Percentage for CUET UG

The starting point of this chapter is simple: percentage means per hundred. So 35% literally means 35 out of 100, which is written as $\frac{35}{100}$.

Once this idea becomes natural, every other application becomes easier. Percentage is simply a compact language for comparison. We use it when raw differences do not tell the full story.

For example, if two students improve by 20 marks each, the percentage improvement may still be different because the base marks may be different. That is why base selection matters throughout this chapter.

To convert a fraction into percentage, multiply it by 100. To convert a decimal into percentage, move the decimal point two places to the right and attach the percent sign. To convert a percentage back to fraction or decimal, divide by 100.

Mental benchmarks save a lot of time in CUET. You should know the common ones instantly: 50% = 1/2, 25% = 1/4, 20% = 1/5, 12.5% = 1/8, 33 1/3% = 1/3, and 16 2/3% = 1/6.

To find a percentage of a number, convert the percentage into a fraction or decimal and multiply. If 60% of 500 students clear a screening test, then the number who clear is $\frac{60}{100} \times 500 = 300$.

Reverse percentage works in the other direction. If a quantity becomes 156 after a 30% increase, then 156 represents 130%. So the original quantity is $156/1.3 = 120$.

Students often rush into subtraction or addition here. The better method is to ask what the final value represents in percent form, and then work backwards to 100%.

Comparison questions on Percentage are all about the base. If A is 25% more than B, the base is B. If you reverse the question and ask how much B is less than A, the base changes to A, so the answer changes too.

This is why "more than" and "less than" are not mirror images. The numerator may stay the same, but the denominator changes. That one detail is where many students lose marks.

Useful shortcuts to remember are: 25% more and 20% less, 50% more and 33 1/3% less, and 20% more and 16 2/3% less.

This is one of the highest-value applied percentage ideas. If price rises while expenditure remains constant, consumption must fall. If price falls while expenditure remains constant, consumption must rise.

Suppose the price of rice rises by 25% and Sneha wants to keep her monthly budget unchanged. Then her consumption must reduce by $\frac{25}{125} \times 100 = 20\%$.

The logic is easiest if you assume expenditure is fixed at 100 units. Then you only need to compare how many units can still be bought after the price change.

Whenever a quantity changes twice, use the successive percentage change formula instead of adding or subtracting percentages blindly.

Use positive values for increase and negative values for decrease. For example, if a shirt price rises by 20% and then falls by 10%, the net change is $20 + (-10) + \frac{20(-10)}{100} = 8\%$ increase.

Similarly, if a quantity rises by 20% and then falls by 20%, the net effect is not zero. It becomes $20 + (-20) + \frac{20(-20)}{100} = -4\%$, which means 4% decrease.

Percentage change appears in population growth, revenue change, depreciation, and area change questions. The same successive-change logic applies whenever two percentage changes happen on the same quantity over time.

If population rises by 10% and then by 20%, the net increase is $10 + 20 + \frac{10 \times 20}{100} = 32\%$.

If the length of a rectangle increases by a% and breadth changes by b%, then area change is also calculated by $a + b + \frac{ab}{100}$. So if length rises by 25% and breadth falls by 20%, area change is zero.

Revenue questions also use the same product logic. If cafe footfall rises by 15% and average bill value rises by 20%, total revenue rises by $15 + 20 + \frac{15 \times 20}{100} = 38\%$.

The first skill in Percentage is not calculation. It is identification. Ask whether the problem is about direct percentage, reverse percentage, comparison, constant expenditure, or successive change. Once the type is clear, the method becomes obvious.

The second skill is choosing the correct base. Students often use the change as numerator correctly, but divide by the wrong quantity. That leads to a wrong percentage even though the arithmetic is fine.

Finally, revise the common fraction-percent pairs until they become automatic. That alone can save several minutes over a mixed CUET paper.