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.

Can someone please help me understand this. I know the rational equation would 1/16 + 1/18 + 1/x = 1,but I don't know what to do now

Answer 1

I think of these as
rate * time = 1 job
for example, if you work at a rate of 1 job in 18 hours 1 job/ 18 hours
\[ \frac{1}{18} \cdot t= 1 \]
and multiply both sides by 18 you get
\[ t= 18 \ hours\]
just as you should. (that says it will take you 18 hours)

Answer 2

Yes, I get that so far.

Answer 3

in your problem, all three working together will take amount "t"
\[ \left(\frac{1}{18} + \frac{1}{16}+ \frac{1}{x} \right) t = 1 \]

Answer 4

if we simplify that we get
\[ \left( \frac{17}{144} + \frac{1}{x}\right) t = 1 \\
\]

if we are told what "t" is , then we can replace t with that number and solve for x

Answer 5

Right, but we don't know what t is...?

Answer 6

So is that all I need to say?

Answer 7

the question says
*** Explain to Joselyn how this project can tell her how long it would take Charles working alone****

the explanation is we use the time it takes (once we learn what it is)
and use that in the above equation to solve for x

Answer 8

Oooooooh!!! NOW I get it!!

Answer 9

if you want, we can solve the equation (get x alone on one side)
\[ \left( \frac{17}{144} + \frac{1}{x}\right) t = 1 \\ \left( \frac{17}{144} + \frac{1}{x}\right) = \frac{1}{t}\\
\frac{1}{x}= \frac{1}{t}- \frac{17}{144} \\
\frac{1}{x}= \frac{144-17t}{144t} \\
x= \frac{144t}{144-17t}
\]

Answer 10

Thank you so so much, I completely understand now!

Answer 11

yw