A Fraction Word Problem – Algebra or Not?

Sometime soon I will do a series of posts on word problems, which are a common point of difficulty with students. But here is one recent example from a high school student, where language was the main difficulty, but the algebra is worth discussing as well. We’ll look a little more deeply into the problem itself, and think about its implications.

Writing and solving an equation

Here is the initial question:

1/2 of a number is 2 more than 1/3 of the number. What is 1/3 of the number?

a. 2

b. 4

c. 8

d. 12

Doctor Rick took the question, and first, as we usually do, pointed out that in order to help, we need some information about the student’s knowledge and needs. When we are just given a problem, we can’t tell what kind of help is needed; the best thing is for the student to show his work so we can see where, if at all, they made a mistake. Sometimes they turn out to be entirely correct, and it was the book that was wrong — so it would be a waste of time to write out a full explanation. Other times, the student is not learning algebra at all, so again, an algebraic solution would be wasted.

But, as we often do, he gave a suggestion to get started, in case that was all that was needed:

One way this problem might be solved is to “translate” the statement into an algebraic equation. Is that something you’ve been learning to do? For instance, give “the number” the name “x”. You’re asked to find “1/3 of the number”, which you can write as (1/3)x or x/3. How would you write “1/2 of the number”? How would you write “2 more than 1/3 of the number”?

In this case (as often happens), the student is not fluent in English, which could be the reason for not giving a full explanation. We find that in such a case, it is better to say more rather than less — the more a student says about his thinking, the more likely we will understand it, even if he doesn’t express it clearly. Here is the response (a little cleaned up):

Not good in English sorry. But I’ll try.

To be honest, I don’t know where to start because I don’t know how to approach this problem. It says “more than” so I think it’s an addition so:

x/2(2) + x/3

…

Then… I’ll need to find 1/3 of that number? I can’t understand it.

Seeing this work gave Doctor Rick a good idea of what was needed. He replied,

Thanks for the additional information about how you’re thinking. That’s the sort of thing I needed.

The problem was:

1/2 of a number is 2 more than 1/3 of the number. What is 1/3 of the number?

You said:

It says that “more than” so I think it’s an addition so:

x/2(2) + x/3

OK, I see some good things here. You translated “1/2 of a number” as x/2, and “1/3 of the number” as x/3. You’re correct that “more than” in this sentence refers to addition (sometimes, as you’ll see later, it can mean other things.)

There are two main difficulties with what you wrote. One is that it isn’t an equation — you didn’t get an equal sign (=) into it. That’s how I’d translate the word “is”. The other problem is that you are not taking the grammar of the sentence into account.

Let’s work step by step, translating one part at a time rather than trying to write the equation all at once. Here’s what I mean:

1/2 of a number is 2 more than 1/3 of the number.
\_____________/ \/   \_______/ \_______________/

      x/2       =  2     +            x/3

Wow, I’m done — I’ve written an equation! It won’t always work this neatly — in this case, the sentence is in nearly mathematical language right from the start. The same thing wouldn’t work for other problems you have sent!

OK, now you have an equation to solve. Can you find the value of x that makes the equation true? And then can you figure out what “1/3 of the number” is?

Doctor Rick has done two things here: one is to look at the whole sentence and translate it into an equation, rather than a mere expression; and the other is to not look at the whole sentence at once, but to break it into parts (according to its grammar) in order to build up the equation part by part. In this case, it was easy to do both things at once; more complicated sentences often require working on each piece separately to avoid confusion, and changing the order of parts. For example, if it had said “2 less than …”, we couldn’t just write “2 – …”, because the order would be wrong. We’d have to say \(x/3-2\) instead. So matching up phrase after phrase in order is not always appropriate, but it was convenient here.

This appeared to be the hard part for this student. We like to leave as much as possible for them to solve themselves, so that they get practice doing the work themselves; here it would have been hard to explain the ideas without doing the whole translation, but we can at least leave the solving to the student, who can probably manage that. And Doctor Rick was right about that:

OMG!!! Yah, the ‘is’ is =. Here’s my solution:

x/2 = 2 + x/3

x/2 – x/3 = 2

(3x – 2x)/6 = 2

x = 12

Since the question is what is the 1/3 of the number

x/3 = 12

3(x/3 = 12)

x = 4

Aaaaaaahhhh!!! And guess what! I got it right!!!!!

Well, there are some things to clean up here, but encouragement was appropriate, and Doctor Rick responded encouragingly,

You got it correct, and you are justified in feeling good about yourself! Keep it up!

A closer look

But let’s go back and look at a few details in the solution.

First, the student used a good method, and show work quite well — not too much detail, but including the important steps. His approach was to subtract \(x/3\) from both sides, then combine the left side into a single fraction, which reduces to \(x/6\); then, multiplying both sides by 6, the result is \(x = 12\).

Another approach, often taught in books and saving some work (or at least some writing, or some opportunities to make mistakes in using fractions, is to “clear fractions” before solving. The LCD of all the fractions in \(x/2 = 2 + x/3\) is 6, so we multiply each term by 6 in order to cancel with each denominator:

\(x/2 = 2 + x/3\)

\(6*x/2 = 6*2 + 6*x/3\)

\(3x = 12 + 2x\)

Now, with no fractions to trip over, we can just subtract x from each side and solve:

\(3x – 2x = 12\)

\(x = 12\)

Second, the student was wise enough not to stop there, but to see that the question was not to give the number, but rather 1/3 of the number. So the answer is 1/3 of 12, which is 4.

But the work he wrote was wrong! It is not uncommon for a student to think correctly, but not yet be skilled in writing what he is thinking. Here, he changed \(x = 12\) to \(x/3 = 12\), replacing \(x\) with the \(x/3\) that is desired; what he should have done is to divide both sides by 3, \(x/3 = 12/3 = 4\). I can only guess that, as he approached the finish line, he started thinking and writing too fast, and stumbled. We’ll give him credit.

Let’s take one more step, and check the answer. Recall again that the problem was, “1/2 of a number is 2 more than 1/3 of the number. What is 1/3 of the number?” We found that the number is 12. What is 1/2 of the number? 6. What is 1/3 of the number? 4. Is 6 2 more than 4? Yes. If we had misinterpreted “2 more than” in translating the equation, we probably would interpret it correctly in this more familiar process, so we would have a chance to discover our error.

And, again, part of checking is seeing whether we answered the question that was asked. What is 1/3 of 12? 4. We got it.

What if he didn’t know algebra?

Let’s take this in yet another direction. I mentioned at the start that this problem might have been given to a student who is not learning algebra, though that tends to be our first assumption. How might we solve it then?

The problem, once again, was, “1/2 of a number is 2 more than 1/3 of the number. What is 1/3 of the number?” The question might lead me to focus my attention not on the number itself, but on 1/3 of the number; I may never need to know what the number itself is! Here is a bar representing the number, with 1/3 of it shaded:

+-----------+-----------+-----------+
|XXXXXXXXXXX|           |           |
+-----------+-----------+-----------+

Here I’ve split each third in halves, so I can mark 1/2 of the number:

+-----+-----+-----+-----+-----+-----+
|XXXXX|XXXXX|     |     |     |     |
+-----+-----+-----+-----+-----+-----+
\_________________/

Since the 1/2 is 2 more than 1/3, that difference must be 2:

+-----+-----+-----+-----+-----+-----+
|XXXXX|XXXXX|  2  |     |     |     |
+-----+-----+-----+-----+-----+-----+
\_________________/

So each piece is 2, and the shaded region (two pieces) is 4. That’s the answer.

This sort of visual thinking is taught commonly today, and is a precursor to algebra. It requires more creativity than algebra, which is just a way to turn any problem into the same kind of symbolic problem, so that you don’t need any special thinking for each problem, but can follow a nearly mindless routine. The thinking I did here involved focusing deeply on the fractions (using a common denominator, for example, without actually saying so) and on relationships. That makes it a very useful experience. Don’t rush to learn (or teach) algebra — being easier (though students are surprised to realize that), it takes away the chance to learn to think deeply!