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**Hint:**We will assume the number as a constant variable and solve the given condition using that constant. After we get that number, we will sum up its digits to find the final answer as asked in the question. We will use the cross-multiplication method for solving the linear equation with one variable.

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

Let us assume the number as $x$ .

The condition is when $24$ is subtracted from $x$, it reduces to its fourth-seventh.

In arithmetic form the condition can be written as,

$x - 24 = \dfrac{4}{7}x$ ….. (1)

From this equation we get the value of $x$ as,

$x = \dfrac{4}{7}x + 24$

Subtracting $\dfrac{4}{7}x$ from $x$,

$x - \dfrac{4}{7}x = 24$

Taking LCM,

$\dfrac{{7x - 4x}}{7} = 24$

$\dfrac{{3x}}{7} = 24$

Solving for $x$,

$x = \dfrac{{24 \times 7}}{3}$

$x = 56$

We can verify this value of $x$ by substituting it in the L.H.S. and R.H.S. of equation (1),

L.H.S. of equation (1),

$x - 24 = 56 - 24 = 32$

R.H.S. of equation (1),

$\dfrac{4}{7}x = \dfrac{4}{7} \times 56 = 32$

L.H.S. = R.H.S.

Therefore, we get the value of $x$ as $56$.

The digits of $56$ are $5$ and $6$ .

The sum of digits of $56$ is $5 + 6 = 11$ .

**The correct option is option C. $11$ .**

**Note:**A linear equation in one variable is an equation that has a maximum of one variable of order $1$. Linear equation in one variable is solved by taking LCM, removing fractions, isolate the variable, and verification of answer. In real life, a linear equation in one variable has many applications and one prominent one is problems on age difference applications.

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