Another work to paint house problem

[karush]

Clarissa and Shawna, working together, can paint the exterior of a house in $$6$$ days. Clarissa by herself can complete this job in $$5$$ days less than Shawna. How long will it take Clarissa to complete the job by herself?

well if they work equally then

$\frac{1}{12}+\frac{1}{12}=\frac{1}{6}$

but I didn't know how to change this to match what the problem says.

The answer is "Clarissa 10 days by herself".

 

[karush]

how about this if $c =$ Clarissa then

$$\frac{1}{c+5}+\frac{1}{c}=\frac{1}{6}$$

 

[Dethrone]

For this problem, we'll start more generally:

Given that it takes them 6 days to paint the house, we can let $c$ and $s$ be the rate of painting in days/house. Then, one way of setting an equation (such that the units cancel) and works with the question:

$$6 \text{ days }\cdot\left( \frac{1}{c \frac{\text{days}}{\text{house}}}+\frac{1}{s \frac{\text{days}}{\text{house}}}\right) = 1 \text{ house}$$

From the second part of the question, we can formulate another equation. What is it?

how about this if $c =$ Clarissa then

$$\frac{1}{c+5}+\frac{1}{c}=\frac{1}{6}$$

That is correct because we know that $c=s-5$, then $s=c+5$ and we can plug that back into the first equation. How can we solve for $c$? (Wondering)

 

[karush]

ok from this I got by LCD $$c^2-7c-30=0$$ factoring
$$\left(c+3\right)\left(c-10\right)=0$$
so $C$ has to positive $C=10$ days I saw some other solutions to this on the internet but MHB is really the best place to be. some solutions elsewhere really got messy with wrong answers