Question 150509
working together, alice and betty can do a certain job in 4 1/3 days. but alice fell ill after 2 days of working and betty finished the job continuing to work alone in 6 3/4 more days. how long would it take each to do the job if each of them worked alone?
:
Let a = time required by Alice working alone
Let b = time required by Betty alone
Let the completed job = 1
:
equation when they both work 4{{{1/3}}} day
{{{4.33/a}}} + {{{4.33/b}}} = 1
Multiply equation by 3 give us integers to work with
{{{13/a}}} + {{{13/b}}} = 3
:
Equation when one got sick; (b worked a total of 2+6.75 = 8.75
{{{2/a}}} + {{{8.75/b}}} = 1
Multiply equation by 6.5 
{{{13/a}}} + {{{56.875/b}}} = 6.5
:
Use elimination here:
{{{13/a}}} + {{{56.875/b}}} = 6.5
{{{13/a}}} + {{{13/b}}} = 3
-------------------------------subtraction eliminates a, find b
{{{43.875/b}}} = 3.5
3.5b = 43.875
b = {{{43.75/3.5}}}
b = 12.5357 days for Betty alone
:
Use {{{13/a}}} + {{{13/b}}} = 3, substitute for b and find a
{{{13/a}}} + {{{13/12.5357}}} = 3
{{{13/a}}} + 1.037 = 3
{{{13/a}}} = 3 - 1.037
{{{13/a}}} = 1.963
1.963a = 13
a = {{{13/1.963}}}
a = 6.6225 days for Alice alone
;
:
To check solutions, use {{{2/a}}} + {{{8.75/b}}} = 1, substituting for a and b:
 {{{2/6.6225}}} + {{{8.75/12.5357}}} = 1
.302 + .698 = 1; confirms our solution