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A rectangular game board is composed of identical squares ar

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A rectangular game board is composed of identical squares ar [#permalink] New post 17 Feb 2017, 02:33
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55% (02:00) correct 44% (01:39) wrong based on 47 sessions


A rectangular game board is composed of identical squares arranged in a rectangular 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 4th row nor in the 7th column?

A) \(r^2\) — r

B) \(r^2\) — 1

C) \(r^2\)

D) \(r^2\) + 1

E) \(r^2\) + r
[Reveal] Spoiler: OA

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Re: A rectangular game board is composed of identical squares ar [#permalink] New post 17 Feb 2017, 11:08
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Explanation

If we were just counting all of the squares in the board game we could use multiplication: r rows times r+1 columns.

This would give us \(r^2+r\) squares.

If we take one row out (doesn't matter if its the first, second, third, etc) then we would have r-1 rows times r+1 columns.

If we also then take a column out, we would have r-1 rows times r columns.

So the total number of squares without a column and without a row is \(r*(r-1)=r^2-r\).

So option A is correct.
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Re: A rectangular game board is composed of identical squares ar [#permalink] New post 07 Mar 2018, 10:20
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what if we calculate all the cells in total first likeso r*(r +1) = r^2 + r

and then calculate the total cells missing likeso r + r + 1 = 2r + 1

and then substract one value from another

likeso r^2 + r - 2r - 1 = r^2 -r -1 here we have one extra one. Why is that?
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Re: A rectangular game board is composed of identical squares ar [#permalink] New post 07 Mar 2018, 10:21
nevermind... got it
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Re: A rectangular game board is composed of identical squares ar [#permalink] New post 28 Mar 2019, 15:01
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Carcass wrote:


A rectangular game board is composed of identical squares arranged in a rectangular 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 4th row nor in the 7th column?

A) \(r^2\) — r

B) \(r^2\) — 1

C) \(r^2\)

D) \(r^2\) + 1

E) \(r^2\) + r


NOTE: After creating the graphics, I see that I used n instead of r.
Please forgive me! :-D


Let's start with an n by n+1 grid
Image
The number of squares = (n)(n + 1) = n² + n

In the 4th row, we can see there are n+1 squares
Image

In the 7th column, there are n squares.
Image
HOWEVER one of the squares is already shaded blue.
So, there are n-1 red squares

TOTAL number of UNSHADED squares = n² + n - (n + 1) - (n - 1)
= n² - n

Answer: A

Cheers,
Brent
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Re: A rectangular game board is composed of identical squares ar   [#permalink] 28 Mar 2019, 15:01
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