# Proportions

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

For example, `5/3=10/6` is a proportion.

**Example 1**. Determine, whether the ratios `18/4` and `27/6` form a proportion.

Let's reduce first ratio: `18/4=(9*color(red)(2))/(2*color(red)(2))=9/2`.

Now, multiple numerator and denominator of the reduced fraction by 3: `9/2=(9*color(red)(3))/(2*color(red)(3))=27/6`.

This means, that `18/4=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.

`huge (color(green)(a))/(color(blue)(b))=(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 `18/4` and `27/6` form a proportion, because `18*6=4*27`.

**Example 2**. Determine, whether the ratios `0.7/2.3` and `1.2/3.5` form a proportion.

First, write a possible equality: `0.7/2.3?=?1.2/3.5`

Now, calculate cross product: `0.7*3.5?=?2.3*1.2`

Simplify: `2.45!=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 `x/8=4/10`.

Find the cross product: `x*10=8*4`.

Simplify: `10x=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: `212/4=954/n`

Find the cross product: `212*n=4*954`.

Simplify: `212n=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 1/2` feet on the actual car. If the model is `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:

`2/(12 1/2)=(2/3)/h`

Cross-multiple: `2*h=12 1/2*2/3`

Simplify: `2h=25/3`

Solution to this linear equation is `h=25/6`.

Thus, actual height of the car is `25/6` feet.

Now, it is time to exercise.

**Exercise 1**. Determine, whether `7/20` and `35/100` form a proportion.

**Answer**: yes.

**Exercise 2**. Determine, whether `3.5/0.7` and `7.35/1.47` form a proportion.

**Answer**: yes.

**Exercise 3**. Solve the following proportion: `5/m=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**: `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.