Constant of Proportionality: Formula, Rules and Examples

Constant of Proportionality is the fixed value that relates two variable quantities. In direct variation, it is found by dividing one variable by the other. In inverse variation, it is found by multiplying the variables. This page covers the constant of proportionality formula, variation rules, and methods for solving proportion questions.

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What Is the Constant of Proportionality?

The constant of proportionality is a fixed number that connects two proportional quantities. It is usually represented by k.

For direct proportion, y is proportional to x when y = kx, so k = y/x. For inverse proportion, y is proportional to 1/x when y = k/x, so k = xy. The value of k remains unchanged for every pair of corresponding values in the same relationship. For example, if y = 15 when x = 3, then k = 15/3 = 5 and the equation is y = 5x.

Constant of Proportionality Formula & Tricks

Important Formulas

Direct variation
y ∝ x ⇒ y = kx ⇒ k = y/x

In direct variation, the ratio y/x is constant. If x increases by a factor, y increases by the same factor.

Inverse variation
y ∝ 1/x ⇒ y = k/x ⇒ k = xy

In inverse variation, the product xy is constant. If one variable increases, the other decreases in the same reciprocal relationship.

Finding a missing value in direct variation
y₂ = (y₁/x₁) × x₂

First calculate k = y₁/x₁, then multiply k by the new value of x.

Finding a missing value in inverse variation
y₂ = (x₁y₁)/x₂

Since x₁y₁ = x₂y₂, divide the constant product by the new value of x.

Quick Tricks

Use the correct operation to find k

Divide y by x for direct variation. Multiply x and y for inverse variation.

Example: For (x, y) = (4, 20), direct k = 20/4 = 5, while inverse k = 4 × 20 = 80.
Check a relationship using two pairs

For direct variation, compare y/x for both pairs. For inverse variation, compare xy for both pairs. Equal results confirm the same constant.

Example: The pairs (2, 12) and (5, 30) have y/x = 6 in both cases, so they show direct variation with k = 6.

Constant of Proportionality Concepts

Direct Proportionality

Two variables are directly proportional when their ratio remains constant, so y = kx.

The constant is k = y/x. If x becomes twice as large, y also becomes twice as large; if x becomes one-half, y becomes one-half. The graph of y = kx is a straight line passing through the origin.

Example: If 6 notebooks cost ₹90, the cost per notebook is k = 90/6 = ₹15. Therefore, the cost C for n notebooks is C = 15n.

Inverse Proportionality

Two variables are inversely proportional when their product remains constant, so y = k/x.

The constant is k = xy. If x is multiplied by a number, y is divided by the same number. The variables must generally be non-zero because the expression k/x is undefined when x = 0.

Example: If 4 workers complete a task in 15 days, the work constant is k = 4 × 15 = 60 worker-days. For 6 workers, the time is 60/6 = 10 days.

Finding the Constant from Given Values

Substitute any known corresponding pair into the appropriate formula to calculate k.

For direct variation, use k = y/x. For inverse variation, use k = xy. After finding k, substitute it into y = kx or y = k/x to form the complete relationship.

Example: If y varies directly as x and y = 28 when x = 7, then k = 28/7 = 4, so y = 4x. If y varies inversely as x for the same values, k = 7 × 28 = 196, so y = 196/x.

Solving Missing-Value Proportion Questions

Use equality of ratios for direct variation and equality of products for inverse variation.

For direct variation, y₁/x₁ = y₂/x₂, which gives x₁y₂ = x₂y₁. For inverse variation, x₁y₁ = x₂y₂. The type of variation must be identified before selecting the equation.

Example: If y is directly proportional to x and y = 18 when x = 6, then k = 3. When x = 14, y = 3 × 14 = 42.

Constant of Proportionality Video Lessons

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Practice Constant of Proportionality Questions

Practise published questions related to this topic.

1Simplify: (2³)² ÷ 2⁴→ 2What is the largest 4-digit number that is divisible by 72?→ 3What is the square root of 1024?→ 4Find the HCF of 2.4, 1.2 and 0.6.→ 5The cost price of the box of sharpener is ₹1,150. The shopkeeper gave 12% discount on the cost price. What is the selling price (in ₹)?→ 6Two successive discounts of 30 percent and 20 percent are given on an article. What will be the ratio of marked price to selling price?→ 7What is the smallest number which when increased by 17 is exactly divisible by 5, 6 and 8?→ 8What is the square root of 4900?→ 9The square of 27x27:→ 10Selling price of an article is Rs. 3500. If the value of the loss is Rs. 500, then what is the loss percentage?→

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Quick Revision Notes

Constant of Proportionality: Quick Revision

Use these formulas and checks to revise direct and inverse variation.

  • Direct variation: y = kx and k = y/x.
  • Inverse variation: y = k/x and k = xy.
  • In direct variation, y/x must have the same value for every pair.
  • In inverse variation, xy must have the same value for every pair.
  • For direct variation, y₂ = (y₁/x₁)x₂.
  • For inverse variation, y₂ = (x₁y₁)/x₂.
  • Keep units consistent when calculating k; the unit of k depends on the variables.

Constant of Proportionality FAQs

What is the constant of proportionality formula for direct variation?

For direct variation, y = kx, so the constant is k = y/x.

What is the proportionality constant formula for inverse variation?

For inverse variation, y = k/x, so k = xy.

If y = 24 when x = 8 in direct variation, what is k?

k = y/x = 24/8 = 3. The equation is y = 3x.

If x and y are inversely proportional and x = 5, y = 12, what is y when x = 10?

The constant product is k = 5 × 12 = 60. Therefore, y = 60/10 = 6.

How can direct and inverse variation be distinguished?

In direct variation, y/x is constant and the variables increase or decrease together. In inverse variation, xy is constant and one variable increases as the other decreases.

What are the units of the constant of proportionality?

For y = kx, k has units of y divided by x. For y = k/x, k has units of x multiplied by y. For example, in distance = speed × time, speed is the direct-variation constant.

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