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:
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.