The total area of a set of different squares, arranged from smallest to largest, is $35 \mathrm{~km}^{2}$. The smallest square has a side length of $500 \mathrm{~m}$. The next larger square has a side length of $1000 \mathrm{~m}$. In the same way, each successive square has its side length increased by $500 \mathrm{~m}$. What is the total number of squares?
(A) 5
(B) 6
(C) 7
(D) 8
(E) 9
The total area of a set of different squares, arranged from smallest to largest, is $35 \mathrm{~km}^{2}$. The smallest square has a side length of $500 \mathrm{~m}$. The next larger square has a side length of $1000 \mathrm{~m}$. In the same way, each successive square has its side length increased by $500 \mathrm{~m}$. What is the total number of squares?
(A) 5
(B) 6
(C) 7
(D) 8
(E) 9

## Solution

We complete the following chart, one row at a time, until 35 appears in the third column.

| Number of  the square | Length of  the square | Area of  the square | Cumulative sum  of areas |
| :---: | :---: | :---: | :---: |
| 1 | $0.5 \mathrm{~km}$ | $0.25 \mathrm{~km}^{2}$ | $0.25 \mathrm{~km}^{2}$ |
| 2 | $1.0 \mathrm{~km}$ | $1.00 \mathrm{~km}^{2}$ | $1.25 \mathrm{~km}^{2}$ |
| 3 | $1.5 \mathrm{~km}$ | $2.25 \mathrm{~km}^{2}$ | $3.50 \mathrm{~km}^{2}$ |
| 4 | $2.0 \mathrm{~km}$ | $4.00 \mathrm{~km}^{2}$ | $7.50 \mathrm{~km}^{2}$ |
| 5 | $2.5 \mathrm{~km}$ | $6.25 \mathrm{~km}^{2}$ | $13.75 \mathrm{~km}^{2}$ |
| 6 | $3.0 \mathrm{~km}$ | $9.00 \mathrm{~km}^{2}$ | $22.75 \mathrm{~km}^{2}$ |
| 7 | $3.5 \mathrm{~km}$ | $12.25 \mathrm{~km}^{2}$ | $35.00 \mathrm{~km}^{2}$ |

Since there are seven rows, we conclude that there are seven squares.

ANswer: (C)