So here go. No crunching numbers allowed. Let's solve this problem using only (gasp) logic and see how far we can get.
Well what is an average? Let's think about the average numbers of beans in these cups.
Average means an even amount in each cup, so this looks like the average below. I moved around the beans from cup to cup until there was an even number in each one. This is the concept of an average.
Now let's make sure we can add up numbers with like units for our numbers 4 1/3, 3 1/2, 3 5/6 and 4 1/6. Thinking of pizzas, how would you have to slice up a pizza so that 1/2 and 1/3 have the same sized slices?
Ah, if you slice it up into six pieces then 3 slices is one half and 2 slices is 1/3. So 1/6ths works so the numbers play nicely. If a pizza is cut up into 6 pieces and your brother eats half and your Dad eats 1/3 then they ate 1/2 + 1/3 = 3/6 + 2/6 = 5/6 of a pizza! You only get 1 lousy slice!
Now let's work on making even stacks to get our average. Here are our numbers.
Let's move a few blocks around.
Not quite there though. Let's move a few more.
Darn it! So close. There's not an even number of 1/6ths blocks to get exactly even stacks but we are almost there.
This is a close up view of the very top of the stacks of blocks. We have to divide these up evenly 4 ways. Divide them up 4 ways ... 4 ways ... that gives me an idea ...
Um, let's divide each block up 4 ways since each one has to be shared 4 ways. It's worth a shot.
So wait, dividing 1/6th of a pizza 4 ways means that we actually have 1/24th of a pizza. So each itty-bitty sliver is actually 1/24th of a block.
Ok so here we see what the very top looks like when we divide those 1/24th slivers evenly. We don't have enough to make it all the way to 4. We are actually just one small 1/24th sliver short of 4 so we have 3 23/24 as our answer.
So there you have it folks. The average of 4 1/3, 3 1/2, 3 5/6 and 4 1/6
= 3 23/24
visually.
