The current age of a manager, X, is six years greater than that of an analyst, Y, and nine years less than that of a director, Z. If the average of the current ages of X, Y, and Z is 39 years, what will be the ratio of the age of X to the age of Z after 7 years?

The correct option is 5:6.

To find the required ratio, let us break down the problem step-by-step by defining variables and setting up algebraic equations based on the given information.

Step 1: Express ages in terms of manager $X$

Let the current ages of the manager, analyst, and director be represented by $X$, $Y$, and $Z$ respectively.

According to the problem:
- The manager $X$ is 6 years older than the analyst $Y$:
$X=Y+6\Rightarrow Y=X-6$
- The manager $X$ is 9 years younger than the director $Z$:
$X=Z-9\Rightarrow Z=X+9$

Step 2: Formulate an equation using the average age

The average of the current ages of $X$, $Y$, and $Z$ is 39 years.
$\frac{X+Y+Z}{3}=39$
Multiplying both sides by 3 gives the total sum of their current ages:
$X+Y+Z=117$

Step 3: Solve for the current age of $X$

Substitute the expressions for $Y$ and $Z$ in terms of $X$ into the total sum equation:
$X+(X-6)+(X+9)=117$
Simplify the terms:
$3X+3=117$
Subtract 3 from both sides:
$3X=114$
Divide both sides by 3:
$X=38$

So, the current age of manager $X$ is 38 years.

Step 4: Find the current age of director $Z$

Using the relation $Z=X+9$:
$Z=38+9=47$

So, the current age of director $Z$ is 47 years.

Step 5: Calculate their ages after 7 years and determine the ratio

After 7 years:
- Age of $X$ = $38+7=45$ years
- Age of $Z$ = $47+7=54$ years

Now, calculate the ratio of the age of $X$ to the age of $Z$ after 7 years:
$\text{Ratio}=\frac{45}{54}$
Dividing both the numerator and denominator by 9:
$\text{Ratio}=\frac{5}{6}$

Thus, the ratio of the age of $X$ to the age of $Z$ after 7 years will be 5:6.