## Definition

**Word problems** do not come with equations. To solve a word problem, it is often helpful to rewrite the problem using mathematical notation.

## Technique

Here are some strategies for approaching a word problem:

- Always start off by reading the entire problem.
- List information you know and assign variables to anything you don't know.
- Identify what the problem is asking for.
- Look for key words to identify which operations you will need to use. Here are some examples:

\[\begin{equation} \begin{array}{ll} \textbf{Addition: } & \text{more than, increased by, total of, sum} \\ \textbf{Subtraction: } & \text{less than, subtract, difference } \\ \textbf{Multiplication: } & \text{times, product of, multiplied by} \\ \textbf{Division: } & \text{per, quotient of, percent, ratio of} \\ \end{array} \end{equation}\]

(Please note that these examples do not represent a complete list of key words.)

Now you have the basic tools to begin constructing equations from word problems. Take care to make your work clear and organized throughout your solution.

## Application and Extensions

## I want to make an apple-orange fruit salad for a picnic. My recipe calls for 2 apples for every 3 oranges. If I want to use 6 apples, what is the total number of fruit I need for the salad?

"2 apples for every 3 oranges" tells us that for every apple there needs to be \(\frac{3}{2}\) oranges. I'm using \(6\) apples, so I will need \(6 \times \frac{3}{2} = 9\) oranges. The problem asks for the total number of fruit, which would be the number of apples plus the number of oranges: \(6+9=15\). \( _\square\)

## Today Beatrice is 3 times as old as Kelly. In 3 years she will be 4 times as old as Kelly was 1 year ago. How many years from today will Beatrice be twice as old as Kelly?

Let \(x\) be the number of years it takes from today for Beatrice to be twice as old as Kelly. Before we can find \(x\), however, we need to determine how old Kelly and Beatrice are right now. Let \(B\) and \(K\) be Beatrice's and Kelly's ages respectively. Now let's set up a couple equations.

"Today Beatrice is 3 times as old as Kelly" translates into:

\[B = 3K\]

"In 3 years she will be 4 times as old as Kelly was 1 year ago" translates into this equation:

\[B + 3 = 4(K-1)\]

Combining the two equations we get: \[(3K) + 3 = 4(K-1)\]

for which \(K=7\), which then means \(B=21\).Now that we know Beatrice's and Kelly's ages, we can find \(x\) to answer the problem's question "How many years from today will Beatrice be twice as old as Kelly?" This can also be rewritten into an equation, which we can then solve for \(x\): \[\begin{align}

B + x &= 2 (K+x)\\

21+ x &= 2 (7+x)\\

21 + x &= 14 + 2x\\

x &= 7 \quad_\square\

\end{align}\]

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