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Going mental with math.

I'm home alone while everyone else goes to watch a Canuck's hockey game. In contemplating on what to do for the evening, I sit on the front steps of the house to enjoy the sunshine on this first nice warm(ish) day of the year. Saw a bee on some spring primulas and pulled out my phone to take a picture to post on Facebook. I decided to check out what else is happening in Facebook while I'm there.

A math blogger I follow occasionally posts mental math problems. Usually I look at them for a few seconds, see something that makes the question simple, and figure it out fairly quickly. Expecting this one to be similar I took a look at it:

Mental Math problem

And I looked at it, and looked at it. And I said to myself "I'm supposed to do this in my head?"

I'm looking for a trick, but nothing is coming to mind immediately, because it's all addition I can't cancel out any common factors. I tried doing some grouping on top, but nothing seems gained by doing that. I start doing straight forward evaluation. 2^1=2, 3^2=9 so 2+9=11. 4^3=64, and ugh! Who wants to start adding those numbers together and keeping them in their head?

So I turn to the denominator. I notice that 2+5=7 and 3+4 also equals 7, so the total on the bottom is 14. 14???? It comes to the prime factorization of 2x7, and I can't see how that's going to help in the slightest bit. Am I missing something?

I considered that this may not be a "mental" math problem after all. I took a look at the comments, the orginal poster wrote: "This one caught me!" So she had troubles with it too. Looking at the comments, someone wrote "50". Talk about spoiler alert! Don't just tell us the answer! But I've come to learn in math that having the answer is not the same as finding the solution. Something in knowing the answer made me think that the question can't actually be all that hard and to go back and take a look at it again.

This time, I decided I'm going to start at the largest number, namely 5^4. Despite being a big number, it's pretty easy to calculate mentally. I split it into (5^2) x (5^2). 5^2=25, and 25x25=625. I think it's safe to say that's pretty common knowledge (well, if it's not, it should be. I recommend that students memorize perfect squares. There's something comforting in the knowledge that the answer to 25^2 ends in 25, which makes it memorable to me, but I digress...)

So, now I have 625, and adding 11 to it brings the last 2 digits up to 36, which I right away realize if I add 64 to it I get an even 100 number. For some reason I ended up with 300, which I'm pretty sure doesn't divide evenly by 14.

Calculating the top again, starting with 5^4 (oh, haha, it's SIX HUNDRED and 25), adding the rest of the numbers bring the total on the top to 700, remembering that the bottom is 14, 70/14=5, so clearly 700/14=50. My answer matches the other persons! YEAH!

It's funny how the brain works, and it's interesting the times when the answer comes in stages, so I can see my mental processing steps to share. Often the answer comes so fast, it's just like "I see it" then I can't break it down to explain each step of how I arrived at the answer. It could be just doing 2 hours of tutoring a math student had mentally fatigued me a little to slow down my thought process as well.

So my take away tip: when presented with a problem, start with the largest/hardest part, and see if you can make the smaller parts fit in to make the question simpler than it first appears.

I think I knew that already, but forgot in this instance, so now I'm going to store that away in my brain to bring out the next time I'm doing problem solving.