Word problem (mathematics education)

The most common types of word problems in elementary algebra are distance problems, age problems, work problems, percentage problems, mixtures problems and numbers problems.

A typical age problem:

Ann is three times as old as her little brother Bob. In five years, she will be only twice as old. How old are they now?

To solve this by algebra, one first translates the words into mathematical variables, operations, and equations:

• Write Ann's age as the variable A, and Bob's age as B.
• Their ages five years from now are A+5 and B+5.
• Twice as old means one age equals two times another, and similarly for three times.

The problem thus becomes:

Solve the system of equations A = 3B and A+5 = 2(B+5) for variables A,B.

The answer is A = 15, B = 5, or in ordinary language: Ann is fifteen years old, and Bob is five.

Word problems such as the above can be examined on three levels:[3]

• A. The verbal formulation;
• B. The underlying mathematical relations;
• C. The symbolic mathematical expression.

Linguistic properties can include such metrics as the number of words in the problem or the mean sentence length.[4] One scheme to analyze the logico-mathematical properties is to classify the numerical quantities in the problem into known quantities (values given in the text), wanted quantities (values to be found) and auxiliary quantities (values found as intermediate stages of the problem).[4]

Word problems commonly include mathematical modelling questions, where data and information about a certain system is given and a student is required to develop a model. For example:

1. Jane had $5.00, then spent $2.00. How much does she have now?
2. In a cylindrical barrel with radius 2 m, the water is rising at a rate of 3 cm/s. What is the rate of increase of the volume of water?

These examples are intended to lead students to develop mathematical models on their own, and to promote mathematical interest and understanding by analyzing real-life situations. The relevance of these problems to students varies. The first example is known even to primary school students, and is often used to teach the concept of subtraction. The second example presents a high school student with the challenge of translating the problem into symbolic form:

Given ${\displaystyle r=2}$ and ${\displaystyle {\tfrac {dh}{dt}}=3}$ , find ${\displaystyle {\tfrac {d}{dt}}(\pi r^{2}h).}$

Word problems are a common way to train and test understanding of underlying concepts using a descriptive problem, instead of solely exercising algebraic manipulation or other mechanical skills.

The modern notation that enables mathematical ideas to be expressed symbolically was developed in Europe from the sixteenth century onwards. Prior to this, all mathematical problems and solutions were written out in words; the more complicated the problem, the more laborious and convoluted the verbal explanation.

Examples of word problems can be found dating back to Babylonian times. Apart from a few procedure texts for finding things like square roots, most Old Babylonian problems are couched in a language of measurement of everyday objects and activities. Students had to find lengths of canals dug, weights of stones, lengths of broken reeds, areas of fields, numbers of bricks used in a construction, and so on.[5]

Ancient Egyptian mathematics also has examples of word problems. The Rhind Mathematical Papyrus includes a problem that can be translated as:

There are seven houses; in each house there are seven cats; each cat kills seven mice; each mouse has eaten seven grains of barley; each grain would have produced seven hekat. What is the sum of all the enumerated things?[6]

In more modern times the sometimes confusing and arbitrary nature of word problems has been the subject of satire. Gustave Flaubert wrote this nonsensical problem, now known as the Age of the captain:

Since you are now studying geometry and trigonometry, I will give you a problem. A ship sails the ocean. It left Boston with a cargo of wool. It grosses 200 tons. It is bound for Le Havre. The mainmast is broken, the cabin boy is on deck, there are 12 passengers aboard, the wind is blowing East-North-East, the clock points to a quarter past three in the afternoon. It is the month of May. How old is the captain? [7]

Word problems have also been satirised in The Simpsons, when a lengthy word problem ("An express train traveling 60 miles per hour leaves Santa Fe bound for Phoenix, 520 miles away. At the same time, a local train traveling 30 miles an hour carrying 40 passengers leaves Phoenix bound for Santa Fe...") trails off with a schoolboy character instead imagining that he is on the train.[8]

• Word problems that lead to simple linear equations at cut-the-knot
• A+Click Word Problems for grade 1 through 12