Translating Words into Mathematics
To a large extent, the purpose of mathematics is to provide a systematic approach to solving everyday problems. As such, an important mathematical skill is the ability to translate from words into mathematical statements. Mathematics is really a language and once you understand how to speak the language, then solving word problems becomes simplified. The best way to introduce you to this process is to work through a few examples, but becoming a skilled translator requires a great deal of practice.
Express the following statement as an equation and indicate what each variable you use stands for.
If Tim were three years older, then he would be twice as old as Rebecca.
First of all we need to determine the unknown quantities in this statement. The two things we do not know are Tim’s age and Rebecca’s age. So we choose variables to represent those unknown quantities so that later on, if we do know either of them, we can replace the variable with that number. Let’s choose t to represent Tim’s age today and r to represent Rebecca’s age today.
Now let’s try to figure out how to rewrite this statement mathematically. The part, “If Tim were three years older,” is another way of saying, “If we added 3 years to Tim’s age.” So let’s do that mathematically. We generate the expression:
Now we look at the last part of the statement which says, “twice as old as Rebecca.” This is another way of saying 2 times Rebecca’s age. So we write this mathematically as:
The phase, “then he would be,” establishes an equivalence (or equality) between the two quantities (one dealing with Tim’s age and one dealing with Rebecca’s age). This corresponds to an equals sign in the mathematical statement. Thus, the full statement can be written mathematically as
The real power of mathematics is seen when we choose to use our mathematical statement to deduce things that must be true given the statements that are made. For instance, if one were to learn that Tim was 9 years old, then determining how old Rebecca is becomes a simple matter of algebra. One need only replace the variable t with the value 9 and solve the resulting linear equation for r. Like this:
We can conclude that given the information that Tim is 9 years old, Rebecca would have to be 6 years old.
A telephone company charges a monthly base fee of $20.00 for land-line service and an additional $1.50 for each hour of long distance service used during the month. Write a mathematical statement which expresses the relationship between the total monthly bill and the number of hours spent using long distance service.
Again, we first identify the unknowns in the problem and designate variables to stand for those quantities. In this case, the unknowns are the total monthly bill value and the number of hours of long distance service used. Let us assign variables c and t to stand for total cost and long distance time respectively.
Now we know that no matter how many hours of long distance service are used, the bill will always include the $20.00 base fee, so we know that the total cost will be the cost due to the long distance hours plus the $20.00 base fee. Now all we need to know is how much the long distance contributes to the bill. The problem states that there is a $1.50 fee per hour of long distance usage. This suggests multiplication. The amount that the long distance contributes to the bill is then 1.5 times the number of hours spent using long distance service. Mathematically we would say:
All we need to do to express the total monthly bill is to add these two quantities and set that expression equal to our variable for the total cost. Mathematically we get:
This equation give us a concise formula for calculating the total cost of the telephone service given we have information about how many hours of long distance service were used.
Try this riddle. It may seem rather complex at first, but don’t get frustrated. Work at it piece-by-piece like we did in the last example and you’ll see that it reduces to a simple algebra problem once it is written out mathematically.
If each one of Justine’s rabbits were to give birth to two rabbits and Kim were to then give Justine ten of her rabbits, Justine would have half as many rabbits as Molly. How many rabbits does Justine have, given that Molly has 44 rabbits?
The only unknown is the number of rabbits that Justine has. However, we may want to initially write the first statement out mathematically (in which the number of rabbits Molly has is also unknown) and then substitute 44 in for the number of rabbits Molly has. The first statement about Justine’s rabbits giving birth to two each means that we are dealing with 3 times the number of rabbits that Justine has. We’ll write that as:
The statement about Kim giving Justine 10 rabbits means we must add 10 to that quantity.
It is claimed that this quantity is equal to half as much as the number of rabbits Molly has, so we generate the equation,
Now we substitute the known quantity of rabbits that Molly has for the variable m and then solve for j like this:
So, Justine currently has 4 rabbits.