Learning Objectives
- Understand direct proportion and its applications
- Understand inverse proportion and its applications
- Differentiate between direct and inverse proportions
- Solve problems using the unitary method and proportion
Key Concepts
Direct Proportion
Two quantities x and y are in direct proportion if an increase in x causes a proportional increase in y, and vice versa. The ratio x/y remains constant.
Condition: x₁/y₁ = x₂/y₂ (or x/y = k, a constant).
Examples: Cost and quantity of items purchased (at fixed rate), distance and time (at constant speed), number of workers and total wages (at fixed wage rate).
Inverse Proportion
Two quantities x and y are in inverse proportion if an increase in x causes a proportional decrease in y, and vice versa. The product x × y remains constant.
Condition: x₁ × y₁ = x₂ × y₂ (or xy = k, a constant).
Examples: Speed and time (for fixed distance), number of workers and days to complete a job, number of pipes and time to fill a tank.
Unitary Method
In the unitary method, we first find the value of one unit, then the value of the required number of units.
Direct proportion: If 5 pens cost ₹50, then 1 pen costs ₹10, so 8 pens cost ₹80.
Inverse proportion: If 6 workers finish in 12 days, then 1 worker finishes in 72 days, so 9 workers finish in 72/9 = 8 days.
Identifying the Type of Proportion
Ask: "If one quantity increases, does the other increase or decrease?"
- If both increase (or both decrease) → Direct proportion
- If one increases and the other decreases → Inverse proportion
Summary
Direct proportion means both quantities change in the same direction with a constant ratio. Inverse proportion means quantities change in opposite directions with a constant product. The unitary method is a powerful tool for solving both types of problems by finding the value of one unit first.
Important Terms
- Direct Proportion: x/y = constant; both quantities increase or decrease together
- Inverse Proportion: x × y = constant; one increases as the other decreases
- Unitary Method: Finding the value of one unit to calculate the value of many units
- Constant of Proportion: The fixed ratio (direct) or fixed product (inverse)
Quick Revision
- Direct proportion: x/y = constant → x₁/y₁ = x₂/y₂
- Inverse proportion: xy = constant → x₁y₁ = x₂y₂
- More workers, less time = inverse proportion
- More items, more cost = direct proportion
- Unitary method: find value of 1 unit first
- Always check: do quantities change in the same or opposite direction?