The old machine's rate is $\displaystyle\frac{{10}}{{2}}\usepackage{cmbright}\Huge\$ or 5 widgets per hour. The new machine's rate is $\displaystyle\frac{{30}}{{3}}\usepackage{cmbright}\Huge\$ or 10 widgets per hour. Their combined rate is 15 widgets per hour. So, if $\displaystyle{t}=\usepackage{cmbright}\Huge\$ the amount of time, in hours, needed for the two machines to produce 75 widgets, then:

$\displaystyle{15}=\frac{{75}}{{t}}\usepackage{cmbright}\Huge\$ , so $\displaystyle{15}{t}={75}\usepackage{cmbright}\Huge\$ , and $\displaystyle{t}={5}\usepackage{cmbright}\Huge\$