Let $\displaystyle{t}=\usepackage{cmbright}\Huge\$ the time, in hours, it takes the repairman to fix one pair of shoes. Then $\displaystyle{1.5}{t}=\usepackage{cmbright}\Huge\$ the amount of time it takes the assistant. Their rates are $\displaystyle\frac{{1}}{{t}}\usepackage{cmbright}\Huge\$ and $\displaystyle\frac{{1}}{{{1.5}{t}}}\usepackage{cmbright}\Huge\$ pairs per hour.

Their combined rate is $\displaystyle\frac{{1}}{{t}}+\frac{{1}}{{{1.5}{t}}}=\frac{{1.5}}{{{1.5}{t}}}+\frac{{1}}{{{1.5}{t}}}=\frac{{2.5}}{{{1.5}{t}}}\usepackage{cmbright}\Huge\$ pairs per hour.

So $\displaystyle\frac{{2.5}}{{{1.5}{t}}}\cdot{6}={10}\usepackage{cmbright}\Huge\$ or $\displaystyle\frac{{15}}{{{1.5}{t}}}=\frac{{10}}{{1}}\usepackage{cmbright}\Huge\$

$\displaystyle{1.5}{t}\cdot\frac{{15}}{{{1.5}{t}}}=\frac{{10}}{{1}}\cdot{1.5}{t}\usepackage{cmbright}\Huge\$

$\displaystyle{15}={15}{t}\usepackage{cmbright}\Huge\$

Therefore, $\displaystyle{t}={1}\usepackage{cmbright}\Huge\$ hour.