$$\frac{1}{x}=\frac{1}{\frac{1}{5}}+\frac{1}{5}+\frac{1}{5}+\frac{1}{5}+\frac{1}{3}}$$
$x=\frac{15}{89}\approx 0.168539326$
$$15=15x\times \frac{1}{\frac{1}{5}}+15x\times \frac{1}{5}+15x\times \frac{1}{5}+15x\times \frac{1}{5}+15x\times \frac{1}{3}$$
$$15=15x\times 1\times 5+15x\times \frac{1}{5}+15x\times \frac{1}{5}+15x\times \frac{1}{5}+15x\times \frac{1}{3}$$
$$15=15x\times 5+15x\times \frac{1}{5}+15x\times \frac{1}{5}+15x\times \frac{1}{5}+15x\times \frac{1}{3}$$
$$15=75x+15x\times \frac{1}{5}+15x\times \frac{1}{5}+15x\times \frac{1}{5}+15x\times \frac{1}{3}$$
$$15=75x+\frac{15}{5}x+15x\times \frac{1}{5}+15x\times \frac{1}{5}+15x\times \frac{1}{3}$$
$$15=75x+3x+15x\times \frac{1}{5}+15x\times \frac{1}{5}+15x\times \frac{1}{3}$$
$$15=78x+15x\times \frac{1}{5}+15x\times \frac{1}{5}+15x\times \frac{1}{3}$$
$$15=78x+\frac{15}{5}x+15x\times \frac{1}{5}+15x\times \frac{1}{3}$$
$$15=78x+3x+15x\times \frac{1}{5}+15x\times \frac{1}{3}$$
$$15=81x+15x\times \frac{1}{5}+15x\times \frac{1}{3}$$
$$15=81x+\frac{15}{5}x+15x\times \frac{1}{3}$$
$$15=81x+3x+15x\times \frac{1}{3}$$
$$15=84x+15x\times \frac{1}{3}$$
$$15=84x+\frac{15}{3}x$$
$$15=84x+5x$$
$$15=89x$$
$$89x=15$$
$$x=\frac{15}{89}$$
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