Arthur D. answered 10/26/14
Tutor
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Mathematics Tutor With a Master's Degree In Mathematics
let x= Tom's rate
let x-13=Sally's rate
Tom's rate per minute=1/x
Sally's rate per minute=1/(x-13)
Together they finish the job in 42 minutes.
(1/x)(42)+(1/x-13)42=1
(42/x)+(42/[x-13])=1
multiply both sides by x(x-13)
(x)(x-13)(42/x)+(x)(x-13)(42/[x-13])=(x)(x-13)
42(x-13)+42x=x2-13x
42x-546+42x=x2-13x
84x-546=x2-13x
x2-13x-84x+546=0
x2-97x+546=0
546=2*3*7*13, 2*3=6, 7*13=91, and 6+91=97 (coefficient of the middle term)
factor
(x-6)(x-91)=0
x-6=0
x=6 doesn't make sense
x-91=0
x=91 minutes for Tom to complete the job alone (and Sally's time is 91-13=78 minutes)
check: (1/91)42+(1/78)42=1
(42/91)+(42/78)=1
(6/13)+(7/13)=1
(6+7)/13=1
13/13=1