Question

As an apprentice at your uncle's cabinet shop, you are cutting out pieces of wood for a dresser. You need pieces that are \(5 \frac{1}{4}\) inches long. Each saw cut wastes \(\frac{1}{8}\) inch of wood. The board you have is 8 feet long. How many pieces can you make from this board?

Short answer

Answer: 17 pieces.

Step-by-step solution

Convert 8 feet to inches

To convert the length of the board from feet to inches, we multiply the length in feet (8 feet) by the conversion factor (12 inches/foot). So, the length of the board in inches is 8 * 12 inches. \[8 \text{ feet} \times 12 \frac{\text{inches}}{\text{foot}} = 96 \text{ inches}\]

Find the total length used by each piece including the wasted part

Now, we need to find out the total length used by each piece when it is cut, including the wasted part. As each piece is \(5 \frac{1}{4}\) inches long, and each saw cut wastes \(\frac{1}{8}\) inch of wood, the total length used by each piece is: \[5 \frac{1}{4} \text{ inches} + \frac{1}{8} \text{ inch}\] To add fractions, we need a common denominator, which in this case is 8. So, we can rewrite this expression as: \[\frac{21}{4} \text{ inches} + \frac{1}{8} \text{ inch} = \frac{21}{4} \text{ inches} + \frac{1}{8} \text{ inch}\]

Add the fractions

Now, let's add the fractions to get the total length used by each piece: \[\frac{21}{4} \text{ inches} + \frac{1}{8} \text{ inch} = \frac{21 \times 2 + 1}{8} \text{ inches} = \frac{43}{8} \text{ inches}\] So, each piece uses a total of \(\frac{43}{8}\) inches of the board, including the wasted part from the saw cut.

Find the number of pieces

To find out how many pieces can be made from the 96-inch long board, we will divide the total length of the board by the length used by each piece: \[ \frac{96 \text{ inches}}{\frac{43}{8} \text{ inches}}\] In order to divide by a fraction, we will multiply by its reciprocal: \[96 \text{ inches} \times \frac{8}{43}\] \[= \frac{96 \times 8}{43}\] \[= \frac{768}{43}\] Now, we will divide 768 by 43, which gives us 17 with a remainder of 41. The remainder means that there will be some leftover wood after cutting the pieces, but we cannot make another full piece from it. So, we can make 17 pieces from the 8 feet long board.