Question

In a school two students Amit and Deepak of class XII are elected for Head boy. Half of the teachers voted for Amit, \[2/3\] voted for Deepak, \[10\] voted for both the boys and \[6\] did not vote for either Amit nor Deepak. Find how many teachers in all were present there.

Answer 1

Hint: As we are given the fraction of votes of teachers given to them, the number of votes are in common and the number of teachers who have not voted to anyone. So let assume the total teachers be \[t\] as given, votes gained by Amit are \[t/2\] and Deepak got \[2t/3\]. Now the sum of fraction votes gained by both the students is greater than \[1\] means some votes are counted more than one Also, given that there are \[10\] teachers who voted to both means \[10\] votes are common in them. So subtract these common or extra counted votes and finally add the teachers who did not vote. The summation of all these votes will be equal to \[t\] and this mathematical expression leads to a linear equation and on solving this we would get our answer.

Complete Step by Step Solution:
Let the total teachers be \[t\]
Now according to question,
Votes gained by Amit \[=t/2\]
Votes gained by Deepak \[=2t/3\]
The number of teachers who voted both means extra votes \[=10\]
Teachers who did not vote \[=6\]
Since when we add the number of votes gained by Amit and Deepak, these are greater than the total teachers \[t\] which means there are some extra votes that we have counted which are also given.
So, we have to subtract these extract votes from their votes
\[\Rightarrow \dfrac{t}{2}+\dfrac{2t}{3}-10\]
These are the total votes but not total teachers as some of them have note voted which are given as \[6\]
Thus, total teachers are-
\[\Rightarrow \dfrac{t}{2}+\dfrac{2t}{3}-10+6=t\]
\[\Rightarrow \dfrac{7t}{6}-4=t\]
\[\Rightarrow \dfrac{t}{6}=4\]
\[\Rightarrow t=24\]

Hence, there are total \[24\] teachers in school.

Note:
When we have to solve these types of questions just assume any variable such that we can write others too in the term of this variable and then just follow the question what it is saying and do it in a mathematical way and this will lead to an equation and on solving this we are able to get the answer.