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a vessel contains 12 gm of methane gas at t degree celcius temp. and 1 atm. pressure. when the temp. is increased by 10 degree celcius at the same volume, the pressure increases by 10%. calculate the volume and initial temp.

First, note the correct spelling of celsius.
I would do something like this.
You know 12 g CH4 = 12/16 = 0.75 mole CH4. Then use PV = nRT
1atm*V = 0.75*0.08206*(273+t)

Then when t increases by 10 C, p increases by 10%; therefore, use PV = nRT again, this time P =1.1 atm and T is (273+t+10) and it would look like this.
1.1atm*V = 0.75*0.08206*(273+t+10).
The problem tells you that both are at the same volume (but not what the volume is), so set the two equal.
0.75*0.08206*(273+10) = [0.75*0.08206*(273+t+10)]/1.1
and solve for t, the only unknown. Then use PV = nRT and the original numbers to solve for V. Check my thinking. Check for typos. As a check I substituted the value for V and 1.1 for P and solved for the new T and, indeed, obtained 10 higher t.

thank you so much drbob222. u are a genius. could you check out my physics questions too?

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