The average of 11 numbers is 10.9. If the average of the first six numbers is 10.5 and that of the last six numbers is 11.4, then the middle number is :
Solution
Let $$a_1,\ a_2.....\ a_{11}$$ be the 11 number
$$a_1+a_2+.....\ +a_{11}\ =\ 11\times\ 10.9\ =\ 119.9$$ .... (I)
Average of first six is 10.5 thus $$a_1+a_2+.....\ +a_6\ =\ 6\times\ 10.5\ =\ 63$$ ....(II)
Average of last six is 10.5 thus $$a_6+a_7+.....\ +a_11\ =\ 6\times\ 11.4\ =\ 68.4$$ ....(III)
(II)+(III)-(I) gives $$a_6\ =\ 11.5$$
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