相乘性

Example and Comparison: Multiplication & Cross Product

The two methods are similar. In my opinion, cross product is a short cut or condensation of the multiplication property. Let's look at an example. $$\frac{2}{3}=\frac{x}{9}$$

Remember our goal is to isolate $x$.

First, Using Multiplication

Then, Using Cross Product

Here is how the two methods are essentially the same.

Try the following problem!

A different way to solve proportions is the multiplication property.

Say you want to find $x$ in the following proportion: $$\frac{a}{b}=\frac{x}{c}$$ You could cross multiply and solve, or you can just multiply both sides by $c$.

With the multiplication property, you multiply both sides by the reciprocal of the number on the side with the variable.

Example

$$\frac{3}{5}=\frac{3x}{10}$$

In order to get the $x$ by itself, you can multiply both sides by the reciprocal of $\frac{3}{10}$ which is $\frac{10}{3}$. $$\begin{align} \frac{10}{3}\times \frac{3}{5}&=\frac{3x}{10}\times \frac{10}{3}\\ \frac{30}{15}&=\frac{30x}{30}\\ 2&=x\\ \end{align}$$

Using Cross Multiplication

Now let's try this with cross multiplication so you can see that the answers are the same. $$\begin{align} \frac{3}{5}&=\frac{3x}{10}\\ (3)(10)&=(5)(3x)\\ 30&=15x\\ \frac{30}{15}&=\frac{15x}{15}\\ 2&=x \end{align}$$

You can take this a step further by evaluating word problems with ratios, which may require you to convert between different forms of measurement.

(Video) Algebra 1 2-7 Solving Proportions: Problem 1 - Solving Proportions Using the Multiplication Property

by Matthew Richardson

This video by Matthew Richardson covers proportions and how to solve them. $$\frac{a}{b}=\frac{c}{d}$$

$\large{\text{Summary}}$

Example:

Solve a proportion using the multiplication property. Multiply each side of $\frac{7}{8}=\frac{m}{12}$ by $12$. That means considering both the numerator and the denominator as one number. We see,

$$\begin{align} \frac{7}{8}&=\frac{m}{12}\\ 12 \times \frac{7}{8}&=12 \times\frac{m}{12}\\ \frac{84}{8}&=m\\ 10.5&=m \end{align}$$

Now, what is the answer to the last problem in the video?

Multiplying both sides by $7$ will isolate $x$. $$\begin{align} \frac{x}{7} &= \frac45\\ \frac{x}{\cancel{7}} \cdot \frac{\cancel{7}}{1} &= \frac45 \cdot \frac71\\ x&=\frac{28}{5} \end{align}$$

Always check your work! You can plug the solved value back into the original equation.