Answer

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**Hint:**To obtain the score of ${{25}^{th}}$ class member we will form two equations. Firstly we will form an equation of mean of 24 test scores by the given data then we will find the equation of mean score of 25 members. Finally we will solve both the equation and subtract the total score before and after ${{25}^{th}}$ member took a test to get our desired answer.

**Complete step-by-step solution:**

It is given that the mean of 24 test scores is $77.5$.

So let total score of 24 class member is $x$ so we get,

$\dfrac{x}{24}=77.5$

On simplifying we get,

$\begin{align}

& x=77.5\times 24 \\

& \Rightarrow x=1860 \\

\end{align}$…..$\left( 1 \right)$

So total test score of 24 class member is 1860

Next when the ${{25}^{th}}$ class member takes the test the mean goes down by $1.1$ points.

So let new total test score be $y$ so we get,

$\begin{align}

& \dfrac{y}{25}=77.5-1.1 \\

& \Rightarrow \dfrac{y}{25}=76.4 \\

& \Rightarrow y=25\times 76.4 \\

\end{align}$

$\therefore y=1910$….$\left( 2 \right)$

Now to find the test score of ${{25}^{th}}$ class member we will subtract equation (2) by equation (1) as follows:

$\begin{align}

& y-x=1910-1860 \\

& \therefore y-x=50 \\

\end{align}$

So we get that test score of ${{25}^{th}}$ class member is 50.

**Hence score of ${{25}^{th}}$ member is 50.**

**Note:**Mean of a list of things is same as the average where we divide the total sum of the data given by the number of data we have. It is a very common term in statistics and used in many areas of mathematics as well. There are many types of mean such as Pythagorean mean, Arithmetic mean, Geometric mean, Harmonic mean etc.

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