Introduction / Context:
This question extends the leap year counting logic from 100 years to 300 years. Understanding how many leap years occur in a long span is important in more advanced calendar problems and helps you internalise the Gregorian rules. The complication again arises from the special treatment of century years that are not multiples of 400.
Given Data / Assumptions:
- We consider a span of 300 consecutive years.
- Leap year rule in the Gregorian calendar:
- A year divisible by 4 is a leap year.
- If a year is divisible by 100, it is not a leap year.
- If a year is divisible by 400, it is a leap year.
- We must compute how many years in these 300 years are leap years.
Concept / Approach:
In 300 consecutive years, we first count all years that are divisible by 4 because these are candidates for leap years. Then we subtract those years that are divisible by 100, since such century years are not leap years unless divisible by 400. In any 300 year period aligned with centuries, such as years 1 to 300, 101 to 400, and so on, we can systematically apply this counting method. For typical exam questions, the standard answer uses a period like 1 to 300 where none of the century years are divisible by 400.
Step-by-Step Solution:
First, count multiples of 4 in 300 years: 300 / 4 = 75.
These 75 years are potential leap years.
Next, count multiples of 100 in the same span: 100, 200, 300, so there are 3 century years.
In a basic 1 to 300 span, none of these (100, 200, 300) is divisible by 400, so none of them is a leap year.
Therefore, we must subtract all 3 from the 75 candidates.
Number of actual leap years = 75 - 3 = 72.
Verification / Alternative check:
List the structure: every 4th year is a leap year, giving 75 such years, but each century year that is not a 400 multiple is wrongly included.
Because 400 is outside the 1 to 300 range, there is no century year to add back as a leap year.
Thus the net count of leap years remains 72 in these 300 years.
Why Other Options Are Wrong:
Option A (75): Simply counts all multiples of 4 and ignores the century rule.
Option B (74) and Option D (73): Remove fewer than 3 century years, which does not match the actual pattern.
Option C (72): Correct, as it equals 75 minus the 3 non leap century years.
Common Pitfalls:
Forgeting to subtract all relevant century years is the main source of error.
Some candidates confuse the formula for 400 years, where the counts differ slightly.
Misinterpreting the phrase 300 consecutive years as three separate centuries rather than one continuous block can also cause confusion.
Final Answer:
In 300 consecutive years there are 72 leap years.
Discussion & Comments