Question

Joselyn is a manager at a sign painting company. She has three painters, Allen, Brianne, and Charles. Allen can complete a large project in 16 hours. Brianne can complete the same sized project in 18 hours. Charles is new, so no one knows how long it will take him.

Joselyn assigns them all a large project to complete together. Explain to Joselyn how this project can tell her how long it would take Charles if he worked by himself. Use complete sentences.

Answer 1

Let c be the number of hours that Charles would take to complete a large project if he worked by himself, and let t be the number of hours that the three painters would take to complete a large project.
The fraction of the large project that each painter working alone would complete in one hour is as follows:
Allen: 1/16
Brianne: 1/18
Charles: 1/c
The fraction of the large project that would be completed by all three painters working together is 1/t.
The following equation can now be written:
\[\large \frac{1}{16}+\frac{1}{18}+\frac{1}{c}=\frac{1}{t}\ ......(1)\]
When the large project with all three painters is completed, the value of t will be known. This value can be substituted for t in equation (1) and the equation can be solved to find the value of c, which was assigned to be the time Charles would take to complete a large project by himself.