Ratio and Proportion: Formulas, Rules and Solved Examples

Ratio and Proportion compares quantities and establishes equality between two ratios. This chapter covers equivalent ratios, simplification, division of quantities, compound ratios, continued proportion, and direct or inverse proportion. It also includes formulas and calculation methods for solving common ratio questions accurately.

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What is Ratio and Proportion?

A ratio compares two quantities of the same kind using division, while a proportion states that two ratios are equal. A ratio is written as a:b or a/b, and a proportion is written as a:b = c:d.

In the ratio a:b, a is the first term and b is the second term. Both quantities must be expressed in the same units before comparison. If a:b = c:d, then the product of the extremes equals the product of the means: ad = bc. For example, 2:3 = 8:12 because 2 × 12 = 3 × 8 = 24.

Ratio and Proportion Formula & Tricks

Important Formulas

Equivalent ratios
a:b = ka:kb, where k ≠ 0

Multiplying or dividing both terms of a ratio by the same non-zero number gives an equivalent ratio.

Proportion rule
a:b = c:d ⇔ ad = bc

In a proportion, the product of the extremes equals the product of the means.

Division of a quantity in a ratio
If S is divided in m:n, first part = S × m/(m+n), second part = S × n/(m+n)

The total number of parts is m+n. Each part has value S/(m+n).

Compound ratio
Compound ratio of a:b and c:d = ac:bd

Multiply the antecedents together and the consequents together, then simplify the result.

Third proportional
If a:b = b:c, then c = b²/a

The third proportional to a and b is the value c that makes a, b, c continued proportionals.

Fourth proportional
If a:b = c:x, then x = bc/a

Cross-multiplication gives the fourth proportional x.

Quick Tricks

Remove units before forming a ratio

Convert both quantities into the same unit, then cancel common factors.

Example: 3 m : 75 cm = 300 cm : 75 cm = 4:1.
Use the total-parts method

For a total divided in m:n, add the ratio terms first and multiply each fraction by the total.

Example: Divide ₹840 in 3:4: total parts = 7, so the shares are ₹840 × 3/7 = ₹360 and ₹840 × 4/7 = ₹480.
Cross-multiply an unknown proportion

For a:b = c:x, multiply the means and divide by the remaining term: x = bc/a.

Example: 5:8 = 15:x gives 5x = 120, so x = 24.
Check whether variation is direct or inverse

In direct proportion, the ratio of corresponding values remains constant. In inverse proportion, the product of corresponding values remains constant.

Example: If 4 workers finish a task in 12 days, 6 workers take 4 × 12/6 = 8 days when work is fixed.

Ratio and Proportion Concepts

Simplification and comparison of ratios

A ratio is simplified by dividing both terms by their greatest common divisor.

The terms of a ratio must represent quantities in the same units. To compare a:b and c:d, either convert them to a common form or cross-multiply: a:b is greater than c:d when ad > bc. For three or more quantities, use the same unit for every term before simplification.

Example: The ratio 48:72 simplifies by dividing both terms by 24, giving 2:3. Also, 5:7 < 3:4 because 5 × 4 = 20 is less than 3 × 7 = 21.

Dividing a quantity in a given ratio

To divide a quantity S in the ratio m:n, divide S into m+n equal parts and assign m parts and n parts.

The two shares are Sm/(m+n) and Sn/(m+n). For a ratio involving three terms, such as a:b:c, the shares are Sa/(a+b+c), Sb/(a+b+c), and Sc/(a+b+c).

Example: A sum of ₹1,200 is divided in 2:3:5. The total parts are 10, so the shares are ₹240, ₹360, and ₹600.

Proportion and missing terms

In a proportion a:b = c:d, the product of the first and fourth terms equals the product of the second and third terms.

The first and fourth terms are called extremes, while the second and third terms are called means. Thus, ad = bc. If one term is unknown, isolate it after cross-multiplication.

Example: For 7:9 = x:45, 9x = 7 × 45 = 315, so x = 35.

Direct and inverse proportion

Two quantities are directly proportional when their ratio is constant and inversely proportional when their product is constant.

For direct proportion, y ∝ x and y/x = constant, so y₁/x₁ = y₂/x₂. For inverse proportion, y ∝ 1/x and xy = constant, so x₁y₁ = x₂y₂. Direct proportion increases or decreases together; inverse proportion changes in opposite directions.

Example: If 5 notebooks cost ₹150, 8 notebooks cost 150 × 8/5 = ₹240 under direct proportion. If 8 machines complete a job in 15 hours, 12 machines take 8 × 15/12 = 10 hours under inverse proportion.

Compound and continued proportion

The compound ratio of a:b and c:d is ac:bd, while a, b, c are in continued proportion when a:b = b:c.

For continued proportion, b² = ac. The mean proportional between a and c is √(ac), provided the quantities permit a real value. Compound ratios combine two or more ratios by multiplying corresponding terms.

Example: The compound ratio of 2:3 and 4:5 is 8:15. If 4, 8, x are in continued proportion, then 8² = 4x, so x = 16.

Ratio and 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 Ratio and Proportion Questions

Practise published questions related to this topic.

1Two successive discounts of 25 percent and 20 percent are given on an article. What is the net discount?→ 2A shopkeeper buys an article for ₹500 and sells it for ₹575. What is the percentage profit?→ 3A person earns 25 percent profit on selling price. What will be his profit percentage on the cost price?→ 4The LCM of two numbers 216 and 120 is 1080. Find the HCF of these numbers.→ 5When an article is sold for a discount of 25 percent, then there is a profit 60 percent. If the marked price of article is Rs. 800, then what is the cost price?→ 6Let a = 24 &middot; 32 &middot; 5 and b = 22 &middot; 3 &middot; 73. which of the following is true?→ 7Which of the following numbers is divisible by 45?→ 8A baker sells 100 cupcakes and earns a profit equal to the selling price of 20 cupcakes. What is his profit percentage?→ 9A trader sold an article for ₹240 and made a gain of 20%. What was its cost price?→ 10The HCF and LCM of two positive integers are 36 and 7,560, respectively. If one of the numbers is 540, what is the other number?→

Ratio and Proportion Quick Quiz

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Quick Revision Notes

Ratio and Proportion Revision Points

Recall these rules and formulas for quick calculations.

  • Convert all quantities to the same unit before forming a ratio.
  • a:b = c:d is equivalent to ad = bc.
  • Multiplying or dividing both terms by the same non-zero number preserves a ratio.
  • A total S divided in m:n gives shares Sm/(m+n) and Sn/(m+n).
  • Compound ratio of a:b and c:d is ac:bd.
  • For continued proportion a:b = b:c, b² = ac.
  • Direct proportion: y/x is constant.
  • Inverse proportion: xy is constant.

Ratio and Proportion FAQs

How do you convert 2.5:4 into a whole-number ratio?

Multiply both terms by 10 to remove the decimal: 25:40. Divide by 5, so the simplified ratio is 5:8.

How do you divide ₹900 in the ratio 2:3:4?

The total parts are 2+3+4 = 9. The shares are ₹900 × 2/9 = ₹200, ₹900 × 3/9 = ₹300, and ₹900 × 4/9 = ₹400.

What is the missing term in 6:11 = x:44?

Using cross-multiplication, 11x = 6 × 44 = 264. Therefore, x = 24.

What is the difference between ratio and proportion?

A ratio compares two quantities, such as 3:5. A proportion equates two ratios, such as 3:5 = 12:20.

How is the fourth proportional calculated?

If a:b = c:x, then x = bc/a. For 4:7 = 12:x, x = 7 × 12/4 = 21.

How can direct and inverse proportion be identified?

If y/x remains constant, the quantities are directly proportional. If xy remains constant, they are inversely proportional.

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