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Answer 1

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Answer 2

◙ Given ◙

• The average of n numbers x₁, x₂, x₃, ..., xₙ is A.

Next,

• x₁ is replaced by (x + a)x₁ .
• x₂ is replaced by (x + a)x₂ .
• And continues ..

◙ To Find ◙

The new average of the numbers.

◙ Solution ◙

We know,

$\boxed{\bf{Average=\frac{Sum\ of\ all\ observations}{No.\ of\ observations}}}$

◘ According to the question,

$\sf{A=\dfrac{x_1+x_2+x_3+...+x_n}{n} }$

$\sf{\longmapsto x_1+x_2+x_3+...+x_n=An}\:\:\:\: \cdots (i)$

______________________

◘ When the terms were replaced, the equation formed by the condition is :-

$\sf{Average,\ a=\dfrac{(x+a)x_1+(x+a)x_2+...+(x+a)x_n}{n} }$

$\sf{\longmapsto a=\dfrac{(x+a)(x_1+x_2+x_3+...+x_n)}{n} }$

From equation (i),

$\sf{\longmapsto a=\dfrac{(x+a)(An)}{n} }$

$\sf{\longmapsto a=(x+a)\times A}$

Therefore, the new average is (x + a) times the previous average, i.e. A.