3.34 Example Problems

AM and GM of two positive numbers are 25 and 15, respectively. Find HM and the numbers.

Solution (i) (click to see)

We know, \(AM \times HM=(GM)^2\)

• Thus, \(HM=\) 9

Solution (ii) (click to see)

If the numbers are \(a, b; a>b\) \(\frac{a+b}{2}=25\) and \(\sqrt{ab}=15\)

Thus, \(a+b=50\), and \(ab=15^2=225\)

\(\therefore (a-b)^2=(a+b)^2-4ab\) \(\Rightarrow a-b =\) 40

• Thus, a = 45, b = 5

** Example Problem 15**

The mean of 200 numbers was 50. Later it was revealed that two observations were incorrectly given as 92 and 8, instead of 192 and 88, respectively. Find the correct mean.

Solution(click to see)

\(n=200, \bar x = 200\)

\(\therefore\) Incorrect total, \(\sum x = n \times \bar x=\) 10000

Correct total, \(\sum x'=10,000-92-8+192+88=\) 10180

Correct mean, \(\bar x'=\frac{10180}{200}=\) 50.9

** Example Problem**

If \(\sum f_i(x_i-k)=0\), what is the value of k?

Answer (click to see)

Given \(\sum f_i(x_i-k)=0\)

\(\Rightarrow \sum f_i x_i - k \sum f_i =0\)

\(\Rightarrow k = \frac{\sum f_i x_i}{\sum f_i}\)

• = \(\bar x\)