Question 1174770
Bobby's rate of work = {{{1/x}}} of the lawn per minute
Billy's rate of work = {{{1/(x-30)}}} of the lawn per minute
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Rate of work working together = {{{1/x}}} + {{{1/(x-30)}}} = {{{1/63}}}
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Solve for x:
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{{{1/x}}} + {{{1/(x-30)}}} = {{{1/63}}}
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{{{(x-30)/(x(x-30))}}} + {{{x/(x(x-30))}}} = {{{1/63}}}
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{{{(x-30+x)/(x(x-30))}}} = {{{1/63}}}
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{{{(2x-30)/(x(x-30))}}} = {{{1/63}}}
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Cross-multiply:
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{{{x(x-30)}}} = {{{63(2x-30)}}}
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{{{x^2 - 30x}}} = {{{126x - 1890}}}
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{{{x^2 - 156x + 1890}}}= {{{0}}}
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Using the quadratic formula, you get two solutions:
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x = {{{78 + 3sqrt(466))}}} = 142.76
x = {{{78 - 3sqrt(466)}}} = 13.24
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You can eliminate the second solution, x = 13.24, because if you use that solution, Billy's rate of work, {{{1/(x-30)}}} of the lawn per minute, would be {{{1/(-16.76)}}}.  Since that is a negative rate of work, this is impossible.
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We are left with x = 142.76.  We are looking for how long it would take Bobby to mow the lawn himself.  Bobby's rate of work is {{{1/x}}} of the lawn per minute, or {{{1/142.76}}} of the lawn per minute.  <b>This means it takes Bobby 142.76 minutes to mow the lawn himself.</b>