In this video, we will be learning how to solve complex proportions (variables on both sides) using cross-multiplication. After you finish this lesson, view all of our Algebra 1 lessons and practice problems.

## Example of Solving a Complex Proportion

$\dfrac{x}{20}=\dfrac{(x+1)}{30}\leftarrow$ First we cross-multiply

$30x=20(x+1)\leftarrow$ Then we distribute the 20 to the x and 1

$30x-20x=20x +20-20x\leftarrow$ Subtract 20x from both sides to isolate x

$\dfrac{10x}{10}=\dfrac{20}{10}\leftarrow$ Divide by 10 on both sides

$x=2$

### Example 1

$\dfrac{3x}{10}=\dfrac{2(x+5)}{15}$

First, distribute $2$ to each of the terms inside the parenthesis

$\dfrac{3x}{10}=\dfrac{2x+10}{15}$

Then, cross multiply

$15(3x)=10(2x+10)$

Distribute $10$ to each of the terms inside the parenthesis

$45x=20x+100$

Subtract $20x$ from both sides to isolate $x$

$45x-20x=20x+100-20x$ $25x=100$

Divide by $25$ on both sides

$\dfrac{25x}{25}=\dfrac{100}{25}$

Therefore,

$x=4$

### Example 2

$\dfrac{3x+6}{5}=\dfrac{3(x+4)}{6}$

First, distribute $3$ to each of the terms inside the parenthesis

$\dfrac{3x+6}{5}=\dfrac{3x+12}{6}$

Then, cross multiply

$(6)(3x+6)=5(3x+12)$

Distribute constants to each of the terms inside the parenthesis

$18x+36=15x+60$

Subtract $15x$ from both sides

$18x+36-15x=15x+60-15x$ $3x+36=60$

Subtract $36 from both sides to iolate [latex]x$

$3x+36-36=50-26$ $3x=24$

Divide $3$ on both sides

$\dfrac{3x}{3}=\dfrac{24}{3}$

Therefore,

$x=8$

## Video-Lesson Transcript

Let’s go over complex proportions.

For example:

$\dfrac{x}{20} = \dfrac{x + 1}{30}$

What makes this complex is that we have one variable on both sides of the equation.

We’re just going to do the same method which is to cross-multiply.

$x \times30 = 30x$

To solve the equation on the right, we have to distribute $20$ on each of term.

${x + 1} \times20$ $20x + 20$

Now we have

$30x = 20x + 20$

Let’s get all the variables on one side by subtracting $20x$ on both sides.

$30x - 20x = 20x - 20x + 20$ $10x = 20$

Simply solve $x$ by dividing both sides by $10$.

$\dfrac{10x}{10} = \dfrac{20}{10}$

And we’ll have

$x = 2$

Complex proportion is different from regular proportion because it has a variable on both sides of the equation.