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In a group of $7$ people, the average age is found to be $17$ years. Two more people joined with an average age $19$ years. One person left the group  whose age was $25$ years. What is the new average age of the group?

1. $17.5$ years
2. $16.5$ years
3. $18$ years
4. $16$ years

In a group of $7$ people, the average age is found to be $17$ years.

$\dfrac{x_{1} + x_{2} + \dots + x_{7}}{7} = 17$

$\implies x_{1} + x_{2} + \dots + x_{7} = 119 \rightarrow(1)$

Two more people joined with an average age $19$ years.

$\dfrac{x_{8} + x_{9}}{2} = 19$

$\implies x_{8} + x_{9} = 38\rightarrow(2)$

Now, $x_{1} + x_{2} + \dots + x_{7} + x_{8} + x_{9} = 119 + 38 = 157$

One person left the group  whose age was $25$ years.

Lets $x_{9}$ leave the group. So $x_{9} = 25$

$x_{1} + x_{2} + \dots + x_{7} + x_{8} = 157 – 25 = 132$

Average $= \dfrac{x_{1} + x_{2} + \dots + x_{7} + x_{8}}{8} = \dfrac{132}{8} = 16.5$ years.

So, the correct answer is $(B).$
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we can do it with the help of Sum.

Avg= Total Sum / total count    [So, Total Sum= Avg * Count]

Now,

Sum(old)= 17*7 +19*2 =157     (here count is 7+2=9)

Sum(new) = 157-25 = 132 ( here count is 8)

So, New avg of 8 people will be

Avg(new) = 132/8= 16.5
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