This Word Problem Shows That Words Are a Problem, Not Math

This is what a true “word problem” looks like

Photo by Ian Panelo from Pexels

During lockdown period which was a couple of months ago, a slightly tricky word problem was making the rounds on WhatsApp. My friends know that I like mathematics very much so one of them sent me the problem to get my opinions on the solution.

Usually, I wouldn’t bother myself about word problems on social media because they don’t turn out to be as interesting as they initially look. Most of those problems are often meant to annoy people with controversy.

In solving word problems, the procedure is usually:

1. Translate the words from English(or whatever language the problem is stated in) to math
2. Solve in math language(because the language of math is way cooler)
3. Send the solutions to your friends just so you can brag

Step 1 is often the toughest part. Once you’re able to derive the math from the English, solving the problem becomes easier.

What intrigued me about the word problem was that people had as many as 3 different seemingly valid results just from completing step 1. That piqued my interest to try it out as well.

Here’s a true word problem where the problem is the words and not the math.

The Word Problem

Some birds were flying and met a bird on their way. The bird greeted them, “Hello, hundred!”

They said, “We are not hundred. We need half of us plus you to make us hundred.”

Photo by Aleksandar Pasaric from Pexels

Question: How many birds were flying?

A. 198

B. 49

C. 66

D. 51

E. 101

F. 199

Before proceeding to read my discussion on possible solutions, I encourage you to try solving the problem on your own first.

First Piece of Ambiguity

Half of us plus you…

We are going to consider the word problem sentence by sentence. As we do, we will notice some ambiguities which caused the issue among my friends.

For the sake of absolute clarity, I assume that the bird that greeted, “Hello, hundred” was flying alone and was separate from the flock of birds. Let x represent the number of birds in the flock(not including the single bird flying alone).

The third sentence in the problem is where a section of the ambiguity lies. Take note of this phrase: half of us plus you to make us hundred. The problem is that this phrase could have at least 2 different valid interpretations:

• x/2 + 1 = 100

or

• (x + 1)/2 = 100

It’s unclear if we are supposed to halve the expression x or halve the expression x + 1.

From here onwards, just a little elementary algebra gives us the answer we’re looking for.

• The first interpretation gives x = 198
• The second interpretation gives x = 199

Both answers are among the choices so it seems that we’re in the clear. However, I came across a third valid solution. As it turns out, there’s even more ambiguity that wasn’t spotted by some of us. That next piece of ambiguity gives birth to one more solution…or two?

Second Piece of Ambiguity?

Make us hundred

The third solution comes from a weird(in my opinion) and not-so-straightfoward interpretation of the expression “make us hundred” in the third sentence. For most of you, I would like to believe that after you computed half of the birds in the flock plus one, you equated that to 100. However, those with the third solution seem to have interpreted things a different way. They had their answer as 66.

I’m going to explain how I think they came to that conclusion. To understand this, I need you to think about the following hypothetical scenario between a boy and a girl:

After mustering enough courage to talk to the girl he likes, a shy guy walks up to the girl and opens his mouth to reveal the following pickup line: “Girl, you make me complete”.

If you wished to be more mathematical, how would you translate the guy’s pickup line to a math equation? I bet you would translate “you make me complete” to be “you + me = complete”.

That’s where the third solution comes from. For a sentence construction involving the word “make”, we replace “make” with “+” and insert “=” after the next word.

Applying this rule to our original problem, we translate “half of us plus you make us hundred” to be “half of us plus you + us = 100”

Keeping in mind that we still don’t know what “half of us plus you” really means, we yet again have two interpretations stemming from this:

• x/2 + 1 + x = 100
• (x + 1)/2 + x = 100

The first interpretation yields x = 66, but the second yields x ≈ 66.33. We can agree that the second figure isn’t plausible since a fractional number of birds doesn’t make sense here.

So in all, the only values for x that I’ve found to be plausible are 198, 199 and 66.

But guess what? We aren’t done solving the problem.

Oh so There’s a Third Piece of Ambiguity

Here, we consider the last sentence in the word problem: How many birds were flying?

This sentence seemed harmless to everyone…everyone except me.

Do you remember what x represents? x denotes the number of birds only in the flock. We didn’t include the lone bird who greeted the flock. It’s not clear whether or not the answer required includes the single bird that is flying separately from the flock.

• If the question is asking us to find the number of all the flying birds in the story, then the final answer is x + 1
• However, if the question wants the number of birds only in the flock, then the final answer is just x

Putting All This together

Therefore, putting together the solutions from all 3 pieces of ambiguity, we have the solution set to be {66,67,198,199,200}. The number of birds that were flying has to be one of the numbers in the aforementioned set.

Two of the numbers aren’t part of the answer choices but who cares? The one who set the question should have known better than to use ambiguous phrases lol.

And this time, I promise we’re really done solving it. No more ambiguities…at least, not ones I can see.

So why should you care about this?

This is a true word problem because it creates a problem with words

Apart from the fun(or torture?) we get from solving this problem, we learn something valuable about language: language(unless it is the language of math) can often be ambiguous.

Sometimes the hindrance to solving a word problem isn’t a lack of math skills but rather a lack of language clarity.

Or perhaps all this noise about interpretations is much ado about nothing?