Percentages: Master Percent Calculations Step by Step

A percentage is a way of expressing a number as a part of 100. The word comes from the Latin per centum, meaning per hundred. So 45 percent means 45 out of every 100, and it is written as 45%. Every percentage is just a fraction with denominator 100 in disguise.

Percentages matter because they let us compare quantities of different sizes fairly. Saying a student scored 36 out of 40 tells you less at a glance than saying 90%, because 90% instantly compares against the same 100-point scale as everyone else. Discounts, exam results, interest rates and statistics all speak in percentages.

Key Concepts

• Percent symbol (%): means “divided by 100”. So 25% = 25/100 = 0.25.
• Base (whole): the quantity the percentage is taken of. In “20% of 150”, the base is 150.
• Percentage points: the plain difference between two percentages. A rise from 20% to 25% is a 5 percentage-point rise, but a 25% relative increase.
• 100%: the whole thing. 200% means twice the whole.
• Conversions: percent to decimal, divide by 100 (45% = 0.45); decimal to percent, multiply by 100 (0.45 = 45%); percent to fraction, put over 100 and simplify (45% = 45/100 = 9/20).

Formulas

• Percent = (part ÷ whole) x 100.
• Part = (percent ÷ 100) x whole.
• Whole = part ÷ (percent ÷ 100).
• Increase by r%: new value = old x (1 + r/100).
• Decrease by r%: new value = old x (1 − r/100).

Worked Example

Find 15% of 240.

1. Convert 15% to a fraction: 15/100.
2. Multiply by the whole: 240 x 15/100 = (240 x 15)/100 = 3600/100.
3. Simplify: 3600 ÷ 100 = 36. So 15% of 240 is 36. Check with decimals: 0.15 x 240 = 36.0, which matches.

Common Misconceptions

• “A 50% increase followed by a 50% decrease returns to the original.” Wrong. Rs. 100 + 50% = Rs. 150, but 50% off Rs. 150 = Rs. 75, not Rs. 100, because the second percentage acts on a bigger base.
• “Percentages can never exceed 100%.” Wrong for growth and comparisons: profits can grow 200%, meaning they tripled.
• “Percentage points and percent are the same.” Wrong. Going from 10% to 20% is +10 percentage points but a 100% increase.

Real-World Applications

• Shops advertise discounts (“20% off”) and taxes are added as percentages, such as GST slabs of 5%, 12%, 18% and 28%.
• Banks quote savings interest, fixed deposit rates and loan charges in percent per year.
• Exams, sports statistics and survey results use percentages to compare fairly across different totals.

Percentages are fractions out of 100, and converting between percent, fraction and decimal is the master key: divide by 100 to leave percent-land, multiply by 100 to enter it. Watch out for the changing base in successive changes, and you will never be fooled by a discount tag again.