什么是比率？

What Is a Ratio?

A ratio is a statement comparing $2$ (or more) values. In other words, a ratio tells us how much of one thing we have compared to how much we have of another thing.

We have $9$ students and $1$ teacher. We can write this as a ratio:

This can also be read as "For every $9$ students, there is $1$ teacher."

Let's take a look at the same picture, but ask a different question.

The reason why I say "at least" in the previous question is because ratios can be reduced just like fractions. So, a ratio of $26:4$ can be reduced to $13:2$ (by dividing both sides of the ratio by $2$).

Ratios must have the same units, unless specified otherwise. Here is another example. What is the ratio of $2 \text{ feet}$ to $20 \text{ inches}$?

We want a ratio like this to have the same units. So we first convert $2 \text{ feet}$ into inches. We know there are $12 \text{ inches}$ in one foot. So, since we have $2 \text{ feet}$, there are $24 \text{ inches}$ to $20 \text{ inches}$. We simplify that by dividing each number by $4$. We see the reduced ratio is $6:5$.

Ratios

What is a ratio?

Examples of ratios: $$\frac{5}{4}$$ $$9 \text{ cows } : 17 \text{ sheep}$$ $$x \text{ oranges} : y \text{ apples}$$

You can simplify ratios the same way you simplify fractions! $$\dfrac{2 \text{ lions}}{4 \text{ tigers}} = \dfrac{1 \text{ lion}}{2 \text{ tigers}}$$ $$8:26 = 4:13$$ $$15w : 10z = 3w :2z$$

(Video) Ratio Word Problems

by Math Meeting

This video by the YouTube channel Math Meeting has some example ratio word problems.

$\large{\text{Summary}}$

The example the video looks at is:

In a class there are $10$ males and $15$ females. What is the ratio of males to females?

The ratio is a way to compare the number of males to the number of females, or vice versa.

We start by writing down the number of males that are in the classroom. This is given to be $10$. $$10$$ Next, we but a colon: $$10:$$ Finally, write down the number of females. This is given to be $15$. $$10:15$$ We have our ratio now. Said, "Ten to fifteen." This means that for every ten males, there are $15$ females.

However, this ratio is not written in the simplest form. Both of these numbers can be reduced. They can be divided by $5$. Doing this give us: $$2:3$$ So, for every two males, there are three females.

The order matters when writing the ratio. If we asked to find the ratio of females to males, then it would be flipped. $$3:2$$

It's a mistake to think these ratios are equal. The ratio $2:3$ is not equivalent to the ratio $3:2$ because

$$\frac{2}{3} \neq \frac{3}{2}$$

Flipping a fraction certainly changes the value of the fraction. Ratios are no different in this regard.

You may also see ratios used to convert between different measurement systems, such as converting from meters to feet.