scaled up or down proportionally.
Understanding direct and inverse proportions is essential for solving many real-world problems involving relationships between quantities.
Direct and Inverse Proportions Quiz – Tough Application Problems
1. **Work and Time:** 12 workers can complete a task in 20 days. How many workers are needed to complete the same task in 15 days?
2. **Speed and Time:** A car travels a certain distance at a speed of 60 km/h in 4 hours. How long will it take to travel the same distance at a speed of 80 km/h?
3. **Cost of Goods:** If 8 kg of rice costs Rs. 240, what will be the cost of 12 kg of rice?
4. **Pipes and Filling Time:** 6 pipes can fill a tank in 12 hours. How long will it take 9 pipes to fill the same tank?
5. **Scaling a Map:** On a map, 5 cm represents 100 km. How many cm will represent 250 km on the same map?
6. **Sharing Profits:** Two partners share profits in the ratio 3:5. If the total profit is Rs. 8000, how much does each partner receive?
7. **Gears and Speed:** Two gears have 30 teeth and 40 teeth respectively. If the smaller gear makes 100 revolutions per minute, how many revolutions per minute does the larger gear make?
8. **Typing Speed:** A typist can type 60 words in 2 minutes. How long will it take the same typist to type 150 words?