Given below are two statements:

Statement (I): One-fourth of sixty percent of a number is equal to two-fifth of twenty percent of another number. Then the respective ratio of the first number to the second number is 8:15.

Statement (II): A student is studying for a test from 11:00am to 8:00pm on weekdays and one-third of that time on Saturday. On Sunday, he takes a break and goes for a movie. The fractional part of the entire that the student is studying is 3/7.

In the light of the above statements, choose the correct answer from the options given below:

The correct option is: Statement (I) is true, but statement (II) is false.

Let us analyze and verify both statements step-by-step to understand why this option is correct.

Step-by-step Analysis of Statement (I):
Let the first number be represented as x and the second number as y.
According to the statement, one-fourth of sixty percent of the first number is equal to two-fifths of twenty percent of the second number.
We can write this mathematically as:

$$\frac{1}{4}\times (60\%\text{ of }x)=\frac{2}{5}\times (20\%\text{ of }y)$$

Expressing percentages as fractions:

$$\frac{1}{4}\times \frac{60}{100}\times x=\frac{2}{5}\times \frac{20}{100}\times y$$

Simplifying the fractions on both sides:

$$\frac{1}{4}\times \frac{3}{5}��x=\frac{2}{5}\times \frac{1}{5}\times y$$

This simplifies to:

$$\frac{3}{20}x=\frac{2}{25}y$$

To find the respective ratio of the first number to the second number, we solve for x/y:

$$\frac{x}{y}=\frac{2}{25}\times \frac{20}{3}$$

Simplifying by dividing 20 and 25 by their common divisor 5:

$$\frac{x}{y}=\frac{2}{5}\times \frac{4}{3}=\frac{8}{15}$$

Thus, the respective ratio of the first number to the second number is indeed 8:15. Therefore, Statement (I) is true.

Step-by-step Analysis of Statement (II):
Let us calculate the total hours of study and the total hours in a week.
1. Daily study hours on weekdays (Monday to Friday, which is 5 days):
The student studies from 11:00am to 8:00pm.
From 11:00am to 12:00pm (noon) is 1 hour, and from 12:00pm to 8:00pm is 8 hours. So, the total daily study duration is 1 + 8 = 9 hours.
Total study hours on weekdays = 5 days × 9 hours/day = 45 hours.

2. Study hours on Saturday:
The student studies for one-third of the weekday study time on Saturday.
Study hours on Saturday = 9 hours / 3 = 3 hours.

3. Study hours on Sunday:
The student takes a break, so study hours on Sunday = 0 hours.

4. Total study hours in a week = 45 hours (weekdays) + 3 hours (Saturday) + 0 hours (Sunday) = 48 hours.

5. Total hours in a week:
A week has 7 days, and each day has 24 hours.
Total hours in a week = 7 × 24 = 168 hours.

6. Fractional part of the entire week spent studying:

$$\text{Fraction}=\frac{\text{Total study hours}}{\text{Total hours in a week}}=\frac{48}{168}$$

Simplifying the fraction by dividing the numerator and denominator by their greatest common divisor (24):

$$\text{Fraction}=\frac{48/24}{168/24}=\frac{2}{7}$$

The actual fractional part of the week spent studying is 2/7, which is not equal to the 3/7 claimed in the statement. Therefore, Statement (II) is false.