Earlier this evening I was poking around map.mathshell.org looking for bite-sized middle school tasks. I found a limited but great set of questions here and was inspired to ask this:

Which is closer to 1? Does it depend on n? Explain. (n-1)/n or n/(n-1) Inspired by Q2 on this: http://t.co/zUp670WyqS

— Michael Fenton (@mjfenton) April 23, 2014

Just in case Twitter messes up the formatting on whatever device you’re reading on, here’s the question again, as well as my challenge to you and a solution of my own:

# The Question

Which is closer to 1? Does it depend on n? Explain.

# My Challenge

I’m curious to know how others reason through and/or visualize this problem. If you find yourself with 5-10 spare minutes this week, scribble out an approach/solution/representation (or two) on a napkin, take a photo, and drop a link to the image in the comments. I’ll update this little ditty on Friday by embedding images in the post.

Pictures, words, algebra, numbers, proof with words, proof without words… It’s all fair game here.

# A Solution

Don’t peek until you’ve given it a go yourself (and hopefully shared your solution in the comments), but here’s my visual argument, sans words. (As in, “I’m not providing any words.” Let me know if you think it needs a comment to clarify.)

## Comments 3

Michael

My first thought (as a Calculus teacher) is that the first expression is 1 – 1/n and as n gets large this is very close to 1. The second expression can be rewritten (by long division of n/(-1+n)) as -n – n^2 – n^3 … and the graph of the rational function shows that this also nears 1 as n gets large. It also appears that the first expression converges more quickly. So, my next idea would be to simply substitute a couple of manageable n values into these expressions to get a table of values going. Looking at n =1 to n = 8 the first expression is closer for each value. Fun problem to think about. I may toss this out to my BC kids and see what they do with it!

I thought: (n-1)/n and n/(n-1) are each fractions that are one “piece” away from being 1. You’re closer to 1 when the pieces are smaller, and hence when the denominator is “larger”. At first I was just thinking about positive numbers, where (n-1)/n is one piece less than 1 and n/(n-1) is one piece more than one. And so n-1/n, by virtue of its smaller pieces, is closer to 1. But then I thought about negative numbers, and realized that the roles would be reversed. That’s how I knew that it depends.

Later on I thought about “cross multiplying” and comparing n^2 and (n-1)^2. Clearly which is larger depends on n.

@mrdardy: Your way makes me think to write n/(n-1) as n/(n-1) – 1/(n-1) + 1/(n-1) = 1 + 1/(n-1). This might be useful in comparing with (n-1)/n = 1 – 1/n.