Another work to paint house problem

[karush]

Clarissa and Shawna, working together, can paint the exterior of a house in \(\displaystyle 6\) days. Clarissa by herself can complete this job in \(\displaystyle 5\) 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

 

[CellCoree]

To solve this problem, we need to use the concept of "work rate". Work rate is the amount of work that can be completed in a certain amount of time. In this case, we can say that Clarissa's work rate is 1 job per 5 days and Shawna's work rate is 1 job per 6 days.

We can use the formula: Work rate = Amount of work / Time to complete the work

Let's say the amount of work is 1 (painting the exterior of the house). We can set up the equation as:

Clarissa's work rate = 1 job / 5 days
Shawna's work rate = 1 job / 6 days
Together, their work rate = 1 job / 6 days

Now, we can use the formula for working together: Work rate together = Sum of individual work rates

1/6 = 1/5 + 1/x

where x is the number of days it takes for Clarissa to complete the job by herself.

Solving for x, we get x = 10 days.

Therefore, it will take Clarissa 10 days to complete the job by herself.