MHB Another work to paint house problem

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Clarissa and Shawna can paint a house together in 6 days, with Clarissa taking 5 days less than Shawna to complete the job alone. The equation derived from their combined work rate leads to a quadratic equation, which simplifies to find Clarissa's solo work time. Solving the equation reveals that Clarissa can paint the house by herself in 10 days. The discussion emphasizes the clarity and correctness of the solution compared to other sources. Ultimately, the conclusion is that Clarissa's time to complete the job alone is 10 days.
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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".
 
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how about this if $c =$ Clarissa then

$$\frac{1}{c+5}+\frac{1}{c}=\frac{1}{6}$$
 
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?
karush said:
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)
 
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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
 
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