Question 239722
 If my son painted it alone it would take him 4 hours longer than I would take. However, if he worked with me, he would work twice as fast. Develop a formula for how long it would take both of us working together. Simplify/reduce to lowest terms.
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Let x = time required by you working alone
then
(x+4) = time required by son working alone
.5(x+4) = time if he worked twice as fast
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Let y = time working together
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Let the completed job = 1
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{{{y/x}}} + {{{y/(.5(x+4))}}} = 1
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multiply both sides .5x(x+4)
.5(x+4)y + xy = .5x(x+4)
Factor out y
y(.5(x+4) + x) = .5x(x+4)
:
y(.5x + 2 + x) = .5x^2 + 2x
:
y(1.5x + 2) = .5x^2 + 2x
:
y = {{{(.5x^2 + 2x)/(1.5x + 2)}}}
Where 
y = hrs when working together using Dad's time working alone (x)
;
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Prove that, assume dad can do it in 12 hrs, x = 12, find y
y = {{{(.5(12^2) + 2(12))/(1.5(12) + 2)}}}
y = {{{(.5(144) + 24)/(18 + 2)}}}
y = {{{(72 + 24)/(20)}}}
y = 96/20
y = 4.8 hrs working together
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Check this in the shared work equation (son's time .5(12+4) = 8 hrs
{{{4.8/12}}} + {{{4.8/8}}}
.4 + .6 = 1; the completed job!
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Interesting that the son does more work than the dad except when dad does the job in 4 hrs, then they share the work equally, and do it in two hours
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An interesting problem, thanks for submitting it.