Question

Ironing Lenore can do the ironing for her family’s business in h hours. Her daughter would take h + 2 hours to get the ironing done. If Lenore and her daughter work together, using 2 irons, the number of hours it would take them to do all the ironing is $\frac{1}{{\displaystyle \frac{1}{h}}+{\displaystyle \frac{1}{h+2}}}$.

1. Simplify the complex fraction $\frac{1}{{\displaystyle \frac{1}{h}+\frac{1}{h+2}}}$.
2. Find the number of hours it would take Lenore and her daughter, working together, to get the ironing done if$h=4$.

Short answer

1. The simplified form of the fraction $\frac{1}{{\displaystyle \frac{1}{h}}+{\displaystyle \frac{1}{h+2}}}$is, $\frac{h\left(h+2\right)}{2h+2}$.
2. The number of hours it would take Lenore and her daughter, working together, to get the ironing done if$h=4$is,$\frac{12}{5}$.

Step-by-step solution

Part a Step 1. Given information

$\frac{1}{{\displaystyle \frac{1}{h}+\frac{1}{h+2}}}$.

We need to simplify the given complex fraction.

Part a Step 2. Find the LCD of the denominator and then simplify it.

$$\begin{gathered} \frac{1}{{\displaystyle \frac{1}{h}+\frac{1}{h+2}}}=\frac{1}{{\displaystyle \frac{h+2}{h\left(h+2\right)}+\frac{h}{h(h+2)}}} \\ =\frac{1}{{\displaystyle \frac{2h+2}{h\left(h+2\right)}}} \\ =1÷\frac{2h+2}{h\left(h+2\right)} \\ =1\times \frac{h(h+2)}{2h+2} \\ =\frac{h(h+2)}{2h+2} \end{gathered}$$

Part b Step 1. Given information 

$h=4$.

From the part $\left(a\right)$the obtained simplified expression is,$\frac{h(h+2)}{2h+2}$.

Part b Step 2. Substitute the given value in the obtained simplified expression and then simplify it.

$$\begin{gathered} \frac{h(h+2)}{2h+2}=\frac{4(4+2)}{2\left(4\right)+2} \\ =\frac{4\left(6\right)}{8+2} \\ =\frac{24}{10} \\ =\frac{12}{5} \end{gathered}$$