1. Proportions in Everyday Life
- When you double the recipe ingredients, you get twice as much food. This is direct proportion.
- If you increase the speed of your car, the time taken to cover the same distance decreases. This is inverse proportion.
- Buying more apples increases the total cost, showing a direct proportion.
- As more people work together, less time is needed to finish a job, showing an inverse proportion.
2. Direct Proportion
Definition
Two quantities are in direct proportion (or vary directly) if increasing one causes the other to increase at the same rate. Similarly, decreasing one quantity causes the other to decrease in the same ratio.
Mathematical Representation
- If x and y are directly proportional, then x ∝ y (read as "x is directly proportional to y")
- There exists a constant k such that x/y = k or x = ky
- This means: if x1/y1 = x2/y2 = ... = k
Table Example
| x (Apples) | y (Price in ₹) |
|---|---|
| 2 | 40 |
| 4 | 80 |
| 6 | 120 |
| 10 | 200 |
As the number of apples increases, so does the price, and the ratio y/x is always 20.
Graphical Representation
When two quantities are directly proportional, plotting them on a graph gives a straight line passing through the origin.
3. Solving Direct Proportion Problems
Step-by-Step Method
- Set up the ratio: x1/y1 = x2/y2
- Substitute the known values
- Solve for the unknown
Worked Example
If 8 pens cost ₹64, how much will 15 pens cost?
Let x1 = 8, y1 = 64, x2 = 15, y2 = ?
Set up: 8/64 = 15/y2
Cross multiply: 8 × y2 = 64 × 15 → y2 = (64 × 15) / 8 = 120
So, 15 pens cost ₹120.
4. Inverse Proportion
Definition
Two quantities are in inverse proportion (or vary inversely) if increasing one causes the other to decrease in such a way that their product remains constant.
Mathematical Representation
- If x and y are inversely proportional, then x ∝ 1/y
- There exists a constant k such that x × y = k
- This means: x1 × y1 = x2 × y2 = ... = k
Table Example
| x (Workers) | y (Days to Finish Job) |
|---|---|
| 2 | 12 |
| 3 | 8 |
| 4 | 6 |
| 6 | 4 |
As the number of workers increases, the number of days decreases. The product x × y is always 24.
Graphical Representation
The graph of an inverse proportion is a curve that falls as one quantity rises, never touching the axes.
5. Solving Inverse Proportion Problems
Step-by-Step Method
- Set up the product: x1 × y1 = x2 × y2
- Substitute the known values
- Solve for the unknown
Worked Example
If 12 men can finish a wall in 8 days, how long will 16 men take?
Let x1 = 12, y1 = 8, x2 = 16, y2 = ?
Set up: 12 × 8 = 16 × y2
96 = 16 × y2 → y2 = 96/16 = 6
So, 16 men will take 6 days.
6. How to Identify the Type of Proportion
- If both quantities increase or decrease together at the same rate, it is direct proportion.
- If one quantity increases while the other decreases so their product is constant, it is inverse proportion.
- Test by making a table of values and checking if ratios (direct) or products (inverse) are constant.
- Read the problem carefully for phrases like "more... more" (direct) or "more... less" (inverse).
7. Applications of Proportions in Real Life
- Cooking: Changing the amount of ingredients while keeping taste consistent (direct).
- Travel: Speed and time taken to cover a distance (inverse).
- Shopping: Cost and quantity bought (direct).
- Construction: Number of workers and time to finish a task (inverse).
- Physics: Mass and acceleration (inverse in Newton’s second law).
- Chemistry: Gas laws (e.g., Boyle’s Law: pressure and volume are inversely proportional at constant temperature).
8. Important Formulas
- Direct proportion: x/y = k or x1/y1 = x2/y2
- Inverse proportion: x × y = k or x1 × y1 = x2 × y2
9. Common Errors and How to Avoid Them
- Mixing up direct and inverse proportion. Always check the relationship carefully.
- Not keeping units the same (e.g., hours vs minutes). Convert all measurements to the same unit.
- Forgetting to cross-multiply or multiply properly when solving for unknowns.
- Ignoring constant product or ratio checks before assuming the type of proportion.
- Not interpreting the context of word problems correctly.
10. Practice Problems
Basic
- If 5 notebooks cost ₹100, how much will 8 notebooks cost (direct proportion)?
- If 3 kg of rice costs ₹150, what is the cost of 10 kg of rice?
- If 7 pencils cost ₹28, find the cost of 12 pencils.
- Is the relationship between number of hours worked and total earnings at a fixed hourly wage direct or inverse?
Intermediate
- 12 workers can dig a trench in 15 days. How many days will 20 workers take?
- If 4 pumps can fill a tank in 6 hours, how long will 3 pumps take?
- If a car covers a distance at a speed of 60 km/h in 4 hours, how long will it take at 80 km/h?
- The cost of 15 candies is ₹75. What is the cost of 100 candies?
Advanced
- A train covers a certain distance in 6 hours at 75 km/h. If the speed is reduced to 60 km/h, how much time will it take?
- 20 painters can paint a wall in 3 days. How many painters are needed to finish the work in 2 days?
- If 5 bags of cement weigh 200 kg, how much will 13 bags weigh?
- 10 students can clean a park in 8 hours. If 4 students join, how long will it take to clean the park?
11. Solutions
- 5/100 = 8/x; x = (100 × 8)/5 = ₹160
- 3/150 = 10/x; x = (150 × 10)/3 = ₹500
- 7/28 = 12/x; x = (28 × 12)/7 = ₹48
- Direct proportion, as more hours means more earnings.
- 12 × 15 = 20 × x; x = (12 × 15)/20 = 9 days
- 4 × 6 = 3 × x; x = (4 × 6)/3 = 8 hours
- 60 × 4 = 80 × x; x = (60 × 4)/80 = 3 hours
- 15/75 = 100/x; x = (75 × 100)/15 = ₹500
- 75 × 6 = 60 × x; x = (75 × 6)/60 = 7.5 hours
- 20 × 3 = x × 2; x = (20 × 3)/2 = 30 painters
- 5/200 = 13/x; x = (200 × 13)/5 = 520 kg
- 10 × 8 = 14 × x; x = (10 × 8)/14 ≈ 5.71 hours
Conclusion
Understanding direct and inverse proportions helps you solve practical problems quickly and confidently, whether you are cooking for a group, managing time and resources, or analyzing scientific data. By practicing these concepts, you will be able to recognize and work with proportional relationships in many real-life situations and advanced mathematics.