Thoughts on the Accidental Fraction Brainbuster

I really enjoyed reading a recent article on Math With Bad Drawings centered on solving the following problem without a calculator:

I won’t repeat the whole post here, but it’s an excellent exercise in numeracy, or developing intuitive understanding of numbers without necessarily doing a ton of computations. It’s also a fun exercise to see how much we can figure out without resorting to plugging into a calculator. I highly recommend reading it.

When I saw this problem, my first reflex wasn’t the technique used in the post. Instead, I thought to try the logic that follows. I don’t claim that this is a better way of solving the problem than the original solution linked above. But I do think that this alternative solution, in its own way, also encourages numeracy as well as what we can quickly determine without using a calculator.

Let’s get a common denominator for the two fractions:

\displaystyle \frac{3997 \times 5001}{4001 \times 5001} \qquad and \displaystyle \qquad \frac{4001 \times 4996}{4001 \times 5001}.

Since the denominators are the same, there is no need to actually compute 4001 \times 5001. Instead, the larger fraction can be determined by figuring out which numerator is largest. At first glance, that looks like a lot of work without a calculator! However, the numerators can both be expanded by cleverly using the distributive law:

3997 \times 5001 = (4000-3)(5000+1) = 4000\times 5000 + 4000 - 3 \times 5000 - 3,

4001 \times 4996 = (4000+1)(5000-4) = 4000\times 5000 - 4 \times 4000 + 5000 - 4.

We can figure out which one is bigger without a calculator — or even directly figuring out each product.

  • Each contains 4000 \times 5000, so we can ignore this common term in both expressions.
  • Also, 4000 - 3\times 5000 and 5000 - 4 \times 4000 are both equal to -11,000, and so we can ignore the middle two terms of both expressions.
  • The only difference is that there’s a -3 on the top line and a -4 on the bottom line.

Therefore, the first numerator is the larger one, and so \displaystyle \frac{3997}{4001} is the larger fraction.

Once again, I really like the original question as a creative question that initially looks intractable that is nevertheless within the grasp of middle-school students. Also, I reiterate that I don’t claim that the above is a superior method, as I really like the method suggested in the original post. Instead, I humbly offer this alternate solution that encourages the development of numeracy.

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1 Comment

  1. Mike T

     /  April 17, 2016

    For some reason, I always feel pleased when I realize I don’t actually have to compute a denominator or otherwise ugly looking term. Maybe this is why I got such a kick out of Bayesian parameter estimation in 4650.


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