Question 22
Context: The calendar designers decided that they will not add 1 extra day in every hundredth year.
Q. Can you write an expression for the number of days in 100 calendar years with this new adjustment?
We will count years with 365 days and years with 366 days in 100 years.
Step 1 — Classify years by rules
Let us consider 100 consecutive years. One year out of these 100 is divisible by 100. This year has 365 days, as per the diagram.
Next, let us look at the other 99 years. Some of these 99 years are divisible by 4. In 100 years, 25 years are divisible by 4. One of these 25 years is divisible by 100. So, 24 years are divisible by 4 but not by 100. These 24 years have 366 days, as per the diagram.
The remaining years are not divisible by 100. They are also not divisible by 4. Total years are 100. We have accounted for 1 year. We have also accounted for 24 years. So, years remain. These 75 years have 365 days.
Let us re-group the years with 365 days. Years with 365 days include 1 year divisible by 100. They also include 75 years not divisible by 4. Total years with 365 days are years. Total years with 366 days are years.
Step 2 — Calculate total days
We will add up the days from all these years. The total number of days is the sum of days. First, let us calculate days for the 76 years. Next, let us calculate days for the 24 years. Now, we add these two numbers together.
Answer
(i) The expression for the number of days is , which equals 36524 days.
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