# The sum of two numbers is 72, and twice their difference is 24. What is the smaller of the two numbers?

##### 2 Answers

The smaller number is

#### Explanation:

Let's call the two numbers

The first sentence translates to

Since the difference is positive and

From this equation we can deduce

Substitute this expression for

We rearrange this equation as

and thus, dividing by

We wouldn't need

The smaller of the two numbers is 30.

#### Explanation:

To solve for one unknown, you need one equation. To solve for two unknown values, you need to set up two equations. To solve for three unknowns, you need three equations. In this case we have two unknowns.

First, you need to translate the given information into algebraic equations.

Let's call the unknown numbers

We are told that the sum of the two numbers is 72 so

**(1)**

That's one equation. We just need one more.

Twice their difference is 24.

Start with their difference

Twice this value is 24, so

Simplify this by dividing both sides by 2

**(2)**

Now that we have two the equations, you can use various methods such as substitution or elimination to solve. I will go with substitution.

From **(2)** we can rearrange it to get an expression of

Now substitute this expression into **(1)**

Simplify this and solve for

Now substitute this value back in to either **(1)** or **(2)** to solve for

As a final check you should test that the sum of the two numbers is 72 and that twice their difference is 24.