Question

Working together two people cut a large lawn in 11 hrs. One person can do it alone in one hour less than the other. How long would it take for the faster person to do the job.?

Answer 1

Thus, the faster person would do the job alone in 22.5 hours.

Explanation:

The faster worker can do it in x hours, therefore:

`\text{ The rate of the faster worker}=(1)/(x)`

The slower worker can do it in 1 hour less, therefore:

`\text{ The rate of the slower worker}=(1)/(x-1)`

Working together, they cut the lawn in 11 hours.

Add the rates:

`\begin{gathered} (11)/(x)+(11)/(x-1)=1 \\ \text{ Multiply all through by }(1)/(11) \\ (1)/(x)+(1)/(x-1)=(1)/(11) \end{gathered}`

We solve the equation for x:

We solve the resulting quadratic equation for x using the quadratic formula.

`\begin{gathered} x=(-b\pm√(b^2-4ac) )/(2a) \\ \text{ From the equation: }a=1,b=-23,c=11 \\ x=(-(-23)\pm√((-23)^2-4(1)(11)))/(2*1) \\ =(23\pm√(485))/(2) \\ \implies x=(23+√(485))/(2)\text{ or }x=(23-√(485))/(2) \\ x=22.5\text{ or }x=0.49 \end{gathered}`

From our values of x, if it takes the two workers 11 hours, it cannot take one of the workers just 0.49 hours.

Thus, the faster person would do the job alone in 22.5 hours.