Four extra-large sandwiches of exactly the same size were ordered for \(m\) students where \(m>4\). Three of the sandwiches were evenly divided among the students. Since 4 students did not want any of the fourth sandwich, it was evenly divided among the remaining students. If Carol ate one piece from each of the four sandwiches, the amount of sandwich that she ate would be what fraction of a whole extra-large sandwich?

(A) \(\frac{m+4}{m(m-4)}\)

(B) \(\frac{2m-4}{m(m-4)}\)

(C) \(\frac{4m-4}{m(m-4)}\)

(D) \(\frac{4m-8}{m(m-4)}\)

(E) \(\frac{4m-12}{m(m-4)}\)

It’s a VIC (variables in choices), and there is only one variable, so, obviously, it’s time to back-solve!

STEP 1: Pick a nice value for \(m\) that is greater than 4. This is our pick.

Anything should work, so let’s say \(m = 6\).

STEP 2: Solve the problem when \(m = 6\). The result is our target number.

So, six students divided three sandwiches equally. That means each of these students got \(\frac{1}{6}\) of each of three sandwiches. Now, because 4 students didn’t want any of the last sandwich is was divided equally between the other 2 students, one of whom was Carol. So, Carol ate \(\frac{1}{6}\) of each of the three sandwiches that everyone shared and \(\frac{1}{2}\) of the forth sandwich.

Carol ate \(\frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{2}\) sandwiches. That’s 1 sandwich. So, our target number is 1.

STEP 3: Plug our pick into the answer choices until we find our target number.

A nice thing to notice is that all of our answers have the same denominator: \(m(m-4)\). If we plug 6 into that denominator we get 12. Since our target number is 1, we only need to find an answer with a numerator equal to 12.

(A) 6 + 4 = 10. Nope

(B) 2(6) – 4 = 8. Nope

(C) 4(6) – 4 = 20. Nope

(D) 4(6) – 8 = 16. Nope

(E) 4(6) – 12 = 12. Yep

The correct answer is E.

An important caveat – this pick worked really well, but it’s always possible that more than one answer will turn your pick into your target number. It’s rare, but it can happen. If it does happen, don’t panic. Pick a new number and try it on the answers that produced your target number. One of them will work and the others probably won’t. If, by some chance, more than one answer still produces your target number, rinse and repeat.

Now you try this problem with your own pick.