To raise the temperature of a certain mass of gas by 50 ° C at a...

At constant pressure :

$∆ Q = {nC}_{P} ∆ T$ $⇒ 160 = {nC}_{P} . 50$ .....(1)

At constant volume

$∆ Q = {nC}_{v} ∆ T$ $⇒ 240 = {nC}_{v} . 100$......(2)

Equation (1) divided by (2)

$\dfrac{160}{240} = \dfrac{{C}_{P}}{{C}_{V}} \cdot \dfrac{50}{100}$

$\dfrac{{C}_{P}}{{C}_{V}} = \dfrac{4}{3} = 1 + \dfrac{2}{f}$

$f = 6$