Ratio

1. Meaning, notation and simplification

A ratio states how large one quantity is relative to another. If a class has 18 women and 24 men, the ratio of women to men is 18:24 = 3:4. The first term is the antecedent and the second is the consequent. Reversing the order gives 4:3, which answers a different question. Always label the quantities before calculating.

For a direct comparison of quantities of the same kind, convert them to identical units. The ratio of 1.5 metres to 75 centimetres is 150:75 = 2:1, not 1.5:75. A comparison of unlike quantities, such as kilometres per hour, is a rate and retains units. CSAT wording often tests this distinction indirectly.

Reduce an integer ratio by dividing all its terms by their highest common factor. For fractions, multiply every term by the least common multiple of the denominators: 1/2:2/3:3/4 becomes 6:8:9. For decimals, clear decimal places first; 0.24:0.36 becomes 24:36 = 2:3. These operations preserve relative magnitudes.

Ratios can also be compared as fractions. For positive denominators, a/b exceeds c/d when ad exceeds bc. Thus, 7:9 is greater than 10:13 because 91 exceeds 90. Cross-multiplication avoids recurring decimals and is useful when answer options contain close ratios.

• Equivalent ratios: 3:5 = 6:10 = 15:25.
• For ordinary distribution problems, quantities and ratio terms are positive.
• A ratio of 1:1 means equality, not a total of two units.

2. Converting ratios into quantities and shares

The common-multiplier method is the foundation of ratio arithmetic. If A:B = 5:7, write A = 5k and B = 7k. If their total is 96, then 12k = 96, so k = 8 and the quantities are 40 and 56. If their difference is 18 instead, then 2k = 18, giving 45 and 63.

For a total T divided in the ratio m:n, the shares are Tm/(m+n) and Tn/(m+n). The same principle extends to three or more recipients. Dividing ₹7,200 in the ratio 2:3:4 gives nine ratio units; each unit is ₹800. The shares are ₹1,600, ₹2,400 and ₹3,200.

Distinguish a part-to-part comparison from a part-to-whole comparison. If boys:girls = 3:2, boys constitute 3/5, or 60 per cent, of the class. They do not constitute 3/2 of the class. If A is 25 per cent more than B, A:B = 125:100 = 5:4. Conversely, B is 20 per cent less than A because the reference quantity changes.

When a difference is supplied, divide it by the difference between the ratio terms, using the larger term first. When one actual quantity is supplied, divide that quantity by its corresponding ratio term. Before accepting an answer involving people or objects, check that every resulting count is a whole number.

• Total known: k = total ÷ sum of ratio terms.
• Difference known: k = difference ÷ difference of ratio terms.
• One quantity known: k = known quantity ÷ its ratio term.

3. Combining ratios and understanding proportion

To combine A:B and B:C, make the shared quantity B identical in both ratios. If A:B = 3:4 and B:C = 6:5, the least common multiple of 4 and 6 is 12. Rewrite the ratios as 9:12 and 12:10. Therefore, A:B:C = 9:12:10. Simply joining the original numbers would assign two different values to B.

A proportion is an equality of ratios. If a:b = c:d, then ad = bc, provided the denominators are non-zero. In direct proportion, one quantity changes by the same factor as another: at a fixed price per kilogram, cost is directly proportional to mass purchased. Thus, cost divided by mass remains constant.

In inverse proportion, the product of the quantities remains constant. For a fixed amount of work and equally efficient workers, workers multiplied by days is constant. If 12 workers finish a task in 15 days, 20 workers require 9 days. The inverse relationship depends on unchanged productivity and work conditions.

A compound ratio is obtained by multiplying corresponding terms. The compound ratio of 2:3 and 4:5 is 8:15. In partnership problems, profit is commonly distributed in the ratio of capital multiplied by investment duration, unless another agreement is specified. Capitals alone determine the shares only when the investment periods are equal.

• Direct proportion: x/y is constant.
• Inverse proportion: xy is constant.
• Combining linked ratios requires a common scale for the shared quantity.

4. Changing ratios, mixtures and equal increments

When quantities change, express the original quantities using one multiplier and then apply the changes. Suppose two quantities are in the ratio 3:5. Adding 10 to each changes the ratio to 5:7. Then (3k+10)/(5k+10) = 5/7. Cross-multiplication gives 21k+70 = 25k+50, so k = 5. The original quantities are 15 and 25.

Adding the same positive amount to two unequal positive quantities brings their ratio closer to 1:1; it does not preserve the original ratio. Multiplying both quantities by the same positive factor does preserve it. This distinction is especially useful in age problems: after a stated number of years, equal amounts are added to both ages while their difference remains unchanged.

For mixtures, convert the ingredient ratio into fractions of the total. A 40-litre mixture with milk:water = 3:1 contains 30 litres of milk and 10 litres of water. Adding 10 litres of water changes the ratio to 30:20 = 3:2. Removing a uniformly mixed portion removes both ingredients in their existing ratio; replacing it with one ingredient changes the composition.

• Different percentage changes: the new ratio is m(1+p/100):n(1+q/100).
• For percentage decreases, use negative values of p or q.
• In age questions, verify that all ages remain valid at every stated time.

5. CSAT solving strategy and common errors

Begin by naming the quantities and identifying whether the information concerns a total, difference, percentage, rate or change. Use ratio units when only relative information matters. For example, assume quantities are 4k and 7k rather than selecting arbitrary actual values that may conflict with later conditions.

Do not average ratios mechanically. Two classes with boys:girls ratios of 1:1 and 3:1 need not have a combined ratio of 2:1. Their class sizes determine the combined numbers. Convert each ratio into actual counts or weighted fractions before combining groups.

Use estimation and option elimination after forming the correct relationship. A share corresponding to the larger ratio term must be larger; component shares must add to the total. In data-sufficiency questions, distinguish knowing a ratio from knowing a scale. A ratio plus a compatible total usually determines both quantities, whereas two equivalent ratio statements add no independent information.

• Never cancel terms across addition: (a+c)/(b+c) is not generally a/b.
• Retain exact fractions until the final step.
• Substitute the answer into the original conditions, not merely the rearranged equation.

Previous year questions

No UPSC question has been asked directly on this micro-topic yet. Use the practice questions below.

Practice questions