This is a print version of PrepForTests.com's Word problems tutorial.

Last updated: 18 Nov 2008

Understanding what you are being asked and making sense of the information you are given is half the battle in many of the questions in the quantitative section.

This tutorial will give you some basic techniques for understanding and solving word problems.

This tutorial will assume that you have already worked through our fractions and exponents, ratios and percents tutorials.

We will also assume that you a fair knowledge of algebra. For example you should be able to solve the following equation to find \(a\)

\[
\begin{split}
2a + 3 &= 11 \\
a &= ? \qquad \text{solution at the bottom of this page}
\end{split}
\]

Solve this problem by finding the value for \(a\).

This tutorial will begin by giving you two methods for making sense of and solving word problems.

\[
\begin{split}
2a + 3 &= 11 \\
2a + 3 - 3 &= 11 - 3 \\
2a &= 8 \\
a &= \frac{8}{2} \\
a &= 4
\end{split}
\]

Last updated: 13 Oct 2008

We will begin with an example of a word problem and then look at how to use equations to solve it

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?

- $9,600
- $8,000
- $6,400
- $4,000
- $3,200

If you feel brave you can have a go at it now.

My advice would be to follow these guidelines.

- Summarize
- Write equations, which contain the information given to you in the question.
- Use sensible variable names for example the first letter of the thing that the variable represents.
- Identify answer i.e. write down exactly what you are looking for.

- Solve
- Find the solution for the equstions you have written down.

Now lets work through the question together and see how you have done.

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.

Last updated: 18 Nov 2008

We have already done much of the work in solving this problem by changing it from the word problem

'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?'

to the algebraic problem

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

To solve the system of equations you want to reduce the problem from two variables in two equations to one variable in one equation. Usually the easiest way to do this is by substitution i.e. replacing one of the variables by the other.

\[
\begin{split}
(10\% \times S) + (5\% \times B) &= 800 \qquad \text{We know that S = 2B so replace in the equation} \\
(10\% \times 2B) + (5\% \times B) &= 800 \qquad \text{Multiply out } 10\% \times 2B \\
(20\% \times B) + (5\% \times B) &= 800 \qquad \text{20\% of B and 5\% of B are 25\% of B} \\
25\% \times B &= 800 \qquad \text{We know that } 25\% = \frac{1}{4} \\
\frac{1}{4} \times B &= 800 \qquad \text{Multiply both sides by 4} \\
B &= 800 \times 4 \\
B &= 3200
\end{split}
\]

Careful at this point not to assume that you have finished. You have found the amount of money invested in bonds, now you need to use the equation \(S = 2B\) and calculate the amount invested in stocks.

\[S = 2B = 2 \times 3200 = 6400\]

Returning to the question.

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?

- $9,600
- $8,000
- $6,400
- $4,000
- $3,200

The amount invested in stocks was $6,400 and the answer is C.

Last updated: 13 Oct 2008

Now for another example of a word problem that we will use a different technique to solve.

This time we will summarize the information that we are given in the form of a table.

At a football game 50% of the seats are sold to season ticket holders who pay $11 each and 10% are sold to children who pay $5 each. All the remaining tickets are sold to non-members at $15 each. What proportion of the total gate receipts for the game is contributed by non-members?

- 60%
- 52%
- 50%
- 40%
- 5%

If you would like to try the question now if you like.

You should follow these guidelines.

- Summarize
- Organize the information you are given into a table.
- Identify answer i.e. mark on the table exactly what you are looking for.

- Solve
- Keep calculating more elements in the table until you arrive at the answer you need.

Let us answer the question together.

Last updated: 13 Oct 2008

Yet again the question contains a great deal of information.

We can put all the information into a table to make the question simpler to solve.

'At a football game 50% of the seats are sold to club-members who pay $11 each and 10% are sold to children who pay $5 each. All the remaining tickets are sold to non-members at $15 each. What proportion of the total gate receipts for the game is contributed by non-members?'

In the question we have three different types of tickets, 'club-members', 'children' and 'non-members' and three different types of information are given or asked for, '% of tickets sold', 'price of ticket' and '% of total income'.

Therefore we would sketch a 3x3 table.

% tickets sold | price | % total income | |
---|---|---|---|

club-members | |||

children | |||

non-members |

And begin to fill in the information.

'At a football game 50% of the seats are sold to club-members who pay $11 each and 10% are sold to children who pay $5 each.'

% tickets sold | price | % total income | |
---|---|---|---|

club-members | 50% | $11 | |

children | 10% | $5 | |

non-members |

'All the remaining tickets are sold to non-members at $15 each. What proportion of the total gate receipts for the game is contributed by non-members?''

% tickets sold | price | % total income | |
---|---|---|---|

club-members | 50% | $11 | |

children | 10% | $5 | |

non-members | $15 | ? |

Where '?' represents what we need to find to answer the question.

We have the summary now let's answer the question.

Last updated: 18 Nov 2008

We reduced the question to the following table.

% tickets sold | price | % total income | |
---|---|---|---|

club members | 50% | $11 | |

children | 10% | $5 | |

non-members | $15 | ? |

We will add a totals row because we are working with percentages and an income column so that we can later work out the percentages for the income.

We can fill in 100% for the totals of the percentages.

% tickets sold | price | % total income | income | |
---|---|---|---|---|

club members | 50% | $11 | ||

children | 10% | $5 | ||

non-members | $15 | ? | ||

total | 100% | - | 100% |

Then it is a matter of filling in as many cells as we can calculate until we have enough information to find the answer.

In this case we know the % tickets sold will sum to 100% so the percentage sold to club members will be

\[
\begin{split}
\%\text{non-members} &= 100\% - (\%\text{club members} + \%\text{children}) \\
&= 100\% - (50\% + 10\%) \\
&= 40\%
\end{split}
\]

% tickets sold | price | % total income | income | |
---|---|---|---|---|

club members | 50% | $11 | ||

children | 10% | $5 | ||

non-members | 40% | $15 | ? | |

total | 100% | - | 100% |

Now we will work out the amount of income from each group.

The income from each group will be the number of tickets sold multiplied by the price of each ticket.

Since we do not know the total number of tickets sold we can assume that there were 100 tickets because this will make the mathematics easier.

\[
\begin{split}
\text{income} &= \text{number of tickets sold} \times \text{price of ticket} \\
\text{income from club members} &= 50 \times \$11 = $550 \\
\text{income from children} &= 10 \times \$5 = \$50 \\
\text{income from non-members} &= 40 \times \$15 = \$600 \\
\text{total income} &= \$550 + \$50 + \$600 = \$1200
\end{split}
\]

% tickets sold | price | % total income | income | |
---|---|---|---|---|

club members | 50% | $11 | $550 | |

children | 10% | $5 | $50 | |

non-members | 40% | $15 | ? | $600 |

total | 100% | - | 100% | $1,200 |

Now that we have the total income and the income from non-members we can find the percentage we need.

\[
\begin{split}
\frac{\text{non-members}}{\text{total}} &= \frac{600}{1200} \\
&= \frac{1}{2} \\
&= 50\%
\end{split}
\]

% tickets sold | price | % total income | income | |
---|---|---|---|---|

club members | 50% | $11 | $550 | |

children | 10% | $5 | $50 | |

non-members | 40% | $15 | ? = 50% | $600 |

total | 100% | - | 100% | $1,200 |

Returning to the question.

At a football game 50% of the seats are sold to season ticket holders who pay $11 each and 10% are sold to children who pay $5 each. All the remaining tickets are sold to non-members at $15 each. What proportion of the total gate receipts for the game is contributed by non-members?

- 60%
- 52%
- 50%
- 40%
- 5%

50% was contributed by non-members so the answer is C.

Last updated: 13 Oct 2008

In this tutorial we have looked at techniques for solving word problems.

- Summarize
- Organize the information you are given into equations or a table.
- When using equations, use sensible variable names.
- Identify exactly which value you have to find.

- Solve
- Keep working even if you are not sure exactly what to do.
- When using equations, keep eliminating variables until you have only the one you need to find.
- With tables, keep calculating more elements in the table until you arrive at the answer you need.

- Introduction to problem solving by Kim Peppard
- "Age" word problems from Purplemath
- System-fo-equations word problems from Purplemath
- Basic "percent of" word problems from Purplemath