Question

The average height of a group of people is $6ft$. $10$ more people are added with an average height of $5ft$, find the average height of the group of people consisting of $60$ people.

Answer 1

Hint: In this question, we are given the average height of certain number people. We have been asked to find the new average height when 10 more people have joined the group with certain same height. Here, use the concept of weighted average, with number of people being the frequency $(n)$. Put all the given values in the formula of weighted average and then simplify to find the new average of 60 people.

Formula used: ${A_w} = \dfrac{{{n_1}{{\bar x}_1} + {n_2}{{\bar x}_2}}}{{{n_1} + {n_2}}}$

Complete step-by-step solution:
In this question, we are given an average height of a certain number of people and have been asked to find the new average when $10$ more people with average height of $5ft$ join the group. It is mentioned in the question that there are 60 people out of which 10 have recently joined. This means that earlier there were 50 people. Hence, the average height of 50 people is $6ft$.
Using formula ${A_w} = \dfrac{{{n_1}{{\bar x}_1} + {n_2}{{\bar x}_2}}}{{{n_1} + {n_2}}}$ where,
${n_1} = 50$ people,
${\bar x_1} = 6ft$,
${n_2} = 10$ people, and
${\bar x_2} = 5ft$
Putting all the values in the formula,
${A_w} = \dfrac{{50 \times 6 + 10 \times 5}}{{50 + 10}}$
Solving further,
$ \Rightarrow {A_w} = \dfrac{{300 + 50}}{{60}}$
$ \Rightarrow {A_w} = \dfrac{{350}}{{60}} = 5.83ft.$

$\therefore $ The average height of 60 people is $5.833ft.$

Note: The following method can be used instead of using weighted average. In this, we will use the formula of averages.
${\text{Average = }}\dfrac{{{\text{Sum of observations}}}}{{{\text{No}}{\text{. of observations}}}}$
Case 1:
Average height of 50 people is $6ft$. Putting this in the above formula,
$ \Rightarrow 6 = \dfrac{{{\text{sum}}}}{{50}}$
$ \Rightarrow $ sum $ = 50 \times 6 = 300$
Case 2:
Average height of 10 people is $5ft$. Putting this in the above formula,
$ \Rightarrow 5 = \dfrac{{{\text{sum}}}}{{10}}$
$ \Rightarrow $ sum $ = 10 \times 5 = 50$
Now, total sum = $300 + 50 = 350$
Total people = $60$
Hence, ${\text{Average = }}\dfrac{{{\text{Sum of observations}}}}{{{\text{No}}{\text{. of observations}}}}$$ = \dfrac{{350}}{{60}} = 5.83ft$
$\therefore $ The new average = $5.83ft$