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Percent describes a quantity out of one hundred and connects fractions, decimals, rates, discounts, taxes, interest, and data.
A percent is a ratio whose second quantity is $100$. The symbol $\%$ means “per hundred.”
The central idea
Percent gives different quantities a common comparison scale. Saying that $18$ of $25$ students submitted an assignment is useful; saying that $72\%$ submitted makes the result easier to compare with another class of a different size.
The three forms below describe the same quantity:
The form changes, but the value does not.
Moving between forms
To turn a decimal into a percent, multiply by $100$ and attach the percent sign. To turn a percent into a decimal, divide by $100$ and remove the sign.
| Start | Operation | Result |
|---|---|---|
| $0.42$ | multiply by $100$ | $42\%$ |
| $7\%$ | divide by $100$ | $0.07$ |
| $3/5$ | divide, then convert | $0.6=60\%$ |
Moving the decimal point is a shortcut for multiplying or dividing by $100$; it is not a separate rule.
Finding a percent of a quantity
Translate “of” as multiplication and write the percent as a decimal:
Finding the percent
When the part and whole are known, divide the part by the whole:
If $18$ of $24$ questions are correct, then $18/24=0.75$, so the score is $75\%$.
Finding the original whole
If the part and percent are known, rearrange the relationship:
If $27$ students represent $90\%$ of a class, the class size is $27/0.90=30$ students. The whole must be at least as large as the part when the percent is at most $100\%$.
Percent increase and decrease
A percent change compares the change with the original amount:
For a $15\%$ discount, multiply the original price by $1-0.15=0.85$. For a $13\%$ tax, multiply by $1+0.13=1.13$. These multipliers are efficient because they combine the original amount and the change in one step.
Common mistakes
Using the new value as the denominator. Percent change is measured relative to the original value.
Treating $8\%$ as $0.8$. Eight percent is $8/100=0.08$.
Adding successive percentages. A $20\%$ decrease followed by a $20\%$ increase does not return to the start because the second percent uses a different whole. The combined multiplier is $0.8(1.2)=0.96$.
Ignoring reasonableness. If $25\%$ of $60$ is reported as $150$, the result is larger than the whole even though the percent is below $100\%$.
Quick self-check
- Is the percent written as a decimal before multiplication?
- Did you identify the part and the original whole?
- Should the answer be larger or smaller than the starting quantity?
- Can you estimate with $10\%$, $25\%$, or $50\%$?
Related topics
Explore the idea
Number model
Change one quantity at a time and connect what moves to Percent.
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
Find 35% of 80.
- 35% = 0.35
- 0.35 × 80 = 28
End of lesson
Nice work making it this far.
Understanding grows through return visits. Save this lesson, try the practice, or continue when you are ready.
