Working alone at their respective constant rates, machine a and machine b can fill a certain order in 3 hours and 6 hours, respe
ctively. if the two machines work simultaneously at their respective constant rates, how many hours does it take the two machines to fill \small \frac{1}{2} of that order?
Machine A works at a rate of 1 order per 3 hours, which means that it can complete 1/3 of an order per hour. Similarly, it takes machine B 6 hours to fill the same order - its order-filling rate is 1/6 of an order per hour. We can find their combined rate by adding their two rates together, giving us 1/3 + 1/6 = 3/6 = 1/2 order per hour, which happens to contain exactly the value the question was looking for: working together, it takes machine A and machine B exactly 1 hour to fill 1/2 an order.
