Hello, everyone: I had a student email me a pretty good question just now, and thought that everyone could benefit from the explanation:

QUESTION:

A rectangular tiled floor is composed of 70 square tiles. The rectangular floor is being rearranged so that 2 tiles will be removed from each row of tiles and 4 more rows of tiles will be added. After the change in layout, the floor will still have 70 tiles, and it will still be a rectangle. How many rows were in the tile floor before the change in layout?

A. 4
B.6
C. 10
D. 14
E. 28

What is the best way to approach this problem? I figured the best first step is to find all the ways to get 70 (i.e. 70 x1, 35 x2 etc.) but I'm not sure what to do after that.


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EXPLANATION:

Good question – and your approach would probably work pretty well. The answer choices give you a pretty good clue, too – you only really need to try the row numbers given to you in the answer choices (since that’s what you’re solving for). So your options really are:

Rows * Tiles = 70
4 * (it won’t work…70 isn’t divisible by 4, so A is out)
6 * (it won’t work…70 isn’t divisible by 6, so B is out)
10 * 7
14 * 5
28 * (it won’t work…70 isn’t divisible by 28, so B is out)

So you’re down to C and D. Now, try taking away 2 tiles per row and adding 4 rows and see if you stay at 70 tiles:

C: 14 rows * 5 tiles per row = 70. C works, so it’s looking pretty good (and there won’t be two right answers, so you can probably stop here)
D: 9 rows * 12 tiles per row = 108. D doesn’t work, so we know that C must be correct.

You could also do this algebraically if you wanted to. We know the r * t (rows * tiles) = 70, and that (r+4)(t-2) = 70.

Then, you can substitute using the first equation: rt = 70, so t = 70/r

Substitute in the second to get: (r + 4)(70/r – 2) = 70

If you FOIL that out you can set up a quadratic to solve for r, and although you’ll get two values, one will be negative, and you can’t have a negative number of rows, so only the positive value will work. If you don’t love quadratics, you could also stop here to plug in numbers – and since one of your terms is 70/r, you need something divisible by 70, so that limits the choices you’d plug in and streamlines your work, too.