Inverse Proportion: Formula, Rules and Solved Examples

Inverse Proportion describes a relationship in which one quantity increases as another decreases, while their product remains constant. This page explains the inverse proportion formula, inverse variation, standard solving methods, work-time applications and quick calculation tricks with clear numerical examples.

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What Is Inverse Proportion?

Two quantities are in inverse proportion when an increase in one quantity causes a proportional decrease in the other. Their product remains constant.

If x and y are inversely proportional, then x ∝ 1/y or y ∝ 1/x. Therefore, xy = k, where k is a constant. For two corresponding values, x₁y₁ = x₂y₂. For example, if 4 workers complete a task in 12 days, 8 workers complete it in 6 days because 4 × 12 = 8 × 6 = 48.

Inverse Proportion Formula & Tricks

Important Formulas

Basic inverse proportion formula
y ∝ 1/x, so y = k/x

The value of y changes inversely with x, and k is the constant of proportionality.

Constant product rule
xy = k

The product of corresponding values remains unchanged in inverse proportion.

Two-value relation
x₁y₁ = x₂y₂

Use this relation when three values are known and the fourth value must be calculated.

Ratio form
x₁/x₂ = y₂/y₁

The ratio of the first quantity is equal to the reciprocal ratio of the second quantity.

Quick Tricks

Keep the product constant

Multiply the two known corresponding values. Divide this constant by the new value to find the unknown quantity.

Example: If 6 machines take 15 hours, then 10 machines take (6 × 15)/10 = 9 hours.
Reverse the second ratio

In inverse variation, the ratio of one quantity is the reciprocal of the ratio of the other quantity.

Example: If workers change from 5 to 8, time changes in the ratio 8:5, so the new time is the old time × 5/8.
Check opposite movement

The quantities must move in opposite directions. If one quantity increases and the other also increases in a supposed inverse relationship, recheck the model.

Example: More workers require fewer days for the same work, so workers and days are inversely proportional.

Inverse Proportion Concepts

Identifying Inverse Variation

A relationship is inverse variation when the product of the two quantities is constant.

To test the relationship, multiply each pair of corresponding values. Equal products confirm inverse proportion. If x doubles, y becomes half; if x becomes three times, y becomes one-third.

Example: For the pairs (2, 18), (3, 12), and (6, 6), the products are 36, 36, and 36. Hence, the quantities are inversely proportional.

Finding an Unknown Value

Use x₁y₁ = x₂y₂ to calculate an unknown value in an inverse proportion question.

First identify the two corresponding pairs. Multiply the known values on one side, then divide by the known value paired with the unknown. Units must remain consistent.

Example: If 7 books cost a fixed amount of storage space requiring 12 shelves, and the same books are arranged using 9 shelves under an inverse arrangement, the corresponding calculation is 7 × 12 = 9 × x, giving x = 28/3. Such a relation must represent quantities genuinely defined as inversely proportional.

Work and Time as Inverse Proportion

For a fixed amount of work, the number of workers and the time taken are inversely proportional.

Assuming equal efficiency, fixed working hours per day and the same work, workers × days remains constant. Thus, w₁d₁ = w₂d₂. The same rule applies to machines and time or taps and filling time when their rates are comparable.

Example: 12 workers finish a task in 10 days. The time for 15 workers is (12 × 10)/15 = 8 days.

Speed and Time for a Fixed Distance

For a fixed distance, speed and travel time are inversely proportional.

Since distance = speed × time, a constant distance gives speed × time = constant. Therefore, s₁t₁ = s₂t₂. Increasing speed reduces the required time in the same reciprocal ratio.

Example: At 60 km/h, a journey takes 5 hours. At 75 km/h, the time is (60 × 5)/75 = 4 hours.

Inverse Proportion Video Lessons

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Practice Inverse Proportion Questions

Practise published questions related to this topic.

1The HCF of 1/2, 3/4, 27/16 and 81/64 is:→ 2A shopkeeper offers the following schemes on toys of the same marked price. (A) Successive discounts of 21% and 8% on any number of toys bought. (B) Successive discounts of 38%, 42% and 30% on any number of toys bought. (C) 17% discount on the first 8 toys and 20% discount on each toy thereon. (D) 7 toys free of cost on buying 10 toys. A customer wants to buy 10 toys. Which of the above schemes is the least beneficial to her?→ 3Find the HCF and LCM of the fractions 3/8, 5/12, and 7/18.→ 4An article is sold for a profit of 30 percent. What is the ratio of the selling price of the article to the cost price of the article?→ 5A person sells an article at the profit of 30 percent. Cost price and selling price both are increased by 40 percent. What will be the new profit percentage?→ 6A rare book dealer sells a vintage novel to a collector for ₹1,440. The collector later auctions it at a 25% profit. What price does the final buyer pay at the auction?→ 7How many two-digit numbers satisfy this property? The last digit (unit's digit) of the square of the two-digit number is 8.→ 8If x = √100 ÷ √25+√64, then what is the value of 2x?→ 9Simplify: (2√3)/√12→ 10What is the Highest Common Factor of 57 x 113 and 55 x 116?→

Inverse Proportion Quick Quiz

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Quick Revision Notes

Inverse Proportion Quick Revision

Use these rules to identify and solve inverse proportion problems.

  • Inverse proportion is written as y ∝ 1/x.
  • The product xy remains constant.
  • For two pairs, x₁y₁ = x₂y₂.
  • If one quantity is multiplied by n, the other is divided by n.
  • In ratio form, x₁/x₂ = y₂/y₁.
  • For fixed work, workers × time remains constant when worker efficiency is equal.
  • For fixed distance, speed × time remains constant.
  • Check that the two quantities move in opposite directions.

Inverse Proportion FAQs

What is the inverse proportion formula?

The inverse proportion formula is y = k/x, or xy = k. For two pairs of values, use x₁y₁ = x₂y₂.

If one quantity doubles in inverse proportion, what happens to the other?

The other quantity becomes half. For example, if x changes from 4 to 8 and y was 12, the new y is 6.

How do you solve an inverse proportion question with workers and days?

Use w₁d₁ = w₂d₂. For example, 8 workers taking 15 days means 12 workers take (8 × 15)/12 = 10 days.

Are speed and time inversely proportional for every journey?

They are inversely proportional when the distance is fixed. This follows from speed × time = distance, so s₁t₁ = s₂t₂ for the same distance.

How can direct proportion be distinguished from inverse proportion?

In direct proportion, the ratio y/x is constant and both quantities move in the same direction. In inverse proportion, the product xy is constant and the quantities move in opposite directions.

If x changes from 5 to 20 in inverse variation, how does y change?

x becomes four times its original value, so y becomes one-fourth of its original value. If y was 28, the new value is 7.

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