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.