A rectangular game board is composed of identical squares arranged in

Question

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) \(r^2 + r\)

A)  r^2 — r

B)  r^2 — 1

C)  r^2

D)  r^2 + 1

E)  r^2 + r

chetna14 · Accepted Answer

we need to find the number of squares that are neither in the 4th row nor in the 7th column.

so basically if we remove 4th row( 1 row) from r rows, and 7th column (1 column) from (r+1) columns, we will get the number of squares

hence total number of squares = (r-1) * r =  r^2 - r ,

IMO A

Talayva · Answer

Total Number of Squares = r * (r+1)= r^2 + r
Number of squares in 4th Row = r+1
Number of squares in 7th column = r.
One square is counted twice ( Square in 4th row , 7th column)
Required squares = r^2 + r -( (r+1) + r -1)= r^2 -r

Answer A

DavidTutorexamPAL · Answer

As this question asks us to count the number of squares, it is a counting problem and can usually be solved with some logic and very little calculations.
We'll look for such a solution, a Logical approach.

We'll calculate this by breaking it into easier bits
(i) total number of squares = r(r+1) = r^2 + r
(ii) squares in 4th row = r+1
(iii) squares is 7th column = r
(iv) squares in both 4th row and 7th column = 1
we need to calculate (i) -  
so our answer is r^2 + r -   = r^2 + r -2r = r^2 - r

(A) is our answer.

ScottTargetTestPrep · Answer

Solution:

First, let’s calculate the total number of squares on the board: r(r + 1) = r^2 + r

Now, we note that there are (r + 1) squares in the 4th row and r squares in the 7th column. Additionally, there is one square that has been double-counted (at the intersection of row 4 and column 7). Thus, the number of squares that we will eliminate is (r + 1) + r -1.

The result is r^2 + r - (r + 1 + r - 1) = r^2 + r - r - 1 - r + 1 = r^2 - r.

Answer: A

Fdambro294 · Answer

On the Official GMAT test day, I would probably choose smart Numbers:

Let r = 11

Let (r + 1) = 12

132 Total Squares -   =

-----*Note* remove 1 Square from Column 7 because we already counted it when we Counted Across Row R ----

132 -   = 110

Plug in Number: r = 11

Target Value: 110

-A-

(r)^2 - r = (11)^2 - 11 = 121 - 11 = 110, our Target Value

Correct Answer -A-

Algebra Method:

No. of Squares that are NEITHER in the 4th Row NOR in the 7th Column

=

(Total No. of Squares) - (No. of Square that ARE IN the 4th Row AND 7th Column)

(1st) Total No. of Squares

there are R - rows of Squares with (R + 1) Columns of Squares

(R) * (R + 1) = Total No. of Squares

(2nd) No. of Squares that ARE IN the 4th Row AND 7th Column

Going Horizontally across Row 4, there will be (R + 1) Squares to account for Each of the (R + 1) Columns

Coming down Vertically from Column 7, there will be a Total of (R) Squares -----HOWEVER, we already counted 1 of these Squares (precisely the Square located at 4-by-7) so we need to Subtract -1

(R + 1) + (R - 1) = 2R = No. of Squares that are in the 4th Row AND 7th Column

(1st) - (2nd)

(R)*(R + 1) - (2R) =

(R)^2 + R - 2R =

(R)^2 - R

-A- is the Correct Answer

Dvaishnav · Answer

Please help. I do not understand the solution fully.

The question says that rows are numbered from 1 to r and columns are numbered from 1 to r+1. It means, we have r squares in each row and r-1 squares in each column. Hence, the total number of squares should be r*(r-1). Not r*(r+1) as others have mentioned.

Just to make my point clear, the following diagram shows a simple grid in which we have rows numbered from 1 to 3 and columns numbers from 1 to 5. As you can see total number of squares are 8 (2*4), not 15 (3*5).

bumpbot · Answer

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