You may have incorrectly tried to find rates by dividing time by work, instead of the other way around. The old machine's rate is $\displaystyle\frac{{24}}{{3}}\usepackage{cmbright}\Huge\$ or 8 widgets per hour. The new machine's rate is $\displaystyle\frac{{48}}{{4}}\usepackage{cmbright}\Huge\$ or 12 widgets per hour. Their combined rate equals 120 widgets divided by the time it takes.