Hey guys,

Just got this question emailed to me from a student and figured everyone could benefit from thinking about it. Enjoy!

Which of the following numbers is the greatest?
A.
1876455/1876452
B.
1883449/1883446
C.
1883456/1883453
D.
1883459/1883456
E.
1883494/1883491


My response:

I love this question – wow, thanks for sending it over!

First, notice that in each of the fractions, the numerator is exactly three greater than the denominator. The differenence between numerator and denominator is always the same.

Here’s, then, where I’d employ the strategy of “prove a relationship using small numbers, then extrapolate it to larger numbers”. I’d just pick a few smaller-number fractions that follow the exact same setup, which is:

1) The numerator is larger than the denominator
2) Both numerator and denominator are positive integers
3) The difference of numerator-minus-denominator is always the same.

So let’s test that relationship with, say:

6/4

8/6

10/8

Note – these fractions all fit the exact same pattern. Numerator > Denominator, and the difference is 2 in each case. But I chose these since they’ll divide pretty nicely and show us what happens.

6/4 = 3/2 = 1.5
8/6 = 4/3 = 1.333
10/8 = 5/4 = 1.25

The smaller the numbers are in the fraction, the larger the result, so the answer should be the one with the smallest numbers. That means A, 1876455/1876452.

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Now, another way to think of it is that, in each case A through E, the remainder is 3. The answer to all of those division problems (Numerator divided by Denominator) is 1, remainder 3, which equals 1 + 3/Denominator. The smaller the denominator, the bigger that 3/Denominator add-on will be, so by this thinking, too, you’re going for the smallest numbers, leading you again to A.