Word Problems

What they are and how to work with them

Steve Sugden

Bond University

10 May 2011

"Word" problems

http://en.wikipedia.org/wiki/Word_problem_(mathematics_education)

Word problems are those where you are given a paragraph or two describing the problem.

You have to then formulate equations from this and then solve them.

These kinds of problems are often found very di¢ cult by beginning algebra students.

From Wikipedia:

The term "word problem" is often used to refer to any math exercise where signi…cant background information on the problem is presented as text rather than in mathematical notation.

Example of a "word problem"

Linear programming

The following word problem is an example of a class of problem known as linear programming. If you go on to do Business Maths at Bond, you will learn how to solve such problems.

Kustom Kars (KK) does van conversions. The custom conversion requires 15 hours of shop time and 8 hours of painting time. The deluxe conversions requires 12 hours of shop time and 12 hours of painting time. There are 108 hours of shop time available and 72 hours of painting time available during the coming two weeks. For integer values only, how many conversions of each type should Kustom Kars perform, assuming that each custom conversion results in $175 pro…t and each deluxe conversion results in $225 pro…t in order to maximize KK’s pro…ts?

Notice there are lots of words here and also lots of numbers. This type of problem is well–suited for a tabulation approach (see next slide).

Tips for word problems

Some general rules to help with word problems.

Draw a diagram to represent crucial parts of the problem.

De…ne variables to represent each quantity in the problem.

Write expressions relating the variables.

Express the goal of the problem in terms of the problem variables.

If you can’t get started, the there are some common strategies to help.

Simplify the problem - try to solve a simpler version.

Work backwards from a known solution.

Tabulate known values and look for a pattern.

The classic reference for problem solving is Polya, G. (1957) How to Solve It, 2nd ed., Princeton University Press, ISBN 0-691-08097-6.

It is in the Bond library.

If you don’t want to read the whole book, a very good one-page summary is at http://www.math.utah.edu/~pa/math/polya.html.

Some "word" problem exercises

1Shonk Shocks supplies wholesale quantities of replacement shock absorbers for cars. Unfortunately their bookkeeping is substandard, and they need to know what prices were charged for two di¤erent products X and Y from invoices several years ago. Invoice A billed ACME Autos for 25 of X and 10 of Y for $1000 and Invoice B billed Jaunty Jalopies for 50 of X and 25 of Y for $2250. Find the cost for one of each product.

2A used-car dealer bought two cars for $29000. He sold one at a pro…t of 10% and the other at a loss of 5% and still made a pro…t of $1850 on the whole transaction. Find the original cost of each car to the dealer.

3A man invests twice as much at 8% as he invests at 5%. His total annual income from the two investments is $84,000. How much is invested at each rate?

Example of tabulation

For the Shonk Shocks problem we tabulate the information

Rows correspond to invoices and columns to quantities and costs

Let each item of X cost $x and each item of Y cost $y

From this table we may write equations

An equation is a claim that two quantities are equal. What are the quantities here?

Units are important: dollars = dollars/unit #units