Let one observation equal to the mean is added to n observations....

Adding mean to the observations does not change the value of $∑ {\left({x}_{i} - \overset{-}{x}\right)}^{2}$, hence,

$\dfrac{\left(\dfrac{\mathop{∑}_{i = 1}^{n}{\left({x}_{i} - \overset{-}{x}\right)}^{2}}{n}\right)}{\left(\dfrac{\mathop{∑}_{i = 1}^{n + 1}{\left({x}_{i} - \overset{-}{x}\right)}^{2}}{n + 1}\right)} = \dfrac{78}{72}$
$⇒ \dfrac{n + 1}{n} = \dfrac{78}{72} ⇒ n = 12$