什么是比例？

Proportions

(Here is a refresher on ratios if you need it!)

Examples of Proportions

$$\begin{align} \dfrac{w}{x}&=\dfrac{y}{z}\\ \\ \dfrac{1}{2}&=\dfrac{2}{4}\\ \\ \dfrac{32}{24}&=\dfrac{4}{3}\\ \end{align}$$

Proportions can include numbers or variables, or even both. We are really just setting two ratios equal to each other. We can use proportions to help with conversions between the same or different measurement systems, such as converting hours to minutes, minutes to days, minutes to seconds, feet to meters, kilometers to miles, Celsius to Fahrenheit, meters to centimeters, and grams to kilograms.

(Video) Proportions

by mathman1024

This video by mathman1024 explains proportions.

$\large{\text{Summary}}$

A proportion is a pair of two equal fractions. This can be seen when looking at equivalent fractions. $$\frac45=\frac8{10}$$

In proportions, the cross products are equal. This means if you take the product of the numbers diagonal to each other, they will be the same number. In other words,

$$\frac{\color{#8925AE}{a}}{\color{#29A603}{b}}=\frac{\color{#29A603}{c}}{\color{#8925AE}{d}}\Rightarrow {\color{#8925AE}{a}}\times {\color{#8925AE}{d}}={\color{#29A603}{b}}\times {\color{#29A603}{c}}$$

In our example above, if we take the cross products we multiply the pairs of positive numbers:

$$4\cdot10=5\cdot8.$$

This simplifies to $40=40$. This tells us that we had a proportion, since the cross products are equal. What we have is a quick way to check whether two fractions are equivalent.

$\large{\text{An Example in Solving Equations}}$

We can use this formula to help solve for unknown numbers or quantities. For example, say you can grade $5$ exams in $10$ minutes. How long would it take to grade $20$ exams?

Let's set up a proportion as follows: $$\begin{align} \frac{5\text{ exams}}{10\text{ minutes}}&=\frac{20\text{ exams}}{x\text{ minutes}}\\\ \frac5{10}&=\frac{20}x \end{align}$$

Now that we have a proportion, we can use the cross product to easily solve for $x$. This first step is also called cross-multiplying.

$$\begin{align} \frac5{10}&=\frac{20}x\\ 5x&=10\cdot20\\ 5x&=200\\ \frac{5x}{\color{#8925AE}{5}}&=\frac{200}{\color{#8925AE}{5}}\\ x&=40 \end{align}$$

In the last step we are dividing both sides of the equation by $5$.

In summary, it would take $40$ minutes to grade $20$ exams.

Here's a sanity check. Based on our calculation, it takes less time to grade $5$ exams than it did to grade $20$ exams.

What are Ratios and Proportions?

A ratio is one thing compared to or related to another thing; it is just a statement or an expression.

A proportion is two ratios that have been set equal to each other, which means that proportions can be solved. For instance: $$\frac{5}{10} = \frac12$$

This is a proportion because these two fractions are equivalent. In this particular case, $\frac{5}{10}$ can be reduced to $\frac{1}{2}$.

Solving Proportions

Solving a proportion means that you are missing one part of one of the fractions, and you need to solve for that missing value. For example: $$\frac12 = \frac{x}{6}$$

These can be solved with cross multiplication. We're going to multiply the numerator of the first fraction by the denominator of the second, and the denominator of the first fraction by the numerator of the second. From there we solve for $x$ the same way we would in any other equation. $$\begin{align} \frac12 &= \frac{x}{6}\\ 6\cdot 1 &= 2x\\ 2x&=6\\ x&=3 \end{align}$$