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Lesson 6: Percent word problems

# Rates and percentages FAQ

## What is a rate?

A rate is a comparison of two quantities that have different units. For example, if we say that a bus travels $240$ kilometers in $15$ hours, we are comparing kilometers and hours. We can write this rate as or $240$ kilometers per $15$ hours. Rates tell us how fast, how often, or how much something happens.

## What is a unit rate?

A unit rate is a special kind of rate that compares a quantity to one unit of another quantity. For example, if we want to know how many kilometers the bus travels in one hour, we can divide the total distance by $15$. We get . This is a unit rate, and we can write it as $16$ kilometers per hour. Unit rates tell us the amount of one quantity for every one unit of another quantity.

## What is a percent?

A percent is a way of expressing a part of a whole as a fraction out of $100$. For example, if $2$ salmon survive to adulthood out of $100$ salmon who hatched, we can write this as $\frac{2}{100}$ or $2\mathrm{%}$. The symbol $\mathrm{%}$ means "out of $100$" or "per $100$". Percents tell us how much of something is in a group or a whole.

## How can we visualize percentages?

We can use different models to help us see what percentages mean. One common model is a $100$-grid, which is a square divided into $100$ smaller squares. Each small square represents $1$. For example, if we shade $25$ squares out of $100$, we can see that this is the same as $25\mathrm{%}$ of the whole square.
Another common model is a circle graph or a pie chart, which is a circle divided into parts that show the percentages of a whole. Each part of the circle represents a percentage of the whole circle. For example, if we cut a circle into $4$ equal parts, each part is $25\mathrm{%}$ of the circle.

## How can we find equivalent representations of percent problems?

Sometimes, we can use different ways to represent the same percent problem. For example, if we want to find $30\mathrm{%}$ of $50$, we can use a fraction, a decimal, or a ratio. Here are some equivalent representations of $30\mathrm{%}$ of $50$:
$\begin{array}{rl}30\mathrm{%}×50& =\frac{30}{100}×50\\ \\ & =15\\ \\ 30\mathrm{%}×50& =\frac{3}{10}×50\\ \\ & =15\\ \\ 30\mathrm{%}×50& =0.3×50\\ \\ & =15\end{array}$
We can also use the formula $\text{part}=\text{percent}×\text{whole}$ to find any missing value in a percent problem. For example, if we know that $15$ is $30\mathrm{%}$ of some number, we can use the formula to find the number:
$\begin{array}{rl}15& =30\mathrm{%}×\text{number}\\ \\ 15& =0.3×\text{number}\\ \\ 15÷0.3& =\text{number}& \text{Rewrite as division equation.}\\ \\ 50& =\text{number}\end{array}$

## Where are rates, unit rates, and percents used in the real world?

Rates, unit rates, and percents are used in many situations in the real world, such as:
• Comparing prices of different items or services that have different units, such as dollars per pound, miles per gallon, or minutes per call. Unit rates help us find the best deal or the most efficient option.
• Calculating tips, taxes, discounts, interest, or commissions that are based on percentages of amounts. Percents help us find how much we pay or earn in different situations.
• Analyzing data, statistics, or surveys that show the percentages of different categories, groups, or outcomes. Percents help us understand and compare the information in different ways.