Direct Proportion: Formula, Rules, Examples and Tricks

Direct Proportion describes a relationship in which two quantities increase or decrease in the same ratio. This page explains the direct proportion formula, constant of proportionality, ratio method, unitary method, graph property and solved direct proportion questions used in quantitative aptitude.

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What Is Direct Proportion?

Two quantities are in direct proportion when their ratio remains constant. If one quantity increases, the other increases in the same ratio; if one decreases, the other also decreases in the same ratio.

If x and y are directly proportional, then y ∝ x and y/x = k, where k is a constant. Therefore, y = kx. For two corresponding values, x₁ and y₁, and x₂ and y₂, the relation is y₁/x₁ = y₂/x₂. For example, if 3 pens cost ₹45, then 5 pens cost ₹75 because the cost per pen remains ₹15.

Direct Proportion Formula & Tricks

Important Formulas

Direct proportion formula
y ∝ x ⇒ y = kx

Here, k is the constant of proportionality. The value of y changes in the same ratio as x.

Ratio form
y₁/x₁ = y₂/x₂

Use this form when two corresponding values of directly proportional quantities are given.

Cross-multiplication form
x₁y₂ = x₂y₁

This is obtained by cross-multiplying y₁/x₁ = y₂/x₂ and is useful for finding an unknown value.

Constant of proportionality
k = y/x

The constant k is found by dividing one value of y by its corresponding value of x.

Quick Tricks

Check the ratio before solving

Divide the second quantity by the first quantity for each pair. Equal ratios confirm direct proportion.

Example: For pairs (4, 12) and (7, 21), 12/4 = 3 and 21/7 = 3, so they are directly proportional.
Use the scale-factor method

Find how many times the known quantity changes, then multiply the corresponding quantity by the same factor.

Example: If 6 kg costs ₹240, then 10 kg costs ₹240 × 10/6 = ₹400.
Keep units consistent

Convert quantities into the same units before applying the ratio formula.

Example: If 2 hours correspond to 120 km, then 30 minutes must first be converted to 0.5 hours before calculating the distance.
Use the origin test for graphs

The graph of y = kx is a straight line passing through the origin. Its slope is the constant k.

Example: For y = 4x, the graph passes through (0, 0), (1, 4) and (2, 8).

Direct Proportion Concepts

Finding an Unknown by Ratio

For directly proportional quantities, set the ratio of corresponding values equal and cross-multiply.

If x₁ corresponds to y₁ and x₂ corresponds to y₂, use y₁/x₁ = y₂/x₂. This method assumes that the order of corresponding quantities remains the same on both sides.

Example: If 8 notebooks cost ₹160, the cost of 13 notebooks is x. Then 160/8 = x/13, so x = 160 × 13/8 = ₹260.

Constant of Proportionality

The constant of proportionality is the fixed ratio between the dependent and independent quantities.

For y = kx, k = y/x. Once k is known, any missing value can be calculated using y = kx. The units of k depend on the units of y and x.

Example: If 5 metres of cloth cost ₹450, then k = 450/5 = ₹90 per metre. The cost of 7 metres is 90 × 7 = ₹630.

Unitary Method in Direct Proportion

The unitary method first finds the value for one unit and then multiplies it by the required number of units.

When more units require more total quantity, divide the known total by the known number of units. Then multiply the unit value by the new number of units.

Example: If 12 workers produce 360 units in a fixed period, one worker produces 360/12 = 30 units in that model. Therefore, 15 workers produce 15 × 30 = 450 units, assuming equal individual productivity.

Direct Proportion and Direct Variation

Direct variation is another name for a relationship represented by y = kx, where the ratio y/x is constant.

In direct variation, doubling x doubles y, and multiplying x by a factor r multiplies y by the same factor r. Its graph is a straight line through the origin. This differs from inverse proportion, where xy remains constant.

Example: If y = 5x, then x = 3 gives y = 15, while x = 6 gives y = 30; both x and y have doubled.

Direct Proportion Video Lessons

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Duplicate, Triplicate and Sub-Duplicate Ratios

Learn the definitions and application of duplicate, triplicate, and sub-duplicate ratios, with clear methods for understanding how these special ratios are formed and used.

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Practice Direct Proportion Questions

Practise published questions related to this topic.

1If we multiply (-6ab) and (11ab²c), then we get:→ 2Aman purchased goods worth Rs. 800 and sold them at a 25% profit. What was the selling price?→ 3A man sells a mobile phone for ₹600 and loses something. If he had sold it for ₹1190, his gain would have been 4-times the former loss. The cost price (in ₹) of the mobile phone is ________.→ 4What is the largest 5 digit number divisible by 89?→ 5The product of two numbers is 300 and their LCM is 150. Find their HCF.→ 6A shopkeeper is offering a 20% discount on a Mixer that originally costs Rs 400. What is the sale price of the Mixer?→ 7Successive discounts of 9% and 6% are equivalent to a single discount of:→ 8An item with a marked price of ₹657 was sold for ₹526. What was the percentage of discount given? (Round off the answer to two decimal places.)→ 9Total marked price of 500 articles is Rs. 400. 25 percent discount is given. What will be the selling price for 15000 such articles? (marked price of each article is same)→ 10An officer computes (√50 + √8).Simplify.→

Direct Proportion Quick Quiz

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Quick Revision Notes

Direct Proportion Quick Revision

Use these rules and formulas to solve direct proportion questions.

  • Direct proportion means y/x remains constant.
  • The direct proportion formula is y = kx.
  • For two pairs, use y₁/x₁ = y₂/x₂ or x₁y₂ = x₂y₁.
  • Find the constant using k = y/x.
  • If one quantity is multiplied by r, the other is also multiplied by r.
  • Convert all measurements to consistent units before forming ratios.
  • The graph of y = kx is a straight line through the origin.
  • Direct proportion has a constant ratio, whereas inverse proportion has a constant product.

Direct Proportion FAQs

What is the direct proportion formula?

The formula is y = kx, where k is constant. For two pairs of values, use y₁/x₁ = y₂/x₂.

How do you find the missing value in direct proportion?

Set the ratios of corresponding quantities equal and cross-multiply. For example, if 4 items cost ₹80, then 7 items cost x: 80/4 = x/7, so x = ₹140.

How can direct proportion be identified from a table?

Divide each value of one quantity by its corresponding value in the other quantity. If the ratio is the same for every pair, the quantities are directly proportional.

What is the constant of proportionality if 9 kg costs ₹540?

Taking cost as y and weight as x, k = y/x = 540/9 = ₹60 per kg.

If one quantity triples in direct proportion, what happens to the other?

The other quantity also triples. For example, if y = 4x, changing x from 2 to 6 changes y from 8 to 24.

What is the difference between direct and inverse proportion?

In direct proportion, y/x is constant and y = kx. In inverse proportion, xy is constant and y = k/x.

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