Percentage for CSAT

Percentage is a way of expressing a fraction as a number of parts out of 100. The word means per hundred: 45 percent is 45 out of every 100, written 45%. It matters for CSAT because Paper II is a qualifying paper that needs a 33 percent score, and percentage questions appear almost every year, often disguised as profit, discount, population or election questions. Once you treat percent as a fraction in disguise, most questions become one-line arithmetic.

Percent as a fraction

Every percent value has an equivalent fraction, and learning the common ones by heart is the single biggest speed gain in this chapter: 50% = 1/2, 25% = 1/4, 75% = 3/4, 20% = 1/5, 10% = 1/10, 33.33% = 1/3, 66.66% = 2/3, 16.66% = 1/6, 14.28% = 1/7, 12.5% = 1/8, 11.11% = 1/9, 9.09% = 1/11, 6.25% = 1/16, 5% = 1/20. To convert a fraction to a percent, multiply by 100; to convert a percent to a fraction, divide by 100 and simplify.

Example 1. Express 3/8 as a percent. Multiply by 100: (3/8) x 100 = 300/8 = 37.5. So 3/8 = 37.5%. Notice this also matches the fraction table: 1/8 = 12.5%, so 3/8 = 3 x 12.5% = 37.5%.

Finding a percent of a quantity

To find x percent of a number, multiply the number by x and divide by 100. Equivalently, convert the percent to its fraction and multiply. Two special cases come up constantly: 10 percent of anything is found by moving the decimal point one place left (10% of 4,680 = 468), and 1 percent by moving it two places left (1% of 4,680 = 46.8). Every other percent can be built from these: 15% = 10% + 5%, 35% = 3 x 10% + 5%, and so on.

Example 2. Find 35% of 240. 10% of 240 is 24, so 30% is 72. 5% is half of 10%, which is 12. Adding: 72 + 12 = 84. So 35% of 240 is 84. By the fraction route: 35% = 35/100 = 7/20, and (7/20) x 240 = 84.

Percent change and reverse percentage

A percent change is the change in a quantity expressed as a percent of the original value: (new value minus old value) divided by old value, multiplied by 100. A reverse percentage question gives the changed value and the percent change, and asks for the original: divide the changed value by (1 plus the change as a decimal) for an increase, or by (1 minus the change as a decimal) for a decrease.

Example 3. A shopkeeper raises the price of an item by 25% and then offers a 20% discount on the raised price. What is the net effect? Take the original price as 100. After the 25% rise it is 125. A 20% discount on 125 removes 25, leaving 100. So the net effect is zero: the price returns to the original. In general, a rise of a% followed by a fall of b% on the new value is not the same as a fall of (a-b)%.

Example 4. After a 15% increase, a salary is Rs. 23,000. What was the original salary? The original times 1.15 equals 23,000, so the original = 23,000 / 1.15 = 20,000. Hence the original salary was Rs. 20,000.

Successive changes and population growth

Successive percent changes multiply rather than add. If a quantity rises by a% and then by b%, the net factor is (1 + a/100)(1 + b/100), and the net percent change is (a + b + ab/100)%. Population growth follows the same idea compounded over years: if a town's population grows at r% per year, then after n years it becomes P x (1 + r/100)^n, and a decline uses (1 - r/100)^n.

Example 5. The price of sugar rises by 20% and then falls by 10%. What is the net change? Net factor = 1.20 x 0.90 = 1.08, which is an 8% net increase. By the formula: 20 + (-10) + (20 x -10)/100 = 20 - 10 - 2 = 8. So the net change is an 8% increase.

Election and marks questions

Many CSAT questions dress percentage up as election or examination scenarios. In an election between two candidates, votes not cast and invalid votes must be subtracted before applying the vote shares. In a marks question, remember that the pass percent applies to the maximum marks, and a student's shortfall must be compared with the pass marks, not with the maximum.

Example 6. In an election between two candidates, 10% of voters did not vote and 5% of the votes cast were invalid. The winner got 60% of the valid votes and won by 1,400 votes. How many voters were on the rolls? Let total voters be V. Votes cast = 0.9V, valid votes = 0.95 x 0.9V = 0.855V. The winner's margin is (60% - 40%) = 20% of valid votes = 0.2 x 0.855V = 0.171V = 1,400. So V = 1,400 / 0.171 = 8,187 (approx). Working with clean numbers in the exam, the same steps apply.

Common traps and speed tips

• Trap 1: percent of what? A 20% rise followed by a 20% fall does not return to the original: 100 rises to 120, then falls by 20% of 120 (= 24) to 96. The base changes at every step.
• Trap 2: percentage points vs percent. If a rate moves from 20% to 25%, it rose by 5 percentage points but by 25 percent. CSAT options exploit this confusion.
• Trap 3: the 33% qualifying bar. When a question mentions the CSAT qualifying mark of 33%, convert it to a fraction (1/3 of the paper's marks) to compute the target quickly.
• Tip 1: build from 10% and 1%. Any percent of a number can be assembled from its 10% and 1%: 23% = 2 x 10% + 3 x 1%.
• Tip 2: learn the fraction table. 1/6, 1/7, 1/8, 1/9, 1/11 as percents (16.66%, 14.28%, 12.5%, 11.11%, 9.09%) appear repeatedly in discount and share questions.