Percentage, Ratio and Average for CSAT

Percentage, ratio and average are the base of quantitative aptitude. Almost every other topic in the CSAT paper, such as profit and loss, interest, mixtures, time and work, and data interpretation, uses these three ideas. If you master them, you can solve many questions in under a minute using mental maths. The UPSC CSAT (GS Paper 2) usually has around 20 to 22 questions of basic numeracy and data interpretation, and since the qualifying mark is only 33 percent, these topics can secure your qualification. This note explains the concepts, formulas and shortcuts in simple language, with solved examples.

Percentage: The Basics

"Per cent" means "per hundred". A percentage is a fraction with denominator 100. To find x percent of a number N, calculate N × x / 100. The key to speed is to know the fraction-percentage equivalents by heart, so that you can convert in seconds.

For example, 25 percent of 480 is one-fourth of 480, which is 120. And 12.5 percent of 640 is one-eighth of 640, which is 80. A very useful trick is that x percent of y equals y percent of x, so 8 percent of 50 is the same as 50 percent of 8, which is 4.

Percentage Change and Successive Change

Percentage change is the change divided by the original value, multiplied by 100. If a price rises from 80 to 100, the increase is 20 and the percentage increase is 20/80 × 100 = 25 percent. For two successive changes of a percent and b percent, the net change is a + b + ab/100 (use a negative sign for a decrease). If a price goes up by 20 percent and then goes down by 20 percent, the net change is 20 − 20 − 400/100 = −4 percent, which is a decrease of 4 percent. This is a very common question in the examination. Another useful rule is that if the price of an item increases by r percent, a person who wants to keep the expenditure the same should reduce consumption by r/(100 + r) × 100 percent. So when the price rises by 25 percent, the consumption must fall by 25/125 × 100 = 20 percent.

Applications of Percentage

Population growth: if a town has 20,000 people and the population grows 10 percent every year, then after 2 years it will be 20,000 × 1.1 × 1.1 = 24,200. Marks and results: if a student gets 120 marks and fails by 30 marks when the pass mark is 40 percent, then the pass mark is 150, and the total marks are 150 × 100/40 = 375. Discount and profit: profit percent is calculated on the cost price, and discount on the marked price. Election and votes: if a candidate gets 55 percent of the valid votes and wins by 2,000 votes, then the difference of 10 percent equals 2,000, so total valid votes are 20,000. These patterns repeat in the CSAT.

Ratio and Proportion

A ratio compares two quantities of the same kind. The ratio a:b means a/b. Ratios can be simplified like fractions, and they have no unit. If a:b is 2:3 and b:c is 4:5, then to join them make the middle term equal: multiply the first ratio by 4 and the second by 3 to get a:b = 8:12 and b:c = 12:15, so a:b:c = 8:12:15. To divide a sum in a given ratio, add the parts and find the value of one part. For example, to divide 2,100 rupees in the ratio 2:3:7, the total parts are 12 and one part is 175, so the shares are 350, 525 and 1,225 rupees. Proportion is the equality of two ratios: a:b = c:d, which gives ad = bc. In direct proportion, if one quantity increases, the other increases at the same rate, as in cost and quantity. In inverse proportion, if one increases, the other decreases, as in the number of workers and the days taken.

Alligation and Mixtures

Alligation is a quick method for mixing two things at different rates to get a mean rate. The ratio of the quantities is (higher price − mean price) : (mean price − lower price). For instance, to mix rice costing 40 rupees and 50 rupees per kg to get a mixture worth 46 rupees per kg, the ratio of cheaper to costlier rice is (50 − 46) : (46 − 40), which is 4:6 or 2:3. This works for mixing milk and water, solutions of different strength and averages of groups.

Average

The average (arithmetic mean) is the sum of the observations divided by their number. So sum = average × number of items. This simple idea solves most questions. For the first n natural numbers, the average is (n + 1)/2, so the average of the first 10 natural numbers is 5.5. For an evenly spaced series, the average is the average of the first and last terms.

Change of average when one item is added. If the average of 5 numbers is 20 and a sixth number is added so that the average becomes 22, the new number is 6 × 22 − 5 × 20 = 132 − 100 = 32. Weighted average. If Class A has 30 students with average marks 60, and Class B has 20 students with average 70, the combined average is (30 × 60 + 20 × 70)/50 = 3,200/50 = 64. It lies closer to the larger group. Average speed. For equal distances at speeds x and y, the average speed is 2xy/(x + y), not the simple average. For 40 km/h and 60 km/h, it is 2 × 40 × 60/100 = 48 km/h. Removal of an item. If the average age of 5 persons is 30 years and a 40-year-old leaves, the new average is (150 − 40)/4 = 27.5 years.

Tricks to Save Time

Convert percentages to fractions when possible, as 37.5 percent is 3/8. Use the multiplier method for successive changes: an increase of 20 percent means multiplying by 1.2 and a decrease of 10 percent means multiplying by 0.9. In ratio questions, assume the common multiplier to be x and form equations. In average questions, think of the "deviation method": choose an assumed average, and add the plus and minus differences. Always check the units and the question's demand (whole amount or only the share). Practise a few questions daily from previous years' papers to see the pattern; the numbers are usually simple.

Common Mistakes

Many students add successive percentages directly, forgetting the product term. Some calculate percentage change on the wrong base (the final value instead of the initial value). In ratio questions, they forget to bring the ratios to the same middle term. In average questions, they use a simple average when a weighted average is needed. Read the question twice, and write the given data in a small table before calculating.

Practice Set With Reasoning

Try these in your head before reading the solutions. First: a shopkeeper marks a product 20 percent above cost and gives 10 percent discount; what is the profit? Take the cost as 100, the marked price becomes 120, the selling price becomes 108, so the profit is 8 percent. Second: the ratio of boys to girls in a class of 45 is 4:5; how many girls? Total parts are 9, so each part is 5 students and the girls are 5 × 5 = 25. Third: the average of three numbers is 15, and two of them are 10 and 20; what is the third? The sum is 45, so the third is 45 − 30 = 15. Fourth: what percent of 40 is 6? It is 6/40 × 100 = 15 percent. These small questions are the building blocks of larger ones. If you solve them mentally, you will have more time for the lengthy comprehension and reasoning questions, and you will protect your score from silly calculation errors under pressure.

Why Practice Beats Memorising

Formulas help, but speed comes from practice. Spend 30 minutes each day on aptitude, set a timer, and check how many questions you solved in the time. Keep a notebook of formulas and mistakes, and revise it every weekend. Do not look at the answer immediately; try at least two approaches before checking. When you learn a trick, understand why it works, so that you can adapt it to a new situation. In the actual paper, attempt the questions in which you are confident first, and leave very lengthy calculations for the end.

Quick Revision

• Percentage change = change/original × 100; successive change = a + b + ab/100.
• Sum = average × number; weighted average depends on group sizes.
• Alligation: (higher − mean) : (mean − lower).
• Average speed for equal distances = 2xy/(x + y).

Percentage, ratio and average form the foundation of aptitude. Learn the equivalents table, practise the five or six standard patterns until they feel automatic, and you will be able to clear the numeracy part of CSAT with confidence.

Related Notes to Read Next

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

• Time, Speed, Distance and Work for CSAT
• Logical Reasoning for CSAT Notes for UPSC
• Data Interpretation for CSAT Notes for UPSC
• All CSAT notes

Frequently Asked Questions (FAQ)

What is percentage?

A percentage expresses a number as a fraction of 100. For example, 25 percent means 25 out of every 100, or one-fourth.

How do I calculate successive percentage change?

For successive changes of a and b percent, the net change is a + b + (a × b)/100, using a negative sign for decreases.

If a price increases 20 percent and then decreases 20 percent, what is the net change?

The net change is a 4 percent decrease, because 20 − 20 − (20 × 20)/100 equals −4.

How much should consumption fall if the price rises by 25 percent?

To keep expenditure unchanged, consumption should fall by 25/125 × 100, which is 20 percent.

What is a ratio?

A ratio is a comparison of two quantities of the same kind by division, written as a:b.

How do I combine two ratios, a:b and b:c?

Make the value of b the same in both ratios by multiplying, and then write a:b:c. For example, 2:3 and 4:5 become 8:12:15.

What is alligation?

Alligation is a shortcut method to find the ratio in which two items of different prices or strengths should be mixed to get a given mean value.

What is the formula for average?

Average equals the sum of all observations divided by the number of observations, so sum equals average times number.

What is weighted average?

A weighted average takes into account the size or importance of each group, such as the number of students in each class, when finding the overall average.

What is the average speed for equal distances at two speeds?

It is 2xy/(x + y), where x and y are the two speeds. It is lower than the simple average of the speeds.

How can I remember percentage and fraction equivalents?

Practise a table of common fractions such as 1/2 = 50%, 1/3 = 33.33%, 1/4 = 25%, 1/5 = 20%, 1/8 = 12.5% until you can recall them instantly.

How many CSAT questions come from these topics?

Basic numeracy questions that use percentage, ratio, average and related ideas make up a significant part of the 20 to 22 quantitative questions in CSAT.