Question

Carl says that there will be less waste if shorter lengths if pieces are cut, no matter which long board is used. Loretta disagrees. Can Loretta show that Car's statement is false using two 6-foot boards, cutting one into as many 5/9-foot lengths as possible and cutting the other into many 3/4-foot lengths as possible? Explain.

Answer 1

Loretta can indeed demonstrate that Carl’s statement is false using two 6-foot boards. Let’s analyze the situation:

Board 1: Cut into as many 5/9-foot lengths as possible.

Board 2: Cut into as many 3/4-foot lengths as possible.

Now, let’s calculate the total waste for each board:

Board 1:

Length of each piece: 5/9 feet

Number of pieces from a 6-foot board: (\frac{6}{5/9} = \frac{6}{5} \cdot \frac{9}{1} = 10.8)

Since we can’t have a fraction of a piece, we’ll take 10 pieces.

Total length used: (10 \cdot \frac{5}{9} = \frac{50}{9} \approx 5.56) feet

Remaining waste: (6 - \frac{50}{9} = \frac{44}{9} \approx 4.89) feet

Board 2:

Length of each piece: 3/4 feet

Number of pieces from a 6-foot board: (\frac{6}{3/4} = \frac{6}{3} \cdot \frac{4}{1} = 8)

Total length used: (8 \cdot \frac{3}{4} = 6) feet

Remaining waste: (6 - 6 = 0) feet (no waste)

Loretta can clearly see that Board 2 results in no waste, while Board 1 leaves a significant amount of waste. Therefore, Carl’s statement is false. Shorter lengths do not necessarily lead to less waste; it depends on the specific measurements and how they fit together.

Answer 2

Loretta can show that Carl's statement is false using two 6-foot boards by comparing the waste generated from cutting 5/9-foot lengths and 3/4-foot lengths.

Step-by-step explanation:

Loretta can show that Carl's statement is false using two 6-foot boards. Let's consider cutting one board into as many 5/9-foot lengths as possible. Since 1 foot is equivalent to 9/9 feet, we can calculate the number of 5/9-foot lengths that can be cut from a 6-foot board:

6 feet × (9/9) × (5/9) = 30/9 feet = 3 3/9 feet = 3 1/3 feet

Now, let's consider cutting the other board into as many 3/4-foot lengths as possible:

6 feet × (9/9) × (3/4) = 54/9 feet = 6 feet

From the calculations above, it is evident that the shorter lengths of 3/4 feet result in less waste compared to the lengths of 5/9 feet. Therefore, Carl's statement is false.

Complete question is:

Loretta is responsible for cutting the small lengths of wood needed for a particular project from long boards. To reduce waste, Loretta must carefully choose the long board that will result in the least amount remaining after she makes the required cuts. The long boards come in lengths of 6, 7, 9, and 10 feet.

1). Carl says that there will be less waste if shorter lengths of pieces are cut, no matter which long board is used. Loretta disagrees. Can Loretta show that Carl's statement is false using two 6-foot boards, cutting one into as many 5/9- foot lengths as possible and cutting the other into many 3/4- foot lengths as possible? Explain.

2). Loretta's boss asks her to cut at least 5 pieces of wood that are 2/3 foot long and some that are 5/7 foot long. He says that she should be able to complete the task with only one long board and with no waste. Which length of board should she choose? How many pieces of length will she be able to cut from the board?