Working alone at their respective constant rates, machine a and machine b can fill a certain order in 3 hours and 6 hours, respectively. 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.

geometry: triangles

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call the larger triangle abc

then the smaller triangle cde is similar to abc (all angles are equal, aaa).  

ab is the side that is 36, and we wish to find de, the side corresponding to ab

area_cde = area_abc - area_cde --> area_abc / area_cde = 2

let x=length of de

ratio of areas = (ratio of sides)^2

+2+=+%2836%2fx%29%5e2+

+2+=+1296%2fx%5e2+

++x%5e2+=+1296%2f2+=+648+

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