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

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?

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: $${{\rm amount\;expelled}\over{\rm final\;volume}} = {3\over 5}$$