Question

Working together, Abe and Bianca cleaned their yard in 6 hours. Last year, Abe cleaned the yard by himself. The year before, Bianca cleaned it by herself, but took 5 hours less than Abe took. How long did Bianca take for cleaning the yard alone?

Answer 1

10 hours

Explanation:

Given that time taken to clean the garden by Abe and Bianca together = 6 hours.

So, in 1 hour both can clean 1/6 of the garden.

Let Abe can clean the garden in x hours, as Bianca took 5 hours less than Abe took, time taken by Bianca alone to clean the garden = x-5 hours.

Therefore, in 1 hour Abe can clean 1/x portion of the garden while Bianca can clean 1/(x-5) portion of the garden.

Together, Abe and Biana can clean \frac 1 x + \frac {1}{x-5} portion of garden in 1 hour.

As both can clean 1/6 portion of the garden in 1 hour together, so

`\frac 1 x + \frac {1}{x-5} = \frac 1 6 \\\\\Rightarrow \frac {2x-5}{x(x-5)}= \frac 1 6 \\\\\Rightarrow x^2 - 5x = 6(2x-5) \\\\\Rightarrow x^2 - 5x = 12x-30 \\\\\Rightarrow x^2 - 17x +30= 0 \\\\\Rightarrow x^2 - 15x-2x +30= 0 \\\\\Rightarrow x(x-15) -2(x +15)= 0 \\\\\Rightarrow (x-2)(x-15)= 0 \\\\\Rightarrow x-2=0 \;or\; x-15=0 \\\\\Rightarrow x = 2 \;or\; x=15.`

Now, if x=2, then the time taken by Bianca x-5 = 2-5=-3 hours (negative) which is not possible.

For x= 15, time taken by Bianca = x=5=15-5=10 hours.

Hence, the time taken by Bianca alone to clean the garden is 10 hours.