Brett usually takes 50 minutes to groom the horses. After working 10 minutes he was joined by Kathy and they finished the grooming in 15 minutes. How long would it have taken Kathy working alone?

Question

Abigail Allen · Accepted Answer

#30\ ext{minutes}# Let #x# minutes be the time taken alone by Kathy to groom the horses then Brett completed his work in ten minutes. #=10/50=1/5# The work left after Brett has worked for #10# minutes #=1-1/5=4/5# Now, the left work #4/5# will be done by Kathy & Brett together in #15# minutes hence we have #15(1/50+1/x)=4/5# #15(\frac{50+x}{50x})=4/5# #x=30# hence Kathy alone takes #30# minutes to groom horses

Eva Abramson · Answer

undefined Let x be the time it would have taken Kathy working alone.
Brett's rate of grooming is 1/50 per minute.
Brett's work in 10 minutes is 10/50 = 1/5.
Kathy's rate of grooming is 1/x per minute.
Combined, their rate is (1/50 + 1/x) per minute.
Together they worked for 15 minutes, so (1/50 + 1/x) * 15 = 1.
Solve for x: 15/50 + 15/x = 1.
Multiply through by 50x to clear the denominator: 15x + 750 = 50x.
Subtract 15x from both sides: 750 = 35x.
Divide both sides by 35: x = 750/35 = 21.43 minutes.
Kathy would have taken approximately 21.43 minutes working alone.

Kennedy Adams · Answer

undefined Let's denote the time it takes Kathy to groom the horses alone as $ K $ minutes.

Brett takes 50 minutes to groom the horses alone, and they worked together for $ 15 $ minutes. So, in the $ 15 $ minutes they worked together, Brett completed $ \frac{15}{50} $ of the work, which is $ \frac{3}{10} $ of the total work.

Since Kathy joined after $ 10 $ minutes, she only worked for $ 5 $ minutes (out of the $ 15 $ minutes they worked together). So, in the $ 15 $ minutes they worked together, Kathy completed $ \frac{5}{K} $ of the work.

Combining their efforts, we have:

$$ \frac{3}{10} + \frac{5}{K} = 1 $$

Now, we can solve for $ K $:

$$ \frac{3}{10} + \frac{5}{K} = 1 $$

$$ \frac{5}{K} = 1 - \frac{3}{10} $$

$$ \frac{5}{K} = \frac{10}{10} - \frac{3}{10} $$

$$ \frac{5}{K} = \frac{7}{10} $$

$$ K = \frac{10 \times 5}{7} $$

$$ K = \frac{50}{7} $$

$$ K \approx 7.14 $$

So, it would have taken Kathy approximately 7.14 minutes to groom the horses alone.