Another Peek Beyond the Point | IT

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?

Solution

Understand the Question

In a normal 100-year span, every $4^{\text{th}}$ year is typically a leap year ( $366$ days), giving $\dfrac{100}{4} = 25$ leap years.
With the new rule, the $100^{\text{th}}$ year does not get an extra day, so it remains a standard year of $365$ days.
This leaves $25 - 1 = 24$ leap years ( $366$ days) and $100 - 24 = 76$ common years ( $365$ days).
The total number of days is obtained by multiplying each group of years by its number of days and adding the products.

Step 1 · Classify Years into 365-Day and 366-Day Years

Diagram 1

In $100$ consecutive years:

Total years divisible by $4 = 25$
Years divisible by $100 = 1$ (adjusted to have $365$ days)

Number of years with $366$ days (leap years): $25 - 1 = 24$

Number of years with $365$ days (ordinary years): $100 - 1 - 24 = 75 \text{ years (not divisible by 4)}$ $\text{Total 365-day years} = 1 + 75 = 76$

Step 2 · Write Expression and Calculate Total Days

The expression for the total number of days is: $\text{Total days} = (76 \times 365) + (24 \times 366)$

Calculating each term: $76 \times 365 = 27740$ $24 \times 366 = 8784$

Adding the days together: $27740 + 8784 = 36524$

Answer

$(76 \times 365) + (24 \times 366) = 36524 \text{ days}$

Common Mistakes

Assuming 25 Leap Years: Forgetting the adjustment and calculating $(75 \times 365) + (25 \times 366)$ , which adds an extra day.
Omitting the 100th Year: Subtracting the 100th year from the leap years but forgetting to include it in the count of 365-day years ( $75$ instead of $76$ ).

More questions in IT

Jonali and Pallabi play a game. Jonali says a fraction and Pallabi gives the equivalent decimal. Write Pallabi's answer in the blank spaces.

Jonali goes to the market to buy spices. She purchases $50\text{ g}$ of Cinnamon, $100\text{ g}$ of Cumin seeds, $25\text{ g}$ of Cardamom and $250\text{ g}$ of Pepper. Express each of the quantities in kilograms by writing them in terms of fractions as well as decimals.

Write the following fractions as a sum of fractions and also as decimals:

Can you give a simple rule to divide any number by a number of the form 1 followed by zeroes — 10, 100, 1000, etc.? For example, $\dfrac{123}{10}$ , $\dfrac{24}{100}$ or $\dfrac{678}{1000}$ ? Look for a pattern in the previous problems.

Can the product of two decimals be a natural number?

Can the product of a decimal and a natural number be a natural number?

Observe the number of digits after the decimal point in the multiplier, the multiplicand and the product. Also note the number of zeroes in the denominator.

Suppose we know that $596 \times 248 = 147808$ , can you immediately write down the product of $5.96 \times 24.8$ ?

Context: Observe the number of digits after the decimal point in the multiplier, the multiplicand and the product. Also note the number of zeroes in the denominator.

Q. By looking at the above examples, can you frame a rule to multiply two decimals?

Q10

When is the product of two decimals greater than both the numbers? When is it less than both the numbers?

Q11

What is $0.039 \text{ m}$ in centimetres and millimetres?

Q12

Complete the table by dividing the decimals:

Q13

Can you find the quotients of $10 \div 9$ , and $100 \div 11$ ?

Q14

Context: Let us consider the number $142857$ that arose when dividing $1$ by $7$ . Multiply $142857$ by numbers from $1$ to $6$ .

Q. What are the products? What do you notice?

Multiply $142857$ by $7$ . What do you observe?

Q15

To find one such number, you can find $1 \div 17$ in decimal, and use the repeating block of digits.

Q16

Will the quotient be always greater than the dividend when the divisor is a decimal? Try it out with different values of the divisor.

Describe the relationship between the dividend, divisor, and the quotient. Create a table for capturing this relationship in different situations, like we did for multiplication.

Q17

What pattern do you observe? Why are 2 and 5 related in this way?

Q18

Do you know which month has this extra day?

Q19

With this new scheme of adding one extra day every 4th year, what is the number of days in 100 calendar years? Can you write an expression to calculate that number?

Q20

How many years are divisible by 4 in 100 years?

Q21

Can you form different expressions for the same question?

Q22

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?

Q23

With this final scheme of leap years can you calculate the number of calendar days in 10,000 years and the number of actual days the Earth will take to make 10,000 revolutions around the Sun? What is the difference? If there is a big difference, can you suggest a way to fix this problem?

Q24

Do you wonder how people figured out that the Earth completes one revolution around the Sun in exactly 364.2422 days?