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

Jun 26, 2018

The smaller number is $30$

#### Explanation:

Let's call the two numbers $x$ and $y$, and assume $x$ is the largest.

The first sentence translates to $x + y = 72$.

Since the difference is positive and $x > y$, the second equation translates to $2 \left(x - y\right) = 24$, which can be rewritten as $x - y = 12$

From this equation we can deduce $x = y + 12$

Substitute this expression for $x$ in the first equation to get

$\setminus \textcolor{red}{x} + y = 72 \setminus \to \setminus \textcolor{red}{y + 12} + y = 72$

We rearrange this equation as

$2 y = 60$

and thus, dividing by $2$,

$y = 30$.

We wouldn't need $x$, but for completeness sake: since we knew that $x = \setminus \textcolor{red}{y} + 12 = \textcolor{red}{30} + 12 = 42$

Jun 26, 2018

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 $x$ and $y$.

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

$x + y = 72$ (1)

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

Twice their difference is 24.

$x - y$

Twice this value is 24, so

$2 \cdot \left(x - y\right) = 24$

Simplify this by dividing both sides by 2

$x - y = 12$ (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 $x$ in terms of $y$

$x - y = 12$

$\Rightarrow \textcolor{b l u e}{x = y + 12}$

Now substitute this expression into (1)

$\textcolor{b l u e}{x} + y = 72$

$\Rightarrow \textcolor{b l u e}{y + 12} + y = 72$

Simplify this and solve for $y$

$2 y + 12 = 72$

$\Rightarrow 2 y = 60$

$\Rightarrow y = 30$

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

$x + y = 72$

$\Rightarrow x + 30 = 72$

$\Rightarrow x = 72 - 30 = 42$

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