READ THIS Before Solving Word Problems

Many of my tutoring clients who work with me on Algebra tell me that they have difficulty with word problems.  They can do the math, they say, they just have a hard time figuring out what the question is asking them.  If this applies to you, please read on because I will show you a way to make sense of these types of problems.

What I tell my tutoring clients is that a word problem is really just a translation problem. Just like in foreign language class, you are translating a sentence in English into “math language”.

If math is a language that you can translate to, then it makes sense that it would include:

• Common nouns: we call these variables: x, y, z
• Proper nouns: refer to a specific number: 2 3 27
• Verbs: = < > ≤ ≥
• Conjunctions: + – × ÷
• and Prepositional phrases: +3 -1 ×4

Let’s look at an example: 

Currently, Bob is three times John’s age. In six years, Bob will be twice John’s age. What are their ages now?

Before I start translating, I need a dictionary of terms, and that’s where the variable definitions come in. The objective is to identify quantities in the problem whose value is not known, and make a variable name for each one. Since the question is asking about their ages (which are numbers that are in the problem but whose values we don’t know), I can create a variable for each one:

x = Bob’s age today

y = John’s age today

Now I am ready to start translating. I take the first sentence, which I have re-worded just slightly: “Bob’s age is three times John’s age”.

• “Bob’s age” translates to “x”
• “is” translates to “=”
• “three times John’s age” translates to “3y”

So the first sentence translates to x = 3y.

I can do the same with the second sentence: “Bob’s age in six years will be twice John’s age in six years”:

• “Bob’s age in six years” translates to “(x + 6)”
• “will be” translates to “=”
• “twice” translates to “2 × “
• “John’s age in six years” translates to “(y + 6)”

So (x + 6) = 2 (y + 6)

And now I am in a position where we can solve.  Let me know what you think the answer is in the comments.

Let’s try another:

A collection of 34 coins, consisting of nickels, dimes, and quarters, has a total value of $3.55.  If the number of nickels is twice the number of dimes, how many coins of each type are there?

Again, I need to start with the dictionary of terms.  Since the problem is asking how many coins there are of each type, I will use the quantities of each type as the variables:

n = number of nickels

d = number of dimes

q = number of quarters 

The first part of the question says that the total number of coins is equal to 34.  I read this as, the number of nickels and the number of dimes and the number of quarters is equal to 34:

• “The number of nickels” translates to “n”
• “and” becomes “+”
• “the number of dimes” translates to “d”
• “the number of quarters” translates to “q”
• “is equal to” becomes “=”

n + d + q = 34

How do we figure out the value?  Well, since each nickel is worth five cents, the value of a set of nickels is five cents times the number of nickels.  So the value of n nickels is 5n.  Likewise, the value of d dimes is 10d, and the value of q quarters is 25q.  If I expand the sentence as I have with the others, and say that the value of the nickels plus the value of the dimes plus the value of the quarters is equal to 355 cents:

• “The value of the nickels” translates to “5n”
• “the value of the dimes” translates to “10d”
• “the value of the quarters” translates to “25q”
• “is equal to” becomes “=”

5n + 10d + 25q = 355

Finally, the number of nickels is twice the number of dimes:

• “The number of nickels” translates to “n”
• “Is” becomes “=”
• “Twice” becomes “2 ×”
• “The number of dimes” translates to “d”

n = 2d

And now I can solve.  Do you know the answer?

Let me know if this is helpful. If you share a problem you are working on, please share it in the comments!  I will illustrate how you could translate that one also.