Question

A class had \[14\] boys with an average height of \[5.3\] feet. A new boy joined the class. Now the average of the \[15\] boys in the class was \[5.25\] feet. How tall was the new boy?

Answer 1

Hint: We have to find the height of the new boy. For this, we will first find the total height of \[14\] boys using the concept of average. Then we will find the total height of \[15\] boys using the same concept. Then we will subtract the old total height of all the \[14\] boys from the new total height of all the \[15\] boys to find the height of the new boy.

Complete step by step answer:
As we know, \[{\text{average}} = \dfrac{{{\text{sum of the numbers}}}}{{{\text{total number of values}}}}\]
So, we can write in this case,
\[ \Rightarrow {\text{average height}} = \dfrac{{{\text{Total height of }}14{\text{ boys}}}}{{14}}\]
Average height is given in the question as \[5.3\] feet. Putting this value, we get
\[ \Rightarrow 5.3{\text{ feet}} = \dfrac{{{\text{Total height of }}14{\text{ boys}}}}{{14}}\]
On cross multiplication and on rearranging, we get
\[ \Rightarrow {\text{Total height of }}14{\text{ boys}} = {\text{14}} \times 5.3{\text{ feet}}\]
\[ \Rightarrow {\text{Total height of }}14{\text{ boys}} = 74.2{\text{ feet}}\]

A new boy joined the class. Now the average of the \[15\] boys in the class was \[5.25\] feet.So, we can write in a similar way
\[ \Rightarrow {\text{average height}} = \dfrac{{{\text{Total height of }}15{\text{ boys}}}}{{15}}\]
Putting the given value, we get
\[ \Rightarrow 5.25 = \dfrac{{{\text{Total height of }}15{\text{ boys}}}}{{15}}\]
On cross multiplication and on rearranging, we get
\[ \Rightarrow {\text{Total height of }}15{\text{ boys}} = {\text{15}} \times 5.25{\text{ feet}}\]
\[ \Rightarrow {\text{Total height of }}15{\text{ boys}} = 78.75{\text{ feet}}\]

Now, we have to find the height of the new boy who joined. For this, we have to subtract the old total height of all the boys from the new total height of all the boys i.e., \[\left( {78.75 - 74.2} \right){\text{ feet}}\].
So, we get
\[ \Rightarrow {\text{Height of the new boy}} = {\text{New total height of 15 boys}} - {\text{Old total height of 14 boys}}\]
On putting the obtained values, we get
\[ \Rightarrow {\text{Height of the new boy}} = 78.75 - 74.2 = 4.55{\text{ feet}}\]

Therefore, the new boy was \[4.55{\text{ feet}}\] tall.

Note: An average is a single number taken as representative of a non- empty list of numbers. All items count equally in determining their average value and their positions in the list are irrelevant. Also, note that if all numbers of a list are multiplied by the same positive number, then its average changes by the same factor.