# Cracking Dollar Shave Club’s March Rompecabeza

As a subscriber of DollarShaveClub.com, I always like to look at the little problems they put at the back of their monthly publication.^{1} March’s was quite spectacular, and it went as follows:

Brian received a cash gift from his parents on his birthday. He spent half of what he got, plus another $5 at the local Big and Tall store, half of what was left plus $5 on in-appCandy Crushpurchases and three-fourths of what was left plus a $5 tip on all-you-can-eat wings. When he left the wings joint, he was out of cash and experiencing some wicked meat-sweats.How much did he start out with?

For your own benefit, I recommend you copy that problem, paste it into a Word document, work it out, then read the rest of this article.

There are two ways to solve this, and I’ll be interested in seeing on DSC’s blog how they choose to solve it. I’m guessing they’ll use the second method, but let’s go over them in order.

Method 1: mathematical approach. First, we start out by creating some variables:

```
x = original_amount
y = .5x + 5
what_left_1 = x - y
what_left_2 = what_left_1 - ( what_left_1 * .5 + 5 )
what_left_3 = what_left_2 - (what_left_2 * .75 + 5 ) = 0
```

We then replace each instance of `what_left_2`

in the final equation for its actual value:^{2}

```
what_left_1 - ( what_left_1 * .5 + 5 ) - (what_left_1 - ( what_left_1 * .5 + 5 ) * .75 + 5 ) = 0
```

We then replace each instance of `what_left_1`

for its actual value:

```
x - y - ( x - y * .5 + 5 ) - (x - y - ( x - y * .5 + 5 ) * .75 + 5 ) = 0
```

We then replace each instance of `y`

for its actual value:

```
((x - (.5x + 5)) - ( (x - (.5x + 5)) * .5 + 5 )) - (((x - (.5x + 5)) - ( (x - (.5x + 5)) * .5 + 5 )) * .75 + 5 ) = 0
```

We *could* manually simplify this expression to solve it, but that would take a while. So far all we’ve done is declare some variables and copy and paste them around. It’d be nice to keep the problem this simple, all the way to the finish line. How can we do that? Well, we know that our formula looks like this when slightly rearranged:

```
0 = X + some stuff
```

That looks a lot like the formula you learned in high school:

```
Y = MX + B
```

When Y = 0, what is X? Taking the left hand side of our massive equation and just pasting it into Google, we see that it’s 110.

Plugging this number into the original story problem, we see that this is indeed the correct answer.

Method 2: heuristic approach. Using formulas is nice because it feels scientific, but you should be able to solve this problem without a pen and paper or Wolfram Alpha.^{3}

We know that near the end of his spending spree, Brian spent 75% of what he had left, plus $5, on wings. Once he bought his wings, he was penniless. That means that $5 of what he had was 25% of his final amount. His pre-tip wings cost $15. Before he bought his wings, therefore, he had $20. Using this exact same method two more times, you discover that he had $50 before buying Candy Crush, and $110 before the Big and Tall store.

Nicely done, Dollar Shave Club.

- This publication is called
*The Bathroom Minutes*and is complementary to the blades that arrive each month. ↩︎ - You’ll also notice that we’re dropping the left part of the equation,
`what_left_3`

, since we already know that equals 0. ↩︎ - The first method is original with me, but I’m indebted to a friend for the second. ↩︎

*Note: This article has been backdated to its original penning. Drinking Caffeine debuted in February 2016, but this article has been dusted off and placed here so that you, dear reader, may access it without the ensnarements of web.archive.org.*