Toothpicks are used to make rectangular grids, as shown. Note that a total of 31 identical toothpicks are used in the $1 \times 10$ grid. How many toothpicks are used in a $43 \times 10$ grid?
(A) 913
(B) 860
(C) 871
(D) 903
(E) 946
![](https://cdn.mathpix.com/cropped/2024_04_20_46ead6524a8d61e21c51g-045.jpg?height=438&width=426&top_left_y=2079&top_left_x=1294)

Part C: Each correct answer is worth 8.
In the diagram shown, the 31 identical toothpicks used in the

$1 \times 10$ grid are separated into 2 sections.

The top section, $\mathrm{T}$, is made from 11 vertical toothpicks and 10 horizontal toothpicks, or 21 toothpicks in total.

The bottom section, B, is made of 10 horizontal toothpicks.

The $2 \times 10$ and $3 \times 10$ grids are similarly separated into top and bottom sections, as shown.

We observe that the $1 \times 10$ grid consists of 1 top section and 1 bottom.

The $2 \times 10$ grid consists of 2 top sections and 1 bottom.

The $3 \times 10$ grid consists of 3 top sections and 1 bottom.

Continuing in this way, a grid of size $n \times 10$ will consist of $n$ top sections and 1 bottom section, for any positive integer $n$. So then a grid of size $43 \times 10$ consists of 43 top sections and 1 bottom section.

Each top section is made from 21 toothpicks and each bottom

![](https://cdn.mathpix.com/cropped/2024_04_20_748fdafcde2a47412608g-208.jpg?height=388&width=504&top_left_y=1709&top_left_x=1431)
section is made from 10 toothpicks.

Thus, the total number of toothpicks in a $43 \times 10$ grid is $(43 \times 21)+(1 \times 10)$ or $903+10$ or 913 .

![](https://cdn.mathpix.com/cropped/2024_04_20_748fdafcde2a47412608g-208.jpg?height=263&width=507&top_left_y=2189&top_left_x=1427)

ANSWER: (A)