Question 149574
A and B working together can do a job in 24 hours.  After A worked alone for 7 hours, B joined him and together they finished the rest of the job in 20 hours.  How long would it take each working alone to do the job?
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Let a = time required when A works alone
Let b = time required when B works alone
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Let the completed job = 1
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write an equation for each scenario :
"A and B working together can do a job in 24 hours."
{{{24/a}}} + {{{24/b}}} = 1
Multiply equation by ab to get rid of the denominators, results
24b + 24a = ab
24a = ab - 24b
24a = b(a-24)
{{{24a/((a-24))}}} = b  
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"After A worked alone for 7 hours, B joined him and together they finished the rest of the job in 20 hours.
That means a worked 27 hrs
{{{27/a}}} + {{{20/b}}} = 1
Multiply equation by ab to get rid of the denominators, results
27b + 20a = ab
20a = ab - 27b
20a = b(a-27)
{{{20a/((a-27))}}} = b  
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Therefore:
{{{24a/((a-24))}}} = {{{20a/((a-27))}}}
Cross multiply:
24a(a-27) = 20a(a-24)
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24a^2 - 648a = 20a^2 - 480a
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24a^2 - 20a^2 -648a + 480a = 0
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4a^2 - 168a = 0
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4a(a - 42) = 0
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a = +42 A's hrs alone 
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Use {{{24/a}}} + {{{24/b}}} = 1 to find b, substitute 42 for a
{{{24/42}}} + {{{24/b}}} = 1
Multiply equation by 42b
24b + 42(24) = 42b
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1008 = 42b - 24b
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1008 = 18b
b = {{{1008/18}}}
b = 56 hrs is B's time alone
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You can check solution using a calc on both equations, a=42, b=56