A factory manufactures three types of products: X, Y, and Z. The average production rates are 50  units per hour for product X, 80 units per hour for product Y, and 120 units per hour for product Z. The daily working hours on products X, Y, and Z follow a ratio of 3:2:1. After a change in production schedules, the hours worked on product X increased by 20%, the hours on product Y decreased by 25%, while the hours on product Z remained the same. Given that the total production after this change is 2,520 units for the day, determine the total number of hours initially worked on products X, Y, and Z before the change.

Let the number of hours worked on product X = 3n number of hours worked on product Y = 2n number of hours worked on product Z = n total production of product X = 50×3n = 150n total production of product Y = 80×2n = 160n total production of product Z = 120×n = 120n after change in production total production of product X = 120% of 150n = 180n total production of product Y = 75% of 160n = 120n total production of product Z = 120n now, total production of each product = 180n+120n+120n = 420n Now, 420n = 2520 n = 6 number of hours worked on product X = 3n = 18 h number of hours worked on product Y = 2n = 12 h number of hours worked on product Z = n = 6 h