This lesson introduces a new type of relationship between two quantities. Unlike direct proportion (where both quantities increase or decrease together), in an inverse proportion, when one quantity increases, the other decreases proportionally. This is very common in problems involving work, time, and speed.

1. Core Concepts (Short Notes)

• Direct Proportion (Revision): Two quantities are directly proportional if they increase or decrease at the same rate.

  • Example: The more sugar you buy (quantity increases), the more you pay (cost increases).

  • The ratio is constant: x / y = k.

• Inverse Proportion (New Concept): Two quantities are inversely proportional if an increase in one causes a proportional decrease in the other.

  • Example: The more men you hire for a job (quantity increases), the less time it will take to complete it (time decreases).

  • The product is constant: x × y = k.

2. Key Methods & Formulas

Method 1: The Constant Product Method (xy = k)

This is the most reliable algebraic method.

1. Identify the two quantities, x and y.

2. Use the first piece of information to find the constant, k.

3. Use k and the second piece of information to find the unknown value.

Example: It takes 5 men 8 days to complete a task. How many days will it take 10 men?

1. Quantities: x = number of men, y = number of days.

2. Find k: k = x × y = 5 men × 8 days = 40. This constant (40 man-days) represents the total amount of work required for the job.

3. Solve for the unknown: We now have 10 men.

   • x × y = k

   • 10 × y = 40

   • y = 40 / 10 = 4

   • Answer: It will take 4 days.

Method 2: The Ratio Method

In an inverse proportion, the ratio of the second quantity is the inverse of the ratio of the first quantity.

• If x₁ corresponds to y₁ and x₂ corresponds to y₂, then: x₁ : x₂ = y₂ : y₁

Example (Same as above):

• Men ratio: 5 : 10

• Days ratio (inverse): x : 8 (where x is the unknown number of days)

• Set up the proportion: 5 : 10 = x : 8

• Convert to fractions: 5/10 = x/8

• Solve for x: x = (5 × 8) / 10 = 40 / 10 = 4

• Answer: 4 days.

3. Tips & Tricks for Exams

• "Man-Days" is Your Best Friend: For problems involving work and people, always calculate the total "man-days" or "man-hours" needed for the job. This number is the constant k and it does not change for that specific job.

  • Total Work = (Number of Workers) × (Time Taken)

• Multi-Step Work Problems: Questions often involve a change in the number of workers part-way through a job.

  • Calculate Total Work: Find the total man-days for the entire job first.

  • Calculate Work Done: Find the man-days of work completed before the change.

  • Calculate Work Remaining: Work Remaining = Total Work - Work Done.

  • Solve for the Final Stage: Use the new number of workers and the remaining work to find the remaining time. Remaining Time = Work Remaining / New Number of Workers.

• Speed & Time: For a fixed distance, speed and time are inversely proportional.

  • The faster you travel (speed increases), the less time the journey takes (time decreases).

  • The constant k is the distance: Distance = Speed × Time.

4. Important Points to Remember

• Identify the Proportion Type: Before you start any calculation, ask yourself: "If I increase one quantity, does the other one increase or decrease?" This tells you if it's direct or inverse.

  • More workers, less time -> Inverse

  • More items, more cost -> Direct

• In inverse proportion, you are dealing with a fixed total amount (of work, of distance, etc.). This fixed amount is your constant, k.

• Don't mix up the formulas:

  • Direct: x/y = k

  • Inverse: x × y = k