# If 4 is added to twice a number and this sum is multiplied by 2, the result is the same as if the...

## Question:

If 4 is added to twice a number and this sum is multiplied by 2, the result is the same as if the number is multiplied by 3 and 4 is added to the product. What is the number?

## Sentence to Equation:

We can translate a sentence into an equation by using algebraic operations. Here, we use + for "more than", - for "less than, multiplication for "times" etc. The equation can be solved using the techniques of algebra.

Let us assume the number to be {eq}x {/eq}.

"{eq}4 {/eq} is added to twice a number and this sum is multiplied by {eq}2 {/eq}" can be written as the expression {eq}2(2x+4) {/eq}.

"the number is multiplied by {eq}3 {/eq} and {eq}4 {/eq} is added to the product" can be written as {eq}3x+4 {/eq}.

The problem says the results of the above two expressions are the same.

So we set them equal and solve for the variable:

$$2(2x+4) =3x+4 \\[0.4cm] \text{Distributing 2 on the left side},\\[0.4cm] 4x+8=3x+4 \\[0.4cm] \text{Subtracting 3x from both sides},\\[0.4cm] x+8= 4 \\[0.4cm] \text{Subtracting 8 from both sides},\\[0.4cm] x= \color{blue}{\boxed{\mathbf{-4}}}$$ 