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