叉积性

(Video) Proportions

by Math Antics

Here's a video by mathantics that explains proportions.

$\large{\text{Summary}}$

A proportion is two ratios that are equivalent. For this to be true, the two ratios need to have the same value and units.

We use proportions to figure out something unknown from something known. So if you know how many seconds are in a minute then you can figure out how many seconds are in an hour since you know and hour is $60\text{ minutes}$.

You would write this as: $$\frac{60\text{ seconds}}{1\text{ minute}}=\frac{x\text{ seconds}}{60\text{ minutes}}$$ where $x$ represents the number we are trying to find.

To solve this, we can use a procedure called cross multiplying. With cross multiplying, you take the numerator of the first ratio and multiply it by the denominator of the second ratio. Then, you set that equal to the denominator of the first ratio times the numerator of the second ration.

In other words: $$\frac{a}{b}=\frac{c}{d}\quad\rightarrow\quad a\times d=b\times c$$

Using the above proportion as an example, we can solve for $x$ by cross multiplying. $$\begin{align} \frac{60\text{ seconds}}{1\text{ minute}}&=\frac{x\text{ seconds}}{60\text{ minutes}}\\ 60\times60&=1\times x\\ 3{,}600&=x \end{align}$$ We now know there are $3{,}600\text{ seconds}$ in an hour.

To check your work, you can insert the solved answer back into the original ratio, and you should get equivalent fractions.

Example:

A student wants to read $23\text{ books}$. He wants to know how many days it will take him to read all those books. We know he can read $1\text{ book}$ in $2\text{ days}$. Therefore, we can set up a proportion. $$\begin{align} \frac{1 \ \text{ book}}{2 \ \text{ days}}&=\frac{23 \ \text{ books}}{n \ \text{ days}}\\ 1\times n&=2\times23\\ n&=46 \end{align}$$ Therefore, the student can read $23\text{ books}$ in $46\text{ days}$.

For more word problem examples, check out this lesson!

Cross Product Property

The cross product property is also called cross multiplication. It's an approach often used to solve equations involving ratios. It's very helpful in converting between with different units, and even different measurement systems!

To cross multiply, we will be multiplying values across the equals sign, in an X shape. Time to try out some problems!

To solve this problem, start by drawing an imaginary X through the equals sign:

Now, we'll cross multiply the two values connected by the purple arrow. That will go on one side of the equals sign.

Then, we can cross multiply the two values connected by the green arrow. That will go on the other side of the equals sign.

Let's rewrite our equation:

(Video) How to Solve Proportions Using Cross Products

by eHow

This quick video by eHow shows you how to cross multiply. Cross multiplication is done when you have two fractions on opposite sides of an equal sign.

$\large{\text{Summary}}$

Suppose we have the following equation. $$\frac{5}{12}=\frac x7$$

To use cross multiplication, take the first numerator, $5$, and multiply it by the term diagonally across the equal sign (the other fraction's denominator), $7$.

Then, we set this equal to the product of the first denominator, $12$, multiplied by the term diagonally across the equal sign (the other fraction's numerator), $x$. $$\frac{\color{#8925AE}{5}}{\color{#29A603}{12}}=\frac{\color{#29A603}{x}}{\color{#8925AE}{7}}\quad\rightarrow\quad {\color{#8925AE}{5}}\times{\color{#8925AE}{7}}={\color{#29A603}{12}}\times \color{#29A603}{x}$$ This gives us a much easier equation to work with. We can solve for $x$. $$\begin{align} 5\times7&=12\times x\\ 35&=12x\\ \frac{35}{12}&=x \end{align}$$

We can start by cross multiplying the proportion. This gives us an equation we can solve for $x$. $$\begin{align} \frac{\color{#8925AE}{3}}{\color{#29A603}{8}}&=\frac{\color{#29A603}{x}}{\color{#8925AE}{16}}\\ \color{#29A603}{8x} &= \color{#8925AE}{48}\\ x&=6 \end{align}$$

The Cross Product Property

The Cross Product Property is one of the approaches to solving proportions. Let's look at what we can do with this property.

This setup is especially helpful when you need to solve for $x$.

The solution: $$\frac34 = \frac{x}{12}$$ $$3*12=4*x$$ $$36=4x$$ $$9=x$$