Question 403408
First figure out how long it takes them to make ONE bicycle. If you get stuck with problems like this, always remember that you must start with one (if you want to get to 7)



Since "Mike takes 3 hours to make one bicycle", we know that his rate is {{{1/3}}} bikes an hour. In other words, in one hour, he has one-third of a bike done.



Also, we know that "Henry takes 4 hours to make one bicycle", so by similar logic his rate is {{{1/4}}} bikes per hour



Now add their two rates. This is of course assuming ideal conditions.



{{{1/3+1/4=4/12+3/12=(4+3)/12=7/12}}}



So under perfect conditions (ie there's no overlap between the two workers, they're working consistently, etc) their combined rate is {{{7/12}}} bikes an hour. Stated another way, in one hour, together they've completed {{{7/12}}} of a single bike.



But we want a WHOLE bike, not {{{7/12}}} of a bike. So we WANT to obtain the value 1. How do we this? We just let some time pass of course. If we let the two workers keep working, they'll get 1 bike done eventually (no matter how slow they go)



So let's let some unknown time pass. We'll call this time 't'. If t = 1, then 1 hour passes and they've completed {{{7/12}}} of a single bike. Now say 2 hours pass, which means t = 2 now. So this means they've completed ANOTHER {{{7/12}}} of that same bike. Now add the fractions to get {{{7/12+7/12=14/12=1&2/12}}}



Notice how {{{1&2/12}}} is clearly over 1. So the two workers have completed bike #1 and have moved onto bike #2 in two hours. So we know for sure that the first bike took under 2 hours to make.



In general, given 't' hours, the two workers will make {{{expr(7/12)t}}} bikes. We just don't know 't' (yet)



Since we're forcing the number of bikes to be 1, this means that we're setting {{{expr(7/12)t}}} equal to 1 to get 



{{{expr(7/12)t=1}}}



which is the equation we want to solve.



To do that, multiply both sides by 12 and then divide both sides by 7 to get {{{t=12/7}}} (note: you can just flip {{{7/12}}} to get {{{12/7}}})



Note: {{{12/7=1&5/7}}}



So it takes {{{12/7}}} or {{{1&5/7}}} hours to make ONE bike...phew. With me so far?



If not, then go back over all of the above (and ask me about anything before moving on)



Since we want 7 bikes, we can just scale up (again assuming everything stays the same). So just multiply {{{12/7}}} by 7 to get {{{(12/7)(7)=(12/7)*(7/1)=(12*7)/(7*1)=84/7=12}}}



So it will take 12 hours to make 7 bikes. Note: the book most likely chose these numbers to make the answer clean.