Translating Word Problems into Equations

One thing that I’ve noticed as a math tutor over the last 7 years is that many students struggle much more with understanding what a word problem is asking them to do moreso than doing the actual math.

This probably isn’t surprising to a lot of people. Word problems are inherently a bit trickier because you have to read them carefully, figure out what is being asked, and set up the math problem yourself.

Today I’m going to walk through an example word problem and give some tips on how to solve it.

Here’s the problem:

A math test is worth 100 points. There are two point problems and four point problems on the test. There are a total of 45 questions on the test. How many two point problems and how many four point problems are there on the test?

Now, let’s solve the problem.

Step 1: Identify and write out the information you are given.

Here’s what we know:

  1. The test is worth 100 points.
  2. There are 45 questions on the test.
  3. There are both 2 point and 4 point problems.

Step 2: Set up an equation to solve the problem.

We know the test is worth 100 points so it makes sense that our equation should equal 100. Since there are questions worth 2 points and questions worth 4 points, we’ll want something like 2x + 4y = 100, where x is the number of 2 point questions and y is the number of 4 point questions.

We still need to account for the fact that the test has 45 questions, though. Since the test has 45 questions, that means x and y are related. Let’s write out how they are related:

x + y = 45 OR y = 45 – x

If we know x, we know we can just take the total number of questions (45) and subtract x to get y. So, let’s re-write what we have:

1) 2x+ 4y = 100

2) y = 45 – x

3. Substitute equation (2) into equation (1) and solve for x

Now, substitute equation (2) into equation (1):

2x + 4*(45-x) = 100

Solving for x is the easy part.

2x + 4*(45-x) = 100

2x +180 – 4x = 100

-2x = -80

x = 40

Now, we know x, but what about y?

4. Substitute the solution for x back into equation (2) and solve for y

y = 45 – x

x = 40

y = 45 – 40 = 5

5. Check your answer

We got x = 40 and y = 5 so let’s substitute that back into equation (1) and see if the equality holds.

2x + 4y = 100

2*40 + 4*5 = 80 + 20 = 100

Yes! The equality holds so our answer is correct.

6. State your answer

Let’s put that into words now so we get full credit.

There are 40 2-point questions and 5 4-point questions on the test.

Wrapping Up

That might seem like a lot of work to solve such a simple problem, but it’s good practice to learn all the steps in solving a word problem.

We’ve actually just solved a systems of equation problem by the substitution method, a common algebra topic that many students struggle to master.

If you’re faced with a tough word problem that you can’t quite wrap your head around, it can really help to just start writing out what you know and trying to set up the equation(s). When you start to write things out, sometimes it’ll hit you what you need to do to solve the problem. I highly recommend doing that as opposed to staring at your test!