Quantity I. The age of the father is 4 times of the age of son . After 5 years, four times the father's age is equal to 13 times the son's age. Find the present age of father. Quantity II. The sum of the age of the mother and her son is 100 years. $\frac{1}{10}\text{th}$ of the product of their age exceeds the mother's age by 180. Find the present age of mother.

To determine the relationship between Quantity I and Quantity II, we will solve for each quantity step-by-step.

Step 1: Solve for Quantity I
Let the present age of the son be s years, and the present age of the father be f years.
According to the first condition, the age of the father is 4 times the age of the son:
$f=4s$

After 5 years, the father's age will be f + 5, and the son's age will be s + 5.
According to the second condition, 4 times the father's age at that time is equal to 13 times the son's age at that time:
$4(f+5)=13(s+5)$

Substitute f = 4s into the equation:
$4(4s+5)=13(s+5)$
Expand both sides:
$16s+20=13s+65$
Subtract 13s from both sides:
$3s+20=65$
Subtract 20 from both sides:
$3s=45$
Divide by 3:
$s=15$

Now, find the present age of the father:
$f=4\times 15=60$
So, Quantity I = 60.

Step 2: Solve for Quantity II
Let the present age of the mother be m years, and the present age of her son be x years.
The sum of their ages is 100 years:
$m+x=100$
Therefore, the son's age can be written in terms of the mother's age as:
$x=100-m$

According to the second condition, one-tenth of the product of their ages exceeds the mother's age by 180:
$\frac{1}{10}(m\times x)=m+180$

Substitute x = 100 - m into the equation:
$\frac{1}{10}m(100-m)=m+180$
Multiply both sides by 10 to eliminate the fraction:
$m(100-m)=10m+1800$
Expand and rearrange into a standard quadratic equation form:
$100m-m^2=10m+1800$
$m^2-90m+1800=0$

Solve the quadratic equation by factoring:
We need two numbers that multiply to 1800 and add up to -90. These numbers are -60 and -30.
$(m-60)(m-30)=0$
Thus, the possible values for the mother's age are:
$m=60\text{ or }m=30$

Since the mother must be older than her son, and their ages sum to 100:
If m = 60, then the son's age x = 40 (which is a valid scenario).
If m = 30, then the son's age x = 70 (which is mathematically possible but biologically invalid as the mother cannot be younger than the son). Under typical assumptions, the mother's age is 60, which gives Quantity II = 60.
However, because multiple mathematical solutions exist, or if we compare the two values directly:
When Quantity II = 60, Quantity I = Quantity II.
When Quantity II = 30, Quantity I > Quantity II.
Therefore, we obtain two possible relationships, meaning there is no unique relation between Quantity I and Quantity II.

Thus, the correct option is: Quantity I = Quantity II or no relation.