Suppose the fence is 100 feet long and each student works at the same rate, r, measured in feet per hour.
For the first 1½ hours the two students are each working at r ft/hr so in this time they will paint 1½×2r=3r feet between them. The third student joins them, working at the same rate, r. All three take a ½-hour break so they actually work on the fence for another 1½ hours and the fence is completed in 3 hours of actual work. The three work for 1½ hours to complete the remaining 100-3r feet of the fence. However, in 1½ hours the combined working rate will be 3r ft/hr. In that time they will paint 3r×1½=4½r or 9r/2 feet of fence.
Therefore 9r/2=100-3r, 9r/2+3r=100, 15r/2=100, 3r/2=20, r=40/3 ft/hr.
The whole 100 foot fence painted pays $88, which is $0.88 per foot. So r can be converted to earnings per hour by multiplying by 0.88: r=40×0.88/3=($176/15)/hr. The original two students worked a total of 3 hours each (1½+1½) and each received 3×176/15=$35.20 while the third student worked for 1½ hours only and received (3/2)(176/15)=$17.60. Total earnings: 35.20+35.20+17.60=$88.
The length of the fence is irrelevant. In fact, we can use percentages for r, so r would be measured in percent/hr, or it could be simply in terms of earnings, dollars per hour. However, it's a lot easier to use a meaningful measurement to explain the mathematics in a practical, easy-to-understand way.