GRE Math Problem

[viravudhlim]

A rectangular game board is composed of identical squares arranged in array of r rows and r+1 columns. The r rows are numbered from 1 through r, and the r+1 columns are numbered from 1 through r+1. If r>10, which of the following represents the number of squares on the board that are neither in the fourth row nor in the 7th column?

r^2 -r
r^2 -1
r^2
r^2 +1
r^2 + r

The answer is r^2 + 1 however I have no idea how to deduce the answer. Please help!

[tommywallach]

Hey Vira,

There are a few different ways to do this, both algebraic and plugging in. Let's start with the logical algebra way. I wish I could put pictures in here, but it's a nightmare, so I'll just have to ask you to picture it.

Imagine the rectangular board. Even though they say r > 10, let's make a smaller board to help us picture it. If the board is 4x5, there would be 20 squares. If we then wanted to imagine how many squares there are without a 3rd row or a 4th column (the actual row # and column # we pick are irrelevant, because they end up getting rid of the same number of squares), how many squares would we lose? Well, 4 from the row we lost, and 5 from the column we lost. EXCEPT there's one DUPLICATE (when we lose the row, we actually lose one square that was in the 5th column). So the actual number of squares we lost is (4 + 5) - 1, or 8. So if the box originally had 20 squares, there are 12 left.

[NOTE: At this point, we could use plugging in to see the answer. In this example, r = 4, and the answer was 12. Which answer choice gets us 12? r^2 - r = 16 - 4 = 12. That's the answer. You must have had the wrong answer from Powerprep!]

Now let's consider what happened algebraically. Whatever r is, we're going to lose that many squares when we lost that row. Then whatever r+1 is, we're going to lose that many squares when we lose the column. Only there will ALWAYS be one square of overlap. So we always lose r + r + 1 - 1 squares, or 2r. If our square originally has r (r + 1) = r^2 + r squares, we're going to have r ^2 + r - 2r squares when we remove a row and a column, which = r ^ 2 - r.

Hope that helps!

[nareshchowdary28]

There is simple way to find this just the presence of mind is important.

please forgot about the columns and rows initially.
put the length and widths of a rectangle first.

L = r, W = r+1

find the area = L*W = r(r+1) = r^2 +r
but among all the area he doesn't want 4th row and 7th column i.e
he wants the area without one column and one row. so the lenght and width will become like below

L= r-1, W = r

now Area = L*W = (r-1)r = r^2-r

which is correct answer.

hope that helps.

[tommywallach]

Great method from Naresh! I'm always a fan of theoretical stuff, but for those of you who can't do it that way, don't be afraid to try out some numbers/drawing some pictures. Always good to have something physical to manipulate.

But if you've got the algebraic/theoretical chops, Naresh's method is great!

-t