Question 162431
A and B working together can do a job in 20/3 hours. A becames ill after 3 hours
 of working with B and B finished the job, continuing the work alone in 33/4 more
 hours. How long would it take each working alone to do the job????
:
Let a = A's time working alone
Let b = B's time working alone
Let the completed job = 1
:
Equation 1, for the statement,"A and B working together can do a job in 20/3 hours" 
{{{(20/3)/a}}} + {{{(20/3)/b}}} = 1
Which is:
{{{20/(3a)}}} + {{{20/(3b)}}} = 1; (we invert the dividing fraction and multiply)
:
Equation 2 when A gets sick:
B works 3 hr + 33/4 hr = 45/4; hrs total for B
:
{{{3/a}}} + {{{45/(4b)}}} = 1
Multiply the 1st fraction by 3/3 to have same denominator as the 1st equation
{{{9/(3a)}}} + {{{45/(4b)}}} = 1
;
Try to use elimination here with these two equations:
{{{9/(3a)}}} + {{{45/(4b)}}} = 1
{{{20/(3a)}}} + {{{20/(3b)}}} = 1
:
Multiply the 1st equation by 20, and the 2nd equation by 9, results:
{{{180/(3a)}}} + {{{900/(4b)}}} = 20
{{{180/(3a)}}} + {{{180/(3b)}}} = 9
---------------------------------------subtraction eliminates a, leaving:
{{{900/(4b)}}} - {{{180/(3)}}} = 11
{{{225/b}}} - {{{60/b}}} = 11; reduced the fractions
{{{165/b}}} = 11
b = {{{165/11}}}
b = 15 hrs by himself
:
Find a using eq: {{{20/(3a)}}} + {{{20/(3b)}}} = 1;
{{{20/(3a)}}} + {{{20/(3*15)}}} = 1;
{{{20/(3a)}}} + {{{20/45}}} = 1;
{{{20/(3a)}}} + {{{4/9}}} = 1; reduced the fraction
{{{20/(3a)}}} = 1 - {{{4/9)}}}
{{{20/(3a)}}} = {{{5/9)}}}
Cross multiply
5(3a) = 20*9
15a = 180
a = {{{180/15}}}
a = 12 hrs by himself
:
:
Check solution using eq;{{{9/(3a)}}} + {{{45/(4b)}}} = 1
{{{9/(36)}}} + {{{45/(60)}}} = 1
{{{1/4}}} + {{{3/4)}}} = 1; confirms the solutions to this horrible problem!