<s>[INST] Time and Work problems $4$ boys and $5$ girls can do $\frac{1}{2}$ work in $6$ days. after this $1$ boy and $2$ girls are added and $\frac{1}{3}$ work is done in $3$ days. how many boys must be added to complete the remaining work in $1$ day?
My Attempt 
$4$ boys and $5$ girls can do $\frac{1}{2}$ work in $6$ days.
$5$ boys and $7$ girls can do $\frac{1}{3}$ work in $3$ days.
$5+x$ boys and $7$ girls can do $\frac{1}{6}$ work in $1$ day.
I could just reach upto here. can anyone help me further?
 [/INST] Let $b$  be the time (in days) a boy needs alone to complete the entire job and $g$ the time (in days) a girl needs alone to do the same. Then:
$$
4\frac{1}{b}+5\frac{1}{g}=\frac{\frac{1}{2}}{6}
$$
and also
$$
5\frac{1}{b}+7\frac{1}{g}=\frac{\frac{1}{3}}{3}
$$
which give $b=g=108$. Thus:
$$
(5+x)\frac{1}{108}+7\frac{1}{108}=\frac{1-\frac{1}{2}-\frac{1}{3}}{1}=\frac{\frac{1}{6}}{1}\Leftrightarrow 6(x+12)=108\Leftrightarrow x=6
$$
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