Mathematics from zero
Percents
A sign says 50% off. A phone shows 80% battery. A test comes back 90%. Three
different things, one shared scale — and the scale is always out of 100. That is what
a percent is.
After this lesson you can say what a percent is, convert between percents, fractions, and decimals, find a percent of a number, and explain why 100% means the whole.
A percent is a fraction whose denominator is always 100. The word percent and
the sign % both mean “per hundred”. 50% is just 50/100. 7% is 7/100. The
bottom number never changes — it is fixed at 100 — so a percent only ever has to
report its top number.
That fixed denominator is what makes percents comparable. Two fractions like
9/20 and 13/25 are hard to compare at a glance — different denominators. As
percents, 45% and 52%, the comparison is instant, because both are measured
against the same 100. A percent is a common scale everyone agrees on.
A percent, a fraction, and a decimal are three names for one amount. 50% is
50/100, which simplifies to 1/2, which is 0.5. To go from a percent to a decimal,
divide by 100 (move the point two places left): 25% = 0.25. To go from a decimal to a
percent, multiply by 100: 0.6 = 60%.
To find a percent of a number, multiply by the percent written as a fraction or
decimal. “20% of 50” means 20 hundredths of 50: 20/100 × 50 = 10. The word of
here means multiply. And 100% is 100/100 = 1, the whole thing — so 100% of any
number is the number itself, and more than 100% is more than the whole.
Find 25% of 80.
First rewrite the percent. 25% is 25/100, which is the decimal 0.25 — or the
simpler fraction 1/4.
“Of” means multiply, so 25% of 80 is 0.25 × 80. Using the fraction is easy here:
1/4 of 80 is 80 ÷ 4 = 20.
So 25% of 80 = 20. Check the size: 25% is one quarter, and a quarter of 80 should be
clearly less than half of 80 (which is 40). 20 fits.
Why this works
Why fix the denominator at 100 — why not 10, or 1000? Because 100 is a practical balance. With a denominator of 10 you could only report whole tenths — too coarse for a battery or a test score. With 1000 the numbers get long. 100 is fine-grained enough for everyday parts and small enough to keep the numbers short. It is a convention, and its whole value is that everyone shares it.
Common mistake
A common mistake is treating the percent sign as decoration and computing with the
bare number — finding “20% of 50” as 20 × 50 = 1000. The % is not decoration: it
means divide by 100. Always convert the percent to a fraction over 100 or a decimal
before multiplying.
Write 50% as a fraction over 100. Type the numerator.
Find 10% of 200.
Write the fraction 1/4 as a percent. Type the number (without the % sign).
Write the decimal 0.6 as a percent. Type the number.
Find 100% of 45.
What does the % sign actually mean?
A percent is a fraction whose denominator is always 100 — the sign % means “per hundred”. That fixed denominator makes any two parts directly comparable. A percent, a fraction, and a decimal are three names for one amount: divide a percent by 100 for the decimal, multiply a decimal by 100 for the percent. To find a percent of a number, rewrite the percent as a fraction or decimal and multiply. 100% is the whole.