Direct and Inverse Variations
In this chapter, students learn about two important types of relationships between quantities: direct variation and inverse variation. Understanding these helps in solving real-life problems involving proportional reasoning.
1. Direct Variation
Two quantities are said to be in direct variation if an increase in one results in a proportional increase in the other, and vice versa. Mathematically, if y varies directly as x, then y = kx for some constant k.
- If the number of articles increases, the total cost increases (at a fixed price per article).
- If speed increases, distance covered in a fixed time increases.
Example: If 5 pens cost ₹50, how much will 8 pens cost?
Solution: Cost per pen = ₹10. So, 8 pens cost 8 × 10 = ₹80.
Solution: Cost per pen = ₹10. So, 8 pens cost 8 × 10 = ₹80.
2. Inverse Variation
Two quantities are said to be in inverse variation if an increase in one results in a proportional decrease in the other, and vice versa. Mathematically, if y varies inversely as x, then y = k/x for some constant k.
- If the number of workers increases, the time taken to complete a job decreases (assuming all work at the same rate).
- If speed increases, time taken to cover a fixed distance decreases.
Example: If 4 workers can finish a task in 6 days, how long will 8 workers take?
Solution: More workers, less time. 4 × 6 = 8 × x ⇒ x = 3 days.
Solution: More workers, less time. 4 × 6 = 8 × x ⇒ x = 3 days.
3. How to Identify the Type of Variation
- Check if the ratio
y/xis constant (Direct Variation). - Check if the product
x × yis constant (Inverse Variation).
4. Word Problems
-
If 12 notebooks cost ₹240, how much will 20 notebooks cost?
Direct variation: Cost per notebook = ₹20. 20 × 20 = ₹400. -
If 6 men can build a wall in 15 days, how many days will 10 men take?
Inverse variation: 6 × 15 = 10 × x ⇒ x = 9 days.
5. Practice Exercises
- Find whether the following pairs of quantities are in direct or inverse variation:
- Number of hours worked and total wages (at fixed hourly rate)
- Speed and time for a fixed distance
- Number of students and share of prize money (fixed total prize)
- Solve: If 3 taps can fill a tank in 12 hours, how long will 6 taps take?
- Solve: If 7 kg of apples cost ₹350, what is the cost of 10 kg?
6. Summary
- Direct variation: Both quantities increase or decrease together; ratio is constant.
- Inverse variation: One quantity increases as the other decreases; product is constant.
- Identify the type of variation to solve real-life problems easily.