Algebra, Word Problems

Nothing strikes fear in a math student like the dreaded word problem. I have seen many students who are very good at solving equations but do poorly with word problems. The problem is that they lack the skill to convert english into an equivalent math language. In my last post, I started with converting english phrases into algebraic expressions. Let’s graduate to a full word problem and create the equivalent algebraic equation.

Karen is twice as old as Lori. Three years from now, the sum of their ages will be 42. How old is Karen and Lori?

As I suggested in my last post, let’s break this down. So here we have two unknowns: Karen’s and Lori’s ages. So a good first step is to assign letters to these unknowns.  Let’s let K be Karen’s age and L be Lori’s age. Now the first sentence in the problem has a word that means “=” in math. That word is “is”.  In other word problems, you may see words like “the same as”, “equals”, “was”, “will be”.

The first sentence in the word problem directly converts to an equation since we already assigned letters to the two ages:

Karen is twice as old as Lori: K = 2L

Now there are two unknowns here but that’s OK. We can’t solve anything yet, but there is more information in the word problem. As we read it, write down the equivalent math expressions.

Three years from now, what are their ages three years from now? Well three years from now, Karen will be K + 3 and Lori will be L + 3.

the sum of their ages will be 42. Another equation here because of the words “will be”. So we add the ages of Lori and Karen three years from now to get 42:

(K + 3) + (L + 3) = 42

The brackets are not really needed. I just put them there so you can see that I am adding Karen’s age 3 years from now to Lori’s age 3 years from now.

Now we have the equation but there are two unknowns. You usually cannot solve a single equation  with more than one unknown. But remember the first equation we wrote down: K = 2L? This equation means that algebraically, K is exactly the same as 2L. In the second equation, we can replace the K with 2L:

(K + 3) + (L + 3) = 42: (2+ 3) + (L + 3) = 42

Now we can solve this equation to find what is. I covered solving equations before, so I won’t do a lot of explaining here. I will start by removing the brackets and proceed:

2+ 3 + L + 3 = 42

3L + 6 = 42

3L = 42 – 6 = 36

L = 36/3 = 12

So Lori is 12. What about Karen? Again, look at the things we’ve written down so far. We have K = 2L, that is, Karen is twice as old as Lori. Since we already know Lori’s age, Karen must be 24.

So in most word problems, it will help if you first assign letters to the unknowns, then create expressions and/or equations from each part of the word problem. Have these all together and usually, the equation you need to solve will pop out.