If it takes two roofers 4 hours to put a new roof on a portable classroom. If the first roofer can do the job by himself in 12 hours, how many hours will second roofer to do the job by himself?

This is the sort of situation where you have to get used to 'enumerating' effort. For example: 1 roof's worth of work.

Let the work rate per hour of roofer 1 be #W_1#
Let the work rate per hour of roofer 2 be# W_2#

Standardise the total amount of work needed by the unit of: 1 roof

Initial condition: They both work together for 4 hours to complete the building of 1 roof

#4W_1+4W_2=1#roof....................................Equation(1)

We are also told that #" "12W_1=1 larr# 12 hours of work for 1 roof
So his work rate #(W_1)# is #1/12# roof per hour

So

#W_1=1/12#roof .................................................Equation(2)

Using Equation(2) substitute for #W_1# in Equation(1) giving:

#4(1/12"roof")+4W_2=1"roof "............Equation(1_a)#
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#(1/3"roof")+4W_2=1" roof"#

#4W_2=1"roof" - 1/3"roof"#

#4W_2=2/3"roof"#

#=>W_2=2/3xx1/4 = 1/6# of a roof per hour

Remember that #W_2# is the work rate for that person

Multiply both sides by 6

#W_2=1/6"roof"" "->" "6xxW_2=(6xx1/6)" roof"#

#6W_2=1"roof"#

So it would take 6 hours for the other person to complete the roof on there own.

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#color(blue)("Check")#

#(4xx1/12) + (4xx1/6) = 1/3+2/3=1#