Answer
Henry: $4\frac{1}{2}$ hours
Irene: $3$ hours
Work Step by Step
Let $t$ h be the time it takes Henry to do the Job.
It means that the time it takes Irene to do the Job is $(t-1\frac{1}{2})$ h.
The equation corresponding the problem is given by:
$\frac{1}{t_{Henry}}+\frac{1}{t_{Irene}}=\frac{1}{t_{Together}}$
$\frac{1}{t}+\frac{1}{t-1\frac{1}{2}}=\frac{1}{1\frac{48}{60}}$
Solve for $t$:
$\frac{1}{t}+\frac{1}{t-\frac{3}{2}}=\frac{1}{\frac{9}{5}}$
$\frac{1}{t}+\frac{2}{2t-3}=\frac{5}{9}$ (Multiply by $9t(2t-3)$)
$9(2t-3)+18t=5t(2t-3)$ (Simplify)
$18t-27+18t=10t^2-15t$
$10t^2-51t+27=0$ (Factorize)
$(2t-9)(5t-3)=0$
$t=\frac{9}{2}$ or $t=\frac{3}{5}$
$t=4\frac{1}{2}$ or $t=\frac{3}{5}$
Let us verify both.
For $t=\frac{3}{5}$, $t_{Irine}=\frac{3}{5}-1\frac{1}{2}<0$ (Impossible).
For $t=4\frac{1}{2}$, $t_{Irine}=4\frac{1}{2}-1\frac{1}{2}=3$.
So, the time it takes Henry and Irene to wash the windows is $4\frac{1}{2}$ h and $3$ h, respectively.
