Concept of Leap Year Using Decimal Multiplication

Concept of Leap Year Using Decimal Multiplication

A year does not consist of exactly 365 days.

The Earth takes approximately 365.2422 days to complete one revolution around the Sun.

Thus, every year has an extra 0.2422 day.

Accumulation of the extra time: This small difference accumulates over time.

4 × 0.2422 = 0.9688 = 1 day

So, about one extra day is added once every 4 years.

This extra day is added in February, making it 29 days keeping the calendar in sync with Earth's revolution. A year in which one extra day is added is called a leap year.

Step 1: Adding extra days using a simple rule

If 1 extra day is added every 4 years, then the number of extra days added per year is: 0.25 0.2422, this method adds slightly more days than required. 1/4 = 0.25 days. Since

Extending over 100 years

Extra days actually needed: 100 × 0.2422 = 24.22 × day 
Extra days added by adding 1 day every 4 years: 100/4 = 25 days

Thus, 25 days are added instead of 24.22 days.

So, one extra day is added more than required.

To correct this extra addition:

1 day is removed every 100 years

1 day is added back every 400 years

Checking accuracy over 400 years

Step 1: Extra days added every 4 years: 400/4 = 100c

Step 2: Days removed at the end of every 100 years: 400/100 = 4 days

Step 3: Days added back at the end of every 400 years: 400/400 = 1 day

Days added after correction = 100-4 + 1 = 97 days

Extra days actually needed = 400 x 0.2422 = 96.88 days

Thus the difference is very small.

Extending the same idea to 1000 years

Extra days actually needed: 1000 * 0.2422 = 242.2 days
Step 1: Extra days added every 4 years: 1000/4 = 250 days

Step 2: Days removed every 100 years: 1000/100 = 10 days

Step 3: Days added back every 400 years: 1000/400 = 2.5 days But only complete 400-year periods are counted, so we take 1000/400 = 2 complete cycles (not 2.5).

Days added after correction: 250-10 + 2 = 242 days

The difference between 242.2 days and 242 days is very small.

On Your Own!

(a) What is meant by a terminating decimal? Give one example.

(b) Why is the Earth's one complete revolution around the Sun not exactly 365 days?

(c) How many extra days are left each year after counting 365 days in a calendar year?

(d) Why do we add one extra day every 4 years in the calendar?

By adding extra days regularly and making suitable corrections over longer periods, the calendar remains closely aligned with the Earth's actual motion around the Sun, even over hundreds and thousands of years.




Comments

Post a Comment

Popular posts from this blog

Class 09 | Ganita Manjari 01 | Ch. 06 Measuring Space Perimeter and Area Solutions

Image
Class 9 Maths Ganita Manjari Chapter 6 Solutions Measuring Space Perimeter and Area Think and Reflect (NCERT Textbook Page No. 127) Q. 1. What is the difference in radius between the first and second lanes? Use the Fig. 6.11 to find the stagger needed by the runner in the second lane. Will an equal stagger be needed between the third and second lanes? Sol. The extra distance in an outer lane is created only on the curves. The distance of a circular path is C = 2πr For the lane 1, r 1 = 36.5 m and width, w = 1.22 m If Lane 2 has a radius that is w (lane width) metres larger than Lane 1, the extra distance for one full circle (360°) is  = 2π(r1 + w) – 2πr1 =2πr1 + 2πw –2πr1 = 2π w =2× {22}{7} × 1.22  = 7.67 m Yes. Because the width of each lane is constant (1.22 m), the difference in radius between Lane 3 and Lane 2 is the same as the difference between Lane 2 and Lane 1. Think and Reflect (NCERT Textbook Page No. 131) Q. 1. The area of a rectangle can ...
Read more

Class 09 | Ganita Manjari 01 | Ch. 02 Introduction To Linear Polynomial

Image
M 9102. Introduction to Linear Polynomials अचर :- संख्याओं या अंको से व्यक्त राशि को अचर कहते हैं। अचरों का मान अपरिवर्तनशील होता है। इन्हे हम अंक भी कहते हैं क्योंकि ये अंका के रूप में लिखे जाते हैं। जैसे 1, 2, 3, 7, 25, 100, Constant: - The quantity represented by numbers is called constant. These are also called Numerals because these are Numbers. It value is fixed i.e. 1, 2, 3, 7, 25, 100, ............ Numeric Value or counting Numbers are called Constant. चर: अक्षरों से व्यक्त राशि को चर कहते हैं। ये अक्षर अंग्रेजी या रोमन वर्णमाला के हो सकते हैं। इन्हे अक्षर भी कहते हैं क्योंकि ये अक्षरों के रूप में प्रयुक्त होते हैं। चरों का मान परिवर्तनशील होता है। जैसे x, y, z, a, b, c, l, m, n, α, β, γ, θ, λ etc. Variables: The quantity represented by letters is called variables. These letters may be from Roman or English alphabets. These are also called Literals because variables are letters. Its value is not fixed i.e. x, y, z, a, b, c, l, m, n, a, β, γ, θ, ...
Read more

Class 09 | Ganita Manjari 01 | Ch. 01 Orienting Yourself: The Use of Coordinates

Image
M 9101. Orienting Yourself: The Use of Coordinates 🌺🌺🌺🌺🌺🌺🌺🌺🌺🌺🌺🌺🌺🌺🌺🌺 Preparing for Ch. 1 Orienting Yourself: The Use of Coordinates 🌺🌺🌺🌺🌺🌺🌺🌺🌺🌺🌺🌺🌺🌺🌺🌺 Ch. 1 Orienting Yourself: The Use of Coordinates The 2-D Cartesian Coordinate System Cartesian plane –> The  Cartesian plan e  is a two-dimensional coordinate system used in mathematics to locate points and graph equations. It was named after the mathematician , who helped develop this system. We studied about the number line which is one-dimensional. For Cartesian system we took two number lines on a plane. This number line maybe horizontal and vertical. We know about the number line and a plane in mathematics, both of them has no end and they are spreaded in all directions infinitively. In the ‘Number System’ the distances from a fixed point are marked in equal units positively in one direction and negatively in the other on the number line. The fixed point from which th...
Read more