You are given $_5p_{40}=0.8, _{10}p_{45}=0.6,_{10}p_{55}=0.4$. Find the probability that $(40)$ will die between between the ages of $55$ and $65$.
$\frac{l_{55}-l_{65}}{l_{40}}=_{15}p_{40}-_{25}p_{40}$
$=(_5p_{40})({10}p_{45})-(_5p_{40})(_{10+5}p_{45})$
How d0 I simplify $(_{10+5}p_{45})$?
We write the desired probability as $$\begin{align*} {}_{15|10} q_{40} &= ({}_5 p_{40}) ({}_{10|10} q_{45}) \\ &= ({}_5 p_{40}) ({}_{10} p_{45})({}_{10} q_{55}) \\ &= ({}_5 p_{40}) ({}_{10} p_{45})(1 - {}_{10} p_{55}) \\ &= (0.8)(0.6)(1-0.4) \\ &= 0.288. \end{align*}$$
As for your calculation, it is worth observing that $${}_{25} p_{40} = ({}_5 p_{40})({}_{\color{red}{20}} p_{45}) = ({}_5 p_{40})({}_{10} p_{45})({}_{10} p_{55}).$$
