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:

and .

Since the denominators are the same, there is no need to actually compute . 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:

,

.

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

Each contains , so we can ignore this common term in both expressions.

Also, and are both equal to , and so we can ignore the middle two terms of both expressions.

The only difference is that there’s a on the top line and a on the bottom line.

Therefore, the first numerator is the larger one, and so 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.

I'm a Professor of Mathematics and a University Distinguished Teaching Professor at the University of North Texas. For eight years, I was co-director of Teach North Texas, UNT's program for preparing secondary teachers of mathematics and science.
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One thought on “Thoughts on the Accidental Fraction Brainbuster”

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.

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.