Question

In the command hospital Lucknow the sum of the ages of all the 29 people, i.e. physicians, surgeons and nurses is 696. If the age of each physician, each surgeon and each nurse were 1 year, 6 years and 3 years more, then the average age of the whole staff would have been 3 years more. If the number of surgeon is a square root of a two digit number which is also a perfect cube, then the number of nurses in the hospital is:
(a) 12
(b) 15
(c) 16
(d) None of these

Answer 1

Hint: We have to assume the number of physicians, surgeons and nurses as x, y and z respectively. From the given data, the number of surgeons is a square root of a two digit number which is also a perfect cube. This two digit number will be 64 since 64 is a perfect cube of 4 and is also the square of 8. We have to consider the perfect square also since the number of surgeons is the square root of this value and x, y and z cannot be a decimal. Therefore, we will get an equation in terms of x and z by subtracting the number of surgeons from the total number of people. From the given condition, we will also get an equation $\dfrac{1x+6y+3z}{29}=3$ . We have to solve these two equations and find the value of z.

Complete step by step answer:
We have to find the number of nurses in the hospital. Let the number of physicians be x, surgeons be y and nurses be z.
We are given that the number of surgeons is a square root of a two digit number which is also a perfect cube. Therefore, we have to look for a 2 digit number that is a perfect cube. We know that 64 is a perfect cube of 4 and is also the square of 8. There is no other two digit number that is a perfect cube and perfect square. We have to consider the perfect square also, since we have to find the number of surgeons which is the square root of the two digit number and this value cannot be a decimal. Therefore,
$\Rightarrow y=\sqrt{64}=8...\left( i \right)$
Therefore, the number of nurses and surgeons will be the difference of 29 and 64, where 29 is the total number of physicians, surgeons and nurses in the hospital.
$\begin{align}
  & \Rightarrow x+z=29-8 \\
 & \Rightarrow x+z=21...\left( ii \right) \\
\end{align}$
We are given that if the age of each physician, each surgeon and each nurse be 1 year, 6 years and 3 years more, then the average age of the whole staff would have been 3 years more. We can form an equation using this data. We know that the average of a data is the sum of all the values of the elements in the data divided by the total number of elements.
$\Rightarrow \dfrac{1x+6y+3z}{29}=3$
Let us substitute (i) in the above equation.
$\begin{align}
  & \Rightarrow \dfrac{x+6\times 8+3z}{29}=3 \\
 & \Rightarrow \dfrac{x+48+3z}{29}=3 \\
\end{align}$
Let us take the denominator of the LHS to the RHS.
$\begin{align}
  & \Rightarrow x+48+3z=3\times 29 \\
 & \Rightarrow x+48+3z=87 \\
\end{align}$
We have to collect constants in the RHS.
$\Rightarrow x+3z=39...\left( iii \right)$
Let us find the value of x and y by solving equations (ii) and (iii). Let us subtract equation (ii) from (iii).
$\begin{align}
  & \Rightarrow x+3z-\left( x+z \right)=39-21 \\
 & \Rightarrow \require{cancel}\cancel{x}+3z\require{cancel}\cancel{-x}-z=18 \\
 & \Rightarrow 3z-z=18 \\
 & \Rightarrow 2z=18 \\
\end{align}$
We have to take the coefficient of z to the RHS.
$\begin{align}
  & \Rightarrow z=\dfrac{18}{2} \\
 & \Rightarrow z=9 \\
\end{align}$
Therefore, the number of nurses is 9.

So, the correct answer is “Option d”.

Note: Students must deeply understand the rules and properties associated with the algebra. They must learn to solve equations. Students must know to take the average of the data. They can have a chance of making a mistake by taking the average value as the total number of elements in the data divided by the sum of all the values of the elements.