# 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.