Arthur D. answered • 10/26/14

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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=x

^{2}-13x42x-546+42x=x

^{2}-13x84x-546=x

^{2}-13xx

^{2}-13x-84x+546=0x

^{2}-97x+546=0546=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