Word problems

Understanding word problems is probably the most important part of mathematics. Mathematics without applications (‘pure’ mathematics) is a beautiful work of art, part of our culture and good practice for precise thinking, and that's not a bad thing. But only applied mathematics is actually useful.

Almost everything in Algebra can be taught to a computer, and while it's good to know how computer algebra systems work, in the end it's more reliable to use them to solve problems than to solve them yourself, just as your calculator is more likely to do 276,435/6,250 correctly than you are, no matter how good you are at arithmetic. But word problems cannot be taught to a computer, and without word problems, Algebra is ultimately useless.

Before I start Algebra proper, I'll finish the review material by discussing word problems in Arithmetic. Most of the ideas that come up in Algebra already appear here.

The fundamental operations

Obviously, a word like plus or add indicates addition. Literally, ‘plus’ means more or greater, and these words also often indicate addition. For example, 8 more than 7 is 7 + 8, or 15. Compare this with 8 is more than 7, which has a verb (‘is’) and means 8 > 7. Verbs like increase or exceed also often indicate addition; for example, if the height of a fence is 6 feet and you increase it by 2 feet, then the new height is 6 ft + 2 ft, or 8 feet. The formal term for the result of an addition operation is sum; for example, the sum of 8 and 7 is 8 + 7, or 15. More generally, the sum of 8, 7, 12, and −2 is 8 + 7 + 12 − 2, or 25.

The words for subtraction are largely the reverse of those for addition. Of course, minus and subtract, but also fewer and less (which is the literal meaning of ‘minus’). Be careful with the order; for example, 8 less than 7 is 7 − 8, or −1, which 8 less 7 (or 8 minus 7) is 8 − 7, or 1. The verbs for subtraction include decrease or reduce; for example, if I decrease the height of an 8-foot fence by 3 feet, then the new height is 8 ft − 3 ft, or 5 feet. The formal term for the result of subtraction is ‘difference’, but this can be tricky, so I save it for later.

The formal term for the result of multiplication is product. For example, the product of 8 and 7 is 8 · 7, or 56. More generally, the product of 8, 7, 12, and 1/2 is 8 · 7 · 12/2, or 336. Of course, the words times and multiply indicate multiplication. Notice that twice and thrice literally mean ‘two times’ and ‘three times’, so they also indicate multiplication. So if one fence is 3 feet high and another fence is twice as high, then the second fence is 2 · 3 ft, or 6 feet, high. Multiplication is also indicated in trickier ways that I discuss below.

The formal term for the result of division is quotient. For example, the quotient of 8 and 2 is 8/2, or 4; and the quotient of 6 and 4 is 6/4, or 3/2. Another word for quotient is ratio; if one fence is 8 feet high and another is 6 feet high, then the ratio of the first height to the second is 8 ft/6 ft, or 4/3. (Notice that the factor of a foot cancels when you reduce this fraction!) Of course, division is also indicated by the verb divide; informally, people also often use the word over, which refers to the use of the fraction slash or horizontal bar.

Tricky terms

Usually, the word total means addition. For example, if one person has $20 and another has $10, then the total amount of money they have is $20 + $10, or $30. Sometimes, however, total means multiplication. For example, if there are 7 people, each of which has $5, then the total amount of money is 7 · $5, or $35. A clue in this case is the word each; every, all and (for multiplication by two) both are also signs that you get a total by multiplying instead of adding.

For fractions and percentages, the word of indicates multiplication. For example, half of 12 is (1/2) · 12, or 6; similarly, 4% of 50 is 4% · 50, or 2. For that matter, the word percent itself means division by 100; in the example above, 4%, or 4 percent, means 4/100, or 0.04. Multplication can even be indicated by nothing at all! For example instead of ‘half of 12’, I might say ‘half a dozen’, which means the same thing.

The word difference indicates subtraction, but sometimes it also indicates taking an absolute value! For example, if one fence is 6 feet high and another is 8 feet high, then the difference between their heights is the absolute value |6 ft − 8 ft|, which is |−2| feet, or 2 feet. If I want the result to be negative, then I should say the difference to 6 feet from 8 feet; this means 6 ft − 8 ft, which is −2 feet. Conversely, the difference from 6 feet to 8 feet means 8 ft − 6 ft, which is 2 feet. But the differnce from 8 feet to 6 feet is −2 feet again.

Algebra

The ideas above work exactly the same way with algebra. You just have to include a letter to stand for ecah unknown quantity, and this means that you can't work out the exression completely.

For example, if a fence is 6 feet high and you raise it, but you don't know how much the height is raised, then let x be the number of feet the fence is raised; then the new height is 6 + x feet (and that's all that you can say). Later if you learn that x is 2 (as in the first fence example on this page), then you can go back to this expression, change it to 6 + 2, and work out that the new height is 8 feet.

This actually gives you a little more flexibility. For example, if you don't know whether the fence will be raised or lowered, still you can let x be the number of feet it is raised; if the fence is actually lowered, then this simply means that x is a negative number. If you later learn that the fence is lowered by 3 feet (as in the second fence example on this page), then you can use −3 for x. The expression 6 + x still works, and you work out 6 + (−3) to get a new height of 3 feet.

In algebra, using absolute values to express absolute differences is even more important. In the last fence example from the previous section, it may seem obvious that you get the difference between 6 feet and 8 feet by working out 8 − 6 instead of 6 − 8, since 6 is smaller than 8. But if you don't know the height of the second fence, and you let h the number of feet in that height, then you don't know whether 6 is larger or smaller than h, so you don't know whether to use h − 6 or 6 − h. However, you can still use |h − 6| (or |6 − h|, which comes to the same thing), and that will always be correct.


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