Question 1174767
Bobby's rate of work = {{{1/x}}} of the lawn per minute
Billy's rate of work = {{{1/(x-20)}}} of the lawn per minute
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Rate of work working together = {{{1/x}}} + {{{1/(x-20)}}} of the lawn per minute = {{{1/96}}} of the lawn per minute
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Solve for x:
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{{{1/x}}} + {{{1/(x-20)}}} = {{{1/96}}}
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{{{(x-20)/x(x-20)}}} + {{{x/x(x-20)}}} = {{{1/96}}}
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{{{(x-20+x)/x(x-20)}}} = {{{1/96}}}
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{{{(2x-20)/x(x-20)}}} = {{{1/96}}}
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Cross-multiply:
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{{{x(x-20)}}} = {{{96(2x-20)}}}
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{{{x^2 -20x}}} = {{{192x - 1920}}}
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{{{x^2 -20x - 192x + 1920}}} = {{{0}}}
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{{{x^2 - 212x + 1920}}} = {{{0}}}
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Use the quadratic formula to solve for x:
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x = {{{106 - 2sqrt(2329)}}} = 9.48
x = {{{106 + 2sqrt(2329)}}} = 202.52
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There are two solutions.  Because Billy's rate of work is {{{1/(x-20)}}} of the lawn per minute, that means the first solution, x = 9.48, would make Billy's rate of work {{{1/(-10.52)}}}.  This is a negative rate and, therefore, impossible.  This means we can eliminate this solution.
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We are left with the solution, x = 202.52.  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.  Plugging in x = 202.52, Bobby's rate of work is {{{1/202.52}}} of the lawn per minute.  <b>That means it would take Bobby 202.52 minutes to mow the lawn himself.</b>