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(Video) Ratio and Rates

by Math Antics

This video by mathantics explains ratios and rates.

$\large{\text{Summary}}$

A ratio is a comparison or relationship of two numbers by division. It's similar to a fraction but you use the top and bottom number of the fraction to refer to two different things. You can use ratios to compare two quantities.

Suppose you went on a picnic and for every $2$ people you only bring $1$ sandwich. So, we can say the ratio of sandwiches to people is $1$ to $2$.

One way to express a ratio, so it doesn't get mixed up with a fraction, is this notation: $$1:2$$ In the case of our last example, this means $1$ sandwich for every $2$ people.

A common example of a ratio in real life is with computer or TV sizes. They call it the aspect ratio. Usually it's $16\,{:}\,9$. This means that $16\,{:}\,9$ is the ratio of the screens width to its height.

Another example of a ratio that we've dealt with often is slope of a line. For example, if a slope has a rise of $3$ and a run of $5$, the slope ratio is $3\,{:}\,5$.

Slopes can also be equivalent to each other, like fractions can. That means that $8\,{:}\,2$ is equal to the ratio $4\,{:}\,1$. Multiplying or dividing ratios is possible to get equivalent ratios.

A rate is a ratio that usually involves a unit of time. Let's look at an example of a ratio in words.

An example of a rate can be found in your car: $40\text{ miles per hour}$. This is the ratio $40\,{:}\,1$. It means you move $40\text{ miles}$ every $1\text{ hour}$.

A common practice with rates is to reduce them so the second number, the unit of time, is $1$. This makes comparing rates much easier. You'll use the same method you used when simplifying fractions.

For example, if you want to find which is larger: $90\text{ meters}$ per $6\text{ seconds}$ or $30\text{ meters}$ per $4\text{ seconds}$, you can find an equivalent rate for each at $1\text{ second}$.

Ratios can be written as:

$$\frac{90\text{ meters}}{6\text{ seconds}}{{\div\,6}\atop{\div\,6}}=\frac{15\text{ meters}}{1\text{ second}}$$ $$\frac{30\text{ meters}}{4\text{ seconds}} {{\div\,4}\atop{\div\,4}}=\frac{7.5\text{ meters}}{1\text{ second}}$$

Rates where the second number is reduced to $1$, are called unit rates.

Ratios

Ratios represent relationships between multiple quantities. Usually, you denote a ratio by using $:$ or $\frac{\phantom{00}}{}.$

Examples

$$\text{2 cats}:\text{5 dogs }$$ $$\frac{\text{60 minutes}}{\text{1 hour }}$$

A ratio is used to compare $2$ or more values. We often use “:” to denote a ratio.

For example, if there are $3$ boys and $5$ girls in a classroom, we could write the ratio as: $3:5$

For every 3 boys there are 5 girls.

We can also write this as a fraction: $\dfrac{3}{5}$

The total number of kids is $8$. Let’s try an example. The ratio of pencils to pens is $7:2$. If Jon has $35$ pencils, how many pens are there?

We know the ratio of pencils to pens is $7:2$ or $\dfrac{7}{2}$ if written as a fraction. Jon has $35$ pencils, and we want to find out the number of pens. We can denote the number of pens as $x$. So, the current ratio is $35:x$, or $\dfrac{35}{x}$ as a fraction.

Since these two ratios have to be equal, we can set them equal to each other and solve for the unknown variable. $$\begin{align} \dfrac{7}{2} &= \dfrac{35}{x} \\ \dfrac {7}{2} \cdot x &= \dfrac{35}{x} \cdot x \\ \dfrac{7x}{2} &= 35 \\ \dfrac{7x}{2} \cdot 2 &= 35 \cdot 2 \\ 7x &= 70 \\ x &= 10 \\ \end{align}$$ Jon has $10$ pens.