What is a Percentage?
The word percent originates from the Latin *per centum*, meaning "by the hundred." A percentage is simply a ratio or fraction whose denominator is fixed at 100.
For example, 25% represents 25 parts out of 100, which can be expressed as:
Fraction: (25 / 100) = (1 / 4)Decimal: 0.25Every percentage calculation involves three core variables:
1Base Value ($B$): The whole or starting amount.
2Rate ($P$): The percentage rate.
3Percentage Amount ($A$): The portion resulting from applying the rate to the base.
To find a specific percentage of a given number, convert the percentage into a decimal (by dividing by 100) and multiply by the base:
Mathematical Equation
Amount = (P / 100) ร B
Worked Example:
Question: What is 15% of \80$?
1Convert percentage to decimal: 15 / 100 = 0.15
2Multiply by base: 0.15 ร 80 = 12
When you know the part (X) and the whole (Y) and need to find the corresponding percentage rate:
Mathematical Equation
Percentage = โค ft((X / Y)) ร 100%
Worked Example:
Question: If you scored 46 correct answers out of 50 on an exam, what is your score percentage?
1Divide part by whole: 46 / 50 = 0.92
2Multiply by 100: 0.92 ร 100 = 92%
To determine the relative change between an Old Value and a New Value:
Mathematical Equation
Percentage Change = โค ft((New Value - Old Value / |Old Value|)) ร 100%
Worked Example:
Question: A stock was priced at \50 and rose to \65. What was the percentage increase?
1Calculate difference: 65 - 50 = +15
2Divide by original base: 15 / 50 = 0.30
3Multiply by 100: 0.30 ร 100 = +30% increase.
Real-World Worked Examples
1. Calculating a Retail Discount
Scenario: A jacket priced at \120 is on a 30\%$ off sale. What is the final price?
Discount Amount: 120 ร 0.30 = \36$Final Sale Price: 120 - 36 = \84$*Shortcut:* Multiply directly by (1 - 0.30) = 0.70: \120 \times 0.70 = \84.2. Calculating Sales Tax
Scenario: A restaurant bill is \75.00 in a jurisdiction with 8.5\%$ sales tax.
Tax Amount: 75.00 ร 0.085 = \6.375 \approx \6.38Total Payable: 75.00 + 6.38 = \81.38$
Common Mistakes to Avoid
1Forgetting to divide by original value when computing change: Always divide the difference by the starting (old) value, not the new value.
2Adding consecutive percentages directly: A 10% increase followed by a 10% decrease does NOT return to the original value (100 ร 1.10 = 110; 110 ร 0.90 = 99, a net 1% loss).
3Confusing Percentage Points with Percent Change: Moving from 4% interest to 5% is a 1 percentage point increase, but a $25\%$ relative increase ( rac{5 - 4}{4} = 0.25).
