At a car show, a salesman earned commissions totaling $1,500 over three days. On the second day, he earned half of what he earned on the first day, and on the third day, he earned one-third of what he earned on the second day. How much did he earn each day?

1. Determine the gains for each day as unknowns:
x (primer día), \frac{x}{2} (segundo día), \frac{x}{6} (tercer día)

2. Set up the total equation:
x + \frac{x}{2} + \frac{x}{6} = 1500

3. Combine the terms into a single fraction:
\frac{6x + 3x + x}{6} = 1500

4. Simplify and solve for x :
\frac{10x}{6} = 1500
10x = 9000
x = 900

5. Calculate the earnings for the second and third days:
\\ Segundo día: \frac{x}{2} = \frac{900}{2} = 450
\\ Tercer día: \frac{450}{3} = 150

So, the earnings each day are:

\\ Primer día: 900
\\ Segundo día: 450
\\ Tercer día: 150