i understand the answer provided, but i feel that i arrived at another answer, and cant prove why it is incorrect. i do see how my answer is more roundabout, but i cant grasp why my math doesnt work or make sense

4 sewing machines can sew shirts in the ration 1:2:3:5. The fastest can sew a shirt in 2 hours. However, the fastest machine breaks. how long will it take the other 3 machines to sew a total of 3 shirts?

First i find how many each other machine (1 through 3) can produce shirts relative to machine 5

I state that machine 5 is the fastest machine

3/5 = x/2 -> machine 5 can sew a shirt in 2 hours, so machine 3 will sew in x hours, solve for x, i get machine 3 can sew 1 shirt in 6/5, or 1 and 1/5 hours.
2/5 = x/2 -> do the same as above, machine 2 can sew a shirt in 4/5 an hour
1/5 = x/2 -> same as above, machine 1 can sew a shirt in 2/5 an hour

i didnt realize at the time i tried solving, but the slower machines are sewing shirts faster than the fastest. could you help explain why this logic is flawed?

thanks

well intentioned, but that won't work.

it's the rates that are in the ratio 1:2:3:5. you're trying to put times, which are inversely proportional to rates, into that ratio.

because times are inversely proportional to rates, if you want to put times into the ratio, the correct ratio would be 1 : 1/2 : 1/3 : 1/5.
these fractions will make your proportion considerably more messy to set up, but, if you try them and work the problem all the way through, you'll find that you get the same answer as does the answer key.