Percent & Mixture Problems

Percent of a number, percent change, simple interest, and mixtures.

The idea

Definition (Percent).

A percent is a rate written per hundred: $r\%$ of a quantity means $\frac{r}{100}$ of it. Every percent computation applies a rate to a base: $\text{amount} = \text{rate} \times \text{base}.$

The work in a percent problem is choosing the base. For a discount or a tip, the base is the listed price. For a percent change, the base is the original value, never the new one: divide the change by the original and multiply by $100$. For simple interest, the base is the principal $P$; at annual rate $r$ for $t$ years the interest is $I = Prt$, where $r$ must be written as a decimal, so a rate of $8\%$ enters the formula as $0.08$.

A mixture problem balances amounts of pure substance. A container holds concentration times volume of the pure substance, and mixing neither creates nor destroys it, so the pure amounts poured in add up to the pure amount in the result. That equality is the equation to solve.

Ways to work on it

• Walkthrough. Percent as a rate: discount, percent change, and simple interest.
• Practice. One randomized percent application.
• Hardest. Set up and solve a mixture/concentration balance.

Not sure where to start? Take the ten-question placement test.