Percentage: Rules, Shortcuts and Worked Examples

Percentage questions appear in almost every Lok Sewa maths and mental ability paper, and the same ideas are used again in profit and loss, interest and statistics. Once you understand that a percent is simply a fraction with 100 at the bottom, most questions can be solved in one or two lines. This note explains the rules step by step with worked examples in rupees.

1. Meaning of percentage

The word percent means “out of every hundred”. The sign % stands for “divided by 100”. So 35% means 35 out of 100, which is the fraction 35/100 or the decimal 0.35.

• 100% of a quantity is the whole quantity.
• 50% is half, 25% is one quarter, 10% is one tenth.
• A percent can be more than 100. For example, 150% of Rs 400 is Rs 600, which is more than the original amount.

2. Converting fractions, decimals and percent

• Fraction to percent: multiply by 100. For example, 3/5 × 100 = 60%.
• Percent to fraction: write over 100 and simplify. For example, 45% = 45/100 = 9/20.
• Decimal to percent: move the decimal point two places to the right. For example, 0.08 = 8%.
• Percent to decimal: move the decimal point two places to the left. For example, 12.5% = 0.125.

Learning the common values below saves a lot of time in the exam:

3. Finding a percent of a number

Rule: x% of N = (x/100) × N.

• Example: Find 15% of Rs 2,400. Working: 15/100 × 2,400 = Rs 360.
• Example: A family spends 40% of a monthly income of Rs 35,000 on food. Food cost = 40/100 × 35,000 = Rs 14,000.

Finding the whole from a part: if x% of a number is known, divide by x and multiply by 100.

• Example: 60% of a sum is Rs 360. The whole sum = 360 × 100 / 60 = Rs 600.

4. What percent one number is of another

Rule: required percent = (part / whole) × 100.

• Example: What percent of Rs 1,800 is Rs 450? Working: 450/1,800 × 100 = 25%.
• Example: A student scores 54 marks out of 75. Percent = 54/75 × 100 = 72%.

Comparing two numbers: be careful about which number is the base.

• If A is r% more than B, then B is less than A by r/(100 + r) × 100 percent.
• If A is r% less than B, then B is more than A by r/(100 − r) × 100 percent.
• Example: Ram’s salary is 25% more than Hari’s. Hari’s salary is less than Ram’s by 25/125 × 100 = 20%.

5. Percent increase and decrease

Rule: percent change = (change / original value) × 100. The base is always the original value.

• Example: The price of a bag rises from Rs 800 to Rs 920. Increase = Rs 120, so percent increase = 120/800 × 100 = 15%.
• Example: A salary of Rs 25,000 is cut by 8%. New salary = 25,000 × 92/100 = Rs 23,000.

A quick way: to increase by r%, multiply by (100 + r)/100; to decrease by r%, multiply by (100 − r)/100. These numbers are called multiplying factors. For example, increase by 20% means × 1.2, and decrease by 30% means × 0.7.

6. Successive percentage change

When one percent change is followed by another, the second change works on the new value, not the original. So the changes cannot simply be added.

Rule: net change = a + b + (a × b)/100. Use a minus sign for a decrease. A positive answer means a net increase; a negative answer means a net decrease.

• Example: A price of Rs 5,000 is raised by 20% and then reduced by 10%. Net change = 20 − 10 + (20 × −10)/100 = 10 − 2 = 8% increase. Check: 5,000 becomes 6,000 and then 5,400, which is 8% more than 5,000.
• Example: A rise of 10% followed by a fall of 10% gives 10 − 10 − 1 = −1, that is a net fall of 1%, not zero.
• Example: Two successive increases of 10% give 10 + 10 + 1 = 21%.

7. Population and price change over years

When a population or a price changes at a fixed rate every year, each year’s change is on the previous year’s value.

• Growth: value after n years = P × (1 + r/100) to the power n.
• Decrease (depreciation): value after n years = P × (1 − r/100) to the power n.
• Example: A village has 40,000 people and grows at 5% per year. After 2 years: 40,000 × 1.05 × 1.05 = 44,100.
• Example: A machine bought for Rs 50,000 loses 10% of its value every year. After 2 years: 50,000 × 0.9 × 0.9 = Rs 40,500.
• Going back: population 2 years ago = present population ÷ (1 + r/100) to the power 2. If a town now has 12,100 people after growing 10% per year, two years ago it had 12,100 ÷ 1.21 = 10,000.

Price rise and consumption: if the price of an item rises by r%, a family must reduce consumption by r/(100 + r) × 100 percent to keep spending the same. If the price falls by r%, consumption can increase by r/(100 − r) × 100 percent.

• Example: Sugar becomes 25% dearer. To keep the same expense, reduce use by 25/125 × 100 = 20%.

8. Useful shortcuts

• 10%: move the decimal point one place left. 10% of Rs 3,670 = Rs 367.
• 5%: half of 10%. 5% of Rs 3,670 = Rs 183.50.
• 1%: move the decimal point two places left, then build other values. 3% of Rs 900 = 3 × 9 = Rs 27.
• Swap trick: x% of y = y% of x. So 8% of 50 = 50% of 8 = 4.
• Use the fraction table: 12.5% of Rs 640 = 1/8 × 640 = Rs 80; 33 1/3% of Rs 270 = 1/3 × 270 = Rs 90.
• Break it up: 35% of 600 = 30% (180) + 5% (30) = 210.