# Problem of the Week Problem B and Solution Wall Construction

## Problem

Moyo wants to build a wall in her backyard. She will build the wall out of bricks, and her measurements show that each brick is $$22.5\,$$cm long and $$5.7\,$$cm high. Moyo’s bricks on the bottom layer of the wall are to be placed lengthwise and spaced $$4.5\,$$cm apart. For example, here is the bottom layer of a wall with three bricks.

1. If she builds a wall with nine layers of bricks, what will the height of the wall be?

2. If Moyo builds a wall with six bricks on the bottom layer, how long will this bottom layer be?

3. If Moyo wanted the wall to be $$130.5 \,$$cm long, how many bricks will Moyo use in the bottom layer of the wall?

## Solution

1. The total height of the nine layers will be $$9\times 5.7=51.3\,$$cm.

2. The bottom layer, which consists of six bricks, will have six bricks and five spaces. Thus the total length will be $$(6 \times 22.5) + (5\times 4.5) = 135 + 22.5=157.5\,$$cm or $$1.575\,$$m.

3. One brick plus one space is $$22.5 + 4.5 = 27\,$$cm long. Since $$27\times 5 = 135$$, and $$135$$ is close to $$130.5$$, we guess that Moyo would use about five bricks, with four spaces between them. Indeed, this gives $$(5\times 22.5) + (4\times 4.5)=112.5+18=130.5\,$$cm, the required length.

Note: Another possible way to solve this is to notice that $$130.5\,$$cm is $$27\,$$cm less than $$157.5\,$$cm. Now, $$27\,$$cm is the length of one brick plus one space. Therefore, the bottom layer has the $$6$$ bricks in part (b) minus $$1$$ brick. Therefore, Moyo will have $$5$$ bricks on the bottom layer.