In ten observations, the mean of all 10 numbers is 15 , the mean of...

Let the mean of the last four observations be ${A}_{2}$. Then, by the formula for combined mean, we get,

$15 = \dfrac{6 \times 16 + 4 \times {A}_{2}}{6 + 4}$

or $150 = 96 + 4{A}_{2}$

$∴ {A}_{2} = \dfrac{54}{4}$

Let the sixth number is $x$, then taking the sixth number as a collection, the combined mean of this collection and the collection of the last four is $12$.

$∴$ By the definition of combined mean

$12 = \dfrac{1 \times x + 4 \times \dfrac{54}{4}}{1 + 4}$

$∴ 60 = x + 54 ∴ x = 6$

Hence, the sixth number $= 6$