# Proportions

An equation, stating that two ratios are equal is called a **proportion.**

For example, $$$\frac{5}{3}=\frac{10}{6}$$$ is a proportion.

**Example 1.** Determine, whether the ratios $$$\frac{18}{4}$$$ and $$$\frac{27}{6}$$$ form a proportion.

Let's reduce first ratio: $$$\frac{{18}}{{4}}=\frac{{{9}\cdot{\color{red}{{{2}}}}}}{{{2}\cdot{\color{red}{{{2}}}}}}=\frac{{9}}{{2}}$$$.

Now, multiple numerator and denominator of the reduced fraction by 3: $$$\frac{{9}}{{2}}=\frac{{{9}\cdot{\color{red}{{{3}}}}}}{{{2}\cdot{\color{red}{{{3}}}}}}=\frac{{27}}{{6}}$$$.

This means, that $$$\frac{{18}}{{4}}=\frac{{27}}{{6}}$$$.

Thus, these two ratios form a proportion.

Another way to determine whether two ratios form a proportion is to** use cross products (cross multiplication).** If the cross products are equal, then the ratios form a proportion.

$$$\frac{\color{green}{a}}{\color{blue}{b}}=\frac{\color{blue}{c}}{\color{green}{d}}\ \text{if}\ \color{green}{ad}=\color{blue}{bc}$$$

$$$a$$$ and $$$d$$$ are called the **extremes**, and $$$b$$$ and $$$c$$$ are called the **means.**

In Example 1 $$$\frac{{18}}{{4}}$$$ and $$$\frac{{27}}{{6}}$$$ form a proportion, because $$${18}\cdot{6}={4}\cdot{27}$$$.

**Example 2.** Determine, whether the ratios $$$\frac{{0.7}}{{2.3}}$$$ and $$$\frac{{1.2}}{{3.5}}$$$ form a proportion.

First, write a possible equality: $$$\frac{{0.7}}{{2.3}}?=?\frac{{1.2}}{{3.5}}$$$

Now, calculate cross product: $$${0.7}\cdot{3.5}?=?{2.3}\cdot{1.2}$$$

Simplify: $$${2.45}\ne{2.76}$$$.

This means, that given two ratios don't form a proportion.

You can write proportions that involve a variable. To solve the proportion, use cross product.

**Example 3.** Solve $$$\frac{{x}}{{8}}=\frac{{4}}{{10}}$$$.

Find the cross product: $$${x}\cdot{10}={8}\cdot{4}$$$.

Simplify: $$${10}{x}={32}$$$.

Solve this linear equation: $$${\color{purple}{{{x}={3.2}}}}$$$.

Proportions are widely used for solving real-world problems.

**Example 4.** John earns $212 in 4 days. How many days will it take him to earn $954?

Let $$${n}$$$ represents number of days it takes him to earn $954.

We need to write proportion for this problem.

Since John works at the same rate, then 212 to 4 is the same as 954 to $$${n}$$$.

Write as proportion: $$$\frac{{212}}{{4}}=\frac{{954}}{{n}}$$$

Find the cross product: $$${212}\cdot{n}={4}\cdot{954}$$$.

Simplify: $$${212}{n}={3816}$$$.

Solve this linear equation: $$${\color{purple}{{{n}={18}}}}$$$.

Therefore, John will earn $954 in 18 days.

Proportions are also used for solving scaling problems. We already saw how to do that (see Ratios), but let's see how to do that, using proportions.

**Example 5.** A collector's model racecar is scaled so that $$${2}$$$ inches on the model equals $$${12}\frac{{1}}{{2}}$$$ feet on the actual car. If the model is $$$\frac{{2}}{{3}}$$$ inches high, how high is the actual car?

Let $$${h}$$$ represents height of the actual car (in feets).

Ratios of scale to actual should be equal:

$$$\frac{{2}}{{{12}\frac{{1}}{{2}}}}=\frac{{\frac{{2}}{{3}}}}{{h}}$$$

Cross-multiple: $$${2}\cdot{h}={12}\frac{{1}}{{2}}\cdot\frac{{2}}{{3}}$$$

Simplify: $$${2}{h}=\frac{{25}}{{3}}$$$

Solution to this linear equation is $$${h}=\frac{{25}}{{6}}$$$.

Thus, actual height of the car is $$$\frac{{25}}{{6}}$$$ feet.

Now, it is time to exercise.

**Exercise 1.** Determine, whether $$$\frac{{7}}{{20}}$$$ and $$$\frac{{35}}{{100}}$$$ form a proportion.

**Answer**: yes.

**Exercise 2.** Determine, whether $$$\frac{{3.5}}{{0.7}}$$$ and $$$\frac{{7.35}}{{1.47}}$$$ form a proportion.

**Answer**: yes.

**Exercise 3.** Solve the following proportion: $$$\frac{{5}}{{m}}=\frac{{3}}{{12}}$$$.

**Answer**: $$${m}={20}$$$.

**Exercise 4.** If Ann drove 58 miles in 5 hours, how many miles will she drive in the next 2 hours?

**Answer**: $$$\frac{{116}}{{5}}={23.2}$$$ miles.

**Exercise 5.** It appears that 4 out of 20 men have a pet. How many men are there, if 10 of them have pet?

**Answer**: There are 50 men.