Percents

The word percent comes from the Latin per centum, meaning “per hundred.” A percent is just a special fraction whose denominator is always $100$. So $35\%$ means $\dfrac{35}{100}$, or “$35$ out of every $100$.”

You can picture a percent as part of a $10\times 10$ grid of $100$ little squares: $35\%$ shades $35$ of the $100$ squares. The whole grid is $100\% = \dfrac{100}{100} = 1$.

These three forms all describe the same amount. To move between them:

• Percent $\to$ decimal: drop the $\%$ and divide by $100$ (move the point two places left).
• Decimal $\to$ percent: multiply by $100$ (move the point two places right) and add $\%$.
• Percent $\to$ fraction: write it over $100$, then simplify.
• Fraction $\to$ percent: divide top by bottom to get a decimal, then multiply by $100$; or rewrite the fraction with denominator $100$.

To find a percent of a number, change the percent to a decimal (or fraction) and multiply. In math, the word “of” almost always means multiply:

To find what percent one number is of another, make a fraction $\dfrac{\text{part}}{\text{whole}}$, then convert it to a percent (divide, then multiply by $100$). This uses the percent proportion:

Sometimes you know the part and the percent, and you must find the whole. Rearrange the percent proportion, or divide the part by the percent (as a decimal):

Percent change tells how much a quantity grew or shrank, compared to where it started:

If the new value is bigger, it is a percent increase; if smaller, a percent decrease. Always divide by the original amount.

Everyday money problems are percent-of-a-number problems in disguise:

• Discount / sale price: discount $=$ percent off $\times$ original price; sale price $=$ original $-$ discount.
• Sales tax: tax $=$ tax rate $\times$ price; total $=$ price $+$ tax.
• Tip: tip $=$ tip rate $\times$ bill.
• Commission: commission $=$ commission rate $\times$ sales.

When money is saved or borrowed, interest is a percent paid for the use of that money. Simple interest is found with:

where $P$ is the principal (starting amount), $r$ is the yearly interest rate (as a decimal), and $t$ is the time in years. The total amount you end with is $A = P + I$.

When one percent change follows another, you multiply their factors; you do not add the percents. A change of $p\%$ turns a quantity into a fraction of itself called the multiplier:

Two changes in a row multiply their multipliers. For example, $+50\%$ then $-50\%$ gives $1.50 \times 0.50 = 0.75$, a net decrease of $25\%$, not a return to the start. The order does not matter, since multiplication commutes.

A percentage point is an absolute gap between two percents; a percent change is relative to the starting percent. These are different! If an interest rate rises from $4\%$ to $6\%$, it went up $2$ percentage points, but the relative increase is $\dfrac{6-4}{4}\times 100\% = 50\%$. Newspapers and ads often blur this, so read carefully: “$2$ points” and “$50\%$ higher” can describe the very same change.

If a price already includes a markup or discount, you cannot just take that percent off the final number to get back. Instead, divide by the multiplier. If a final amount $F$ came from an original $X$ after a $p\%$ increase, then $F = X\left(1 + \tfrac{p}{100}\right)$, so

For a discount, divide by $\left(1 - \tfrac{p}{100}\right)$ instead. The key idea: the percent was taken of the original, which is the unknown, so you must undo the multiplication.

Simple interest ($I = Prt$) is always figured on the original principal. Compound interest instead earns interest on the interest already added, so the balance grows by repeated multiplication. If interest compounds $n$ times per year at yearly rate $r$ (a decimal) for $t$ years, the ending amount is

Each period multiplies the balance by $\left(1 + \tfrac{r}{n}\right)$, and there are $nt$ periods in all. For the same rate and time, compounding always beats simple interest, because later periods earn on a larger balance. (You will study this fully in Algebra; here just notice it is repeated percent increase.)

In science, percent error compares a measured (or estimated) value to the true (accepted) value, relative to that true value:

The absolute value bars keep the answer positive — percent error measures size of the mistake, not its direction. It is structurally the same as percent change, but the denominator is always the true value.

A concentration is a percent: the amount of pure substance divided by the total amount of mixture. The pure amount is

When two solutions are combined, the pure amounts add, and so do the volumes; the new concentration is the total pure amount over the total volume. Concentrations themselves do not simply average unless the two volumes happen to be equal.