Summarize word problems using equations

Last updated: 18 Nov 2008

In the question you are given a great deal of information and you need to be able to summarize it in a more manageable form.

Often it is possible to translate the question into equations. It is important to use variable names that will make sense to you when you are translating these questions into equations.

'Carl has twice as much money invested in stocks as in bonds. Stocks earn 10% interest per year and bonds 5% per year. If Carl earned a total of $800 dollars from his stocks and bonds last year how much money did he have invested in stocks?'

We will use 'S' to represent stocks and 'B' to represent bonds. Using the first letters of each word makes it easy to remember which is which and avoids any confusion that might arise from using more traditional variable names such as 'x' and 'y'.

'Carl has twice as much money invested in stocks as in bonds.'

This translates to

\[S = 2B\]

Note: many people get confused with the phrase 'twice as much' and write \(2S = B\).

This is a very common mistake and must be avoided.

If you find that you get confused writing the equation try replacing the variables with numbers and then read the sentence again to see if it makes sense.

For example in this case if \(S = 2B\), then if \(B = 1\), \(S = 2\). This makes sense because stocks are '2' which is twice as much as bonds which are '1'.

'Stocks earn 10% interest per year and bonds 5% per year. If Carl earned a total of $800 dollars from his stocks and bonds last year...'

I.e. Stocks earned 10% of \(S\) and bonds earned 5% of \(B\) and this totaled $800, which we can write as an equation

\[( 10\% \times S ) + ( 5\% \times B ) = 800\]

It is also important to write down what you are trying to find.

It is all too easy to do the correct working and get to a related or intermediate answer which you find in the list of answers A to E and to choose it in your haste to finish the question.

'...how much money did he have invested in stocks?'

You are trying to find the amount in stocks which we have represented as \(S\), so write down

\[S = ?\]

To summarize we have:

\[ \begin{split} S &= 2B \\ ( 10\% \times S ) + ( 5\% \times B ) &= 800 \\ S &= ? \end{split} \]

Two equations with two unknowns so we can solve them.

Next page: Solving equations

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