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, 2013

Algebra Difficulty 2.7 Junior Find the answer Canada

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?

Pick one

Solution

In the diagram shown, the 31 identical toothpicks used in the $1\times10$ grid are separated into 2 sections.

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The top section, 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 $2\times10$ and $3\times10$ grids are similarly separated into top and bottom sections, as shown.

[[IMAGE1]]

We observe that the $1\times10$ grid consists of 1 top section and 1 bottom.

The $2\times10$ grid consists of 2 top sections and 1 bottom. The $3\times10$ grid consists of 3 top sections and 1 bottom.

Continuing in this way, a grid of size $n\times10$ will consist of $n$ top sections and 1 bottom section, for any positive integer $n$ .

So then a grid of size $43\times10$ consists of 43 top sections and 1 bottom section.

Each top section is made from 21 toothpicks and each bottom section is made from 10 toothpicks.
Thus, the total number of toothpicks in a $43\times10$ grid is $(43\times21)+(1\times10)$ or $903+10$ or 913.

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