What fraction of air gets expelled upon heating an empty flask?


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> A student forgot to add the reaction mixture to the round bottomed flask at $\pu{27 ^\circ C}$ but instead he placed the flask on the flame. After a lapse of time he realised his mistake and using a pyrometer, he found the temperature of the flask was $\pu{477 ^\circ C}$. What fraction of air would have been expelled out? 
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By solving $$\frac{V\_1}{V\_1 + V\_2} = \frac{T\_1}{T\_2}$$ I found $$\frac{V\_2}{V\_1} = \frac{3}{2}.$$ But according to my book it is $\frac 35$. What exactly is this ratio?




Your first equation should be:
$${V\_1\over V\_2} = {T\_1\over T\_2}$$


We want to know what $V\_2$ is in terms of $V\_1$. Here, I call $V\_2$ final volume. Arranging the equation and using the values for temperature in kelvin:
$${\rm final\;volume} = {V\_1 \times 750\over 300}= {5\over 2}V\_1$$


To get the amount that was expelled, we subtract the initial volume ($V\_1$) from the final volume:
$${\rm amount\;expelled} = {5\over 2}V\_1 - V\_1 = {3\over 2}V\_1$$


The fraction expelled is then:
$${{\rm amount\;expelled}\over{\rm final\;volume}} = {3\over 5} $$ 


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