Important Formulas Direct & Inverse Proportion - (Maths) Class 8

Terms

Direct Proportion

Direct proportion (also called direct variation) describes the relationship between two quantities that increase or decrease together so that their ratio remains constant.

If x and y are in direct proportion, then

x / y = k = constant (k)

Equivalently,

y = kx

Here k is the constant of proportionality. If x₁, x₂ are two values of x and y₁, y₂ are the corresponding values of y, then

x₁ / y₁ = x₂ / y₂

To find the constant k, divide a known value of y by the corresponding x:

k = y / x

Graphical property: the graph of y = kx is a straight line passing through the origin with slope k.

Examples and Methods (Direct Proportion)

• When the cost of 1 kg of a commodity is fixed, the total cost is directly proportional to the quantity bought.
• If the speed of a vehicle is fixed, the distance travelled in equal times is directly proportional to the time.
• To solve a direct-proportion problem, find k from a known pair and use y = kx to find the unknown.

Example 1. If 3 pens cost ₹54, what is the cost of 7 pens?

Sol.

Cost is directly proportional to number of pens.

Find the constant k by dividing cost by number of pens.

k = 54 ÷ 3

k = 18

Use y = kx to find cost for 7 pens.

Cost = 18 × 7

Ans. ₹126

Inverse Proportion

Inverse proportion (also called inverse variation) describes the relationship between two quantities where an increase in one quantity causes a proportional decrease in the other, so that the product of their corresponding values remains constant.

If x and y are in inverse proportion, then

xy = k = constant (k)

Equivalently,

y = k / x

For two corresponding pairs (x₁, y₁) and (x₂, y₂), we have

x₁ y₁ = x₂ y₂

Graphical property: the graph of y = k / x is a rectangular hyperbola in the first and third quadrants (for positive and negative k respectively).

Examples and Methods (Inverse Proportion)

• If a given job requires a fixed number of man-days, then the number of days is inversely proportional to the number of workers (when all workers work at the same rate).
• Time taken to cover a fixed distance is inversely proportional to speed.
• To solve an inverse-proportion problem, use the product constant: x₁y₁ = x₂y₂ or find k = x·y from a known pair and use y = k / x.

Example 2. 5 men can complete a piece of work in 12 days. How many men will be required to finish the same work in 8 days?

Sol.

The number of men is inversely proportional to the number of days for fixed total work.

Let m be required number of men.

Using inverse proportion: 5 × 12 = m × 8

60 = 8m

m = 60 ÷ 8

m = 7.5

Ans. 8 men will be needed if only whole men are allowed; otherwise 7.5 men indicates 7 men plus additional labour equivalent to half a man (practically hire 8).

How to Recognise and Solve Proportion Problems

• Look for phrases indicating direct proportion: "as A increases, B increases", "for every", "cost proportional to quantity", "per unit".
• Look for phrases indicating inverse proportion: "as A increases, B decreases", "fixed total", "shared among", "constant product", "for the same work".
• Choose the appropriate model: y = kx for direct, y = k / x for inverse.
• Use a known pair to determine k, then compute the unknown quantity using the model.
• For multiple-step problems (compound proportions), reduce proportion step by step using direct or inverse relations as appropriate.

Worked Example (Compound / Mixed)

Example 3. A machine can produce 240 items in 6 hours. How many similar machines are needed to produce 360 items in 4 hours?

Sol.

Let n be the required number of machines.

Production is directly proportional to number of machines and to time.

Using proportionality: (machines × time) is proportional to items produced.

For the first situation: 1 machine × 6 hours produces 240 items.

So production per machine-hour = 240 ÷ (1 × 6)

Production per machine-hour = 40 items

To produce 360 items in 4 hours, we need total machine-hours = 360 ÷ 40

Total machine-hours = 9

Number of machines = total machine-hours ÷ hours available per machine

Number of machines = 9 ÷ 4

Number of machines = 2.25

Ans. 3 machines are needed if only whole machines can be used; otherwise 2.25 machines means 2 machines plus additional capacity (practically hire 3 machines).

Important Formulas (Summary)

• Direct proportion: y = kx; x / y = constant; y₁/x₁ = y₂/x₂
• Inverse proportion: xy = k; y = k / x; x₁y₁ = x₂y₂
• Finding k (direct): k = y / x
• Finding k (inverse): k = x · y
• Compound problems: Break into direct and inverse parts; combine by multiplication or division as required.

Tips and Common Errors

• Always determine whether the relation is direct or inverse before applying formulas.
• Check units and keep quantities consistent (e.g., hours, days, items, people).
• When answers are fractional but real-world objects must be whole (people, machines), round up if necessary and explain the practical interpretation.
• Use cross-multiplication for direct proportion problems presented as ratios.

If needed, practise with diverse word problems: cost-quantity, speed-time-distance, workers-days-work, machine-time-output, and concentrations where appropriate. These strengthen recognition of direct and inverse relations and build quick problem-solving skills.