Ratio and Proportion Notes: Formulas, Tricks and Solved Examples

Ratio and proportion is a basic chapter of quantitative aptitude, but its ideas appear everywhere: in sharing profits, mixing liquids, ages of family members, maps and scales, speed and time, and data interpretation. A clear understanding of this chapter makes many other chapters easy. This note explains the concepts of ratio, proportion and variation in simple language, gives the important formulas and shortcuts, and walks through solved examples on common question types such as sharing, change in ratio, ages, mixtures, partnership and coins.

What Is a Ratio?

A ratio is the comparison of two quantities of the same kind by division. The ratio of a to b is written as a:b or a/b. The first term is called the antecedent and the second the consequent. A ratio has no unit, and it does not change if both terms are multiplied or divided by the same non-zero number. For example, 6:9 is the same as 2:3. To compare two quantities in a ratio, they must be in the same units: the ratio of 2 hours to 40 minutes is 120:40 = 3:1.

Types of Ratios

Comparing Ratios

To compare a:b and c:d, convert them into fractions and cross-multiply. Compare 3:4 and 5:7. We check 3 × 7 = 21 and 4 × 5 = 20. Since 21 is greater, 3/4 is greater than 5/7. This method is quick and avoids decimals.

Sharing in a Given Ratio

To divide a quantity in the ratio a:b:c, add the parts (a + b + c) and find the value of one part. For example, 2,100 rupees divided in the ratio 2:3:7 has 12 parts, one part is 175, and the shares are 350, 525 and 1,225 rupees. If the ratio is given in fractions, such as 1/2:1/3:1/4, multiply by the LCM of the denominators (12) to get 6:4:3.

Change in a Ratio

Two numbers are in the ratio 3:5. If 10 is added to each, the ratio becomes 5:7. Find the numbers. Let the numbers be 3x and 5x. Then (3x + 10)/(5x + 10) = 5/7, which gives 21x + 70 = 25x + 50, so 4x = 20 and x = 5. The numbers are 15 and 25. Check: 25/35 = 5/7, which is correct.

Proportion

When two ratios are equal, a:b = c:d, we say that a, b, c and d are in proportion, written as a:b::c:d. The first and fourth terms are the extremes, and the second and third terms are the means. The key rule is: product of extremes = product of means, that is, ad = bc. From this, the fourth proportional to a, b and c is d = bc/a. The third proportional to a and b is b²/a, and the mean proportional between a and b is the square root of ab.

Examples: the fourth proportional to 4, 9 and 12 is 9 × 12/4 = 27. The third proportional to 4 and 6 is 36/4 = 9. The mean proportional between 4 and 9 is the square root of 36, which is 6.

Useful Properties: Componendo and Dividendo

If a/b = c/d, then (a + b)/b = (c + d)/d (componendo), (a − b)/b = (c − d)/d (dividendo), and (a + b)/(a − b) = (c + d)/(c − d) (componendo-dividendo). For example, if a/b = 5/3, then (a + b)/(a − b) = (5 + 3)/(5 − 3) = 8/2 = 4. This property is useful in simplifying algebraic ratio problems quickly.

Variation: Direct and Inverse

Two quantities are in direct variation if both increase or decrease together in the same ratio, as the cost of items and the quantity bought. They are in inverse variation if one increases while the other decreases so that the product stays constant, such as speed and time for a fixed distance, or workers and days for a fixed job. If 12 workers can complete a task in 15 days, then 18 workers will take 12 × 15/18 = 10 days.

Ages Using Ratio

The ratio of the ages of a father and his son is 7:2. After 10 years, the ratio will be 9:4. Find their present ages. Let the ages be 7x and 2x. Then (7x + 10)/(2x + 10) = 9/4, so 28x + 40 = 18x + 90, giving 10x = 50 and x = 5. The present ages are 35 and 10 years. Check: after 10 years, 45 and 20 are in the ratio 9:4.

Mixtures

A vessel contains 50 litres of milk and water in the ratio 3:2. Then milk is 30 litres and water is 20 litres. If 10 litres of water is added, the milk and water become 30 and 30 litres, so the ratio becomes 1:1. For mixing two different mixtures, use the alligation rule: (higher − mean):(mean − lower).

Partnership

Profits in a partnership are shared in the ratio of capital × time. If A invests 20,000 rupees for 12 months and B invests 30,000 rupees for 8 months, the ratio of their investments is 240,000:240,000, which is 1:1. Therefore, if the profit is 12,000 rupees, each gets 6,000 rupees. If a partner is a working partner and gets a salary or a commission, first deduct it from the profit and share the rest.

Income, Expenditure and Savings

The incomes of A and B are in the ratio 5:4 and their expenditures are in the ratio 3:2. If each saves 2,000 rupees, find the incomes. Let the incomes be 5x and 4x and the expenditures be 3y and 2y. Then 5x − 3y = 2,000 and 4x − 2y = 2,000. From the second, y = 2x − 1,000. Putting it in the first, 5x − 6x + 3,000 = 2,000, so x = 1,000 and y = 1,000. The incomes are 5,000 and 4,000 rupees, and the expenditures are 3,000 and 2,000 rupees.

Coins Problem

A bag contains one-rupee, fifty-paise and twenty-five-paise coins in the ratio 5:6:8, and the total value is 210 rupees. The values of the coins are in the ratio 5 × 1 : 6 × 0.5 : 8 × 0.25, which is 5:3:2, with a total of 10 parts for each unit k of coins. So 10k = 210 and k = 21. The numbers of coins are 105, 126 and 168.

Tips and Common Mistakes

Convert all quantities to the same unit before forming a ratio. In "after n years" problems, the same number is added to each age, not multiplied. Do not mix up the ratio of numbers and the ratio of their values in coin problems. In partnership, remember to multiply capital by time. In ratio change problems, form the equation using a variable x as the common multiplier. When ratios are given in a chain, make the middle terms equal first. Check the answer by putting it back in the question.

Real-Life Uses of Ratio and Proportion

Ratio and proportion is not only an examination topic; it is a tool used every day. A cook adjusts a recipe for more guests by keeping the ingredients in the same ratio. A map has a scale such as 1:50,000, so a distance of 4 cm on the map is 2 km on the ground. A chemist mixes solutions in a fixed ratio, an architect draws plans to scale, and a business shares profit by the investment of partners. Governments use ratios such as the sex ratio, teacher-student ratio, doctor-population ratio and debt-to-GDP ratio to measure progress. Seeing these examples makes the chapter more interesting and helps you remember the ideas. When you practise, try to create small stories from your own life, such as dividing pocket money or mixing cold drink concentrate, and solve them using the same methods.

How to Practise This Chapter

Start with five questions of each type from the examples above, solving without help. Then increase the difficulty by changing the numbers. Keep a small notebook with formulas and one solved example for each type. Time yourself: aim for under one minute for sharing and comparison questions, and under two minutes for ages and partnership. If you make a mistake, find out whether it is a conceptual or a calculation error. Revise the notebook weekly.

Quick Revision

• a:b = c:d means ad = bc; fourth proportional = bc/a; third proportional = b²/a.
• Share in ratio: add the parts, find one part and multiply.
• Partnership profit ratio = capital × time.
• Inverse variation: product remains constant.

With regular practice, ratio and proportion becomes one of the easiest and most scoring chapters in arithmetic. Solve each type at least five times and write the steps neatly to avoid errors.

Related Notes to Read Next

Continue your preparation with these related notes from the same subject:

• Time and Work Notes for Competitive Exams
• Profit, Loss and Interest Notes for Competitive Exams
• Number System Notes for Competitive Exams
• All Quantitative Aptitude notes

Frequently Asked Questions (FAQ)

What is a ratio?

A ratio is a comparison of two quantities of the same kind by division, written as a:b. It has no units.

What is a proportion?

A proportion is an equality of two ratios. If a:b = c:d, then a, b, c and d are in proportion.

What is the rule of proportion?

In a proportion, the product of the extremes equals the product of the means, that is, ad = bc.

What is the fourth proportional?

The fourth proportional to a, b and c is the number d such that a:b = c:d, given by d = bc/a.

What is the mean proportional between two numbers?

The mean proportional between a and b is the square root of their product, √(ab).

How do I divide a number in a given ratio?

Add the parts of the ratio, divide the number by this sum to get the value of one part and multiply it by each term of the ratio.

What is componendo and dividendo?

If a/b = c/d, then (a + b)/(a − b) = (c + d)/(c − d). This is a quick method to simplify ratio problems.

What is the difference between direct and inverse proportion?

In direct proportion, the quantities increase or decrease together in the same ratio. In inverse proportion, one increases when the other decreases, keeping the product constant.

How do I solve age problems using ratio?

Assume the ages as multiples of x according to the ratio, add or subtract the given years to form the new ratio and solve the resulting equation.

How are profits shared in a partnership?

Profits are shared in the ratio of the product of the capital invested and the time for which it was invested.

What is a compounded ratio?

A compounded ratio is the ratio formed by multiplying the antecedents and consequents of two or more ratios.

How do I compare two ratios quickly?

Cross-multiply. For a:b and c:d, compare ad with bc. The ratio with the larger cross product on its side is the greater.