Question

Loretta is responsible for cutting 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. Loretta's boss aks her to cut at least 5 pieces of wood that are 2/3-foot and some that are 5/7-foot long. He says she should be able to complete the task with only one long board and no waste. Which legth of board should she choose? (Assume that the width of the saw cut is insignificant and type whole numbers.)

Answer 1

In this case, we'll have to carry out several steps to find the solution.

Step 01:

Data:

long boards (wood): 6, 7, 9, and 10 feet

2/3 foot ===> 5 pieces

5/7 foot ===> some pieces

Step 02:

only one long board:

5 pieces * 2/3 ft = 10/3 ft

If long board = 6 ft:

$6\text{ ft - }\frac{10}{3}ft\text{ = }\frac{18\text{ - 10}}{3}ft\text{ = }\frac{8}{3}\text{ ft}$

no waste (whole number of pieces):

$\frac{8}{3}\text{ ft / }\frac{5}{7}\text{ ft = }\frac{8\cdot7\text{ }}{5\cdot3}\text{ = }\frac{56}{15}\text{ = }3.73\text{ pieces }$

It is not the solution

If long board = 7 ft:

$7\text{ ft - }\frac{10}{3}ft\text{ = }\frac{21\text{ - 10}}{3}ft\text{ = }\frac{11}{3}\text{ ft}$

no waste (whole number of pieces):

$\frac{11}{3}\text{ ft / }\frac{5}{7}\text{ ft = }\frac{11\cdot7\text{ }}{5\cdot3}\text{ = }\frac{77}{15}=5.13\text{ pieces }$

It is not the solution

If long board = 9 ft:

$9\text{ ft - }\frac{10}{3}ft\text{ = }\frac{27\text{ - 10}}{3}ft\text{ = }\frac{17}{3}\text{ ft}$

no waste (whole number of pieces):

$\frac{17}{3}\text{ ft / }\frac{5}{7}\text{ ft = }\frac{17\cdot7\text{ }}{5\cdot3}\text{ = }\frac{119}{15}=7.93\text{ pieces }$

7.93 pieces ≅ 8 pieces

If long board = 10 ft:

$10\text{ ft - }\frac{10}{3}ft\text{ = }\frac{30\text{ - 10}}{3}ft\text{ = }\frac{20}{3}\text{ ft}$

no waste (whole number of pieces):

$\frac{20}{3}\text{ ft / }\frac{5}{7}\text{ ft = }\frac{20\cdot7\text{ }}{5\cdot3}\text{ = }\frac{140}{15}=9.33\text{ pieces }$

It is not the solution

The answer is:

long board = 9 ft