Question

In ten observations, the mean of all $10$ numbers is $15$ , the mean of the first six observations is $16$ and the mean of the last five observations is $12$ . The sixth number is

• A. $6$
• B. $9$
• C. $12$
• D. $3$

Answer 1

Answer: (A)

Let the mean of the last four observations be $A_{2}$ . Then, by the formula for combined mean, we get,
$15=\frac{6 \times 16 + 4 \times A_{2}}{6 + 4}$
or $150=96+4A_{2}$
$\therefore A_{2}=\frac{54}{4}$
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 $12$ .
$\therefore $ By the definition of combined mean
$12=\frac{1 \times x + 4 \times \frac{54}{4}}{1 + 4}$
$\therefore 60=x+54\therefore x=6$
Hence, the sixth number $=6$