Another Peek Beyond the Point | IT

Question 19

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?

Solution

We will count the number of normal years and leap years.

Step 1 — Count Years

First, we find the number of leap years. A leap year happens every 4th year. We have 100 years in total. We divide the total years by 4.

$\text{Number of leap years} = \frac{100}{4}$

$= \mathbf{25}$

So, there are 25 leap years. Next, we find the number of normal years. We subtract the leap years from the total years.

$\text{Number of normal years} = 100 - 25$

$\boxed{\mathbf{75}}$

Diagram 1

Step 2 — Calculate Total Days

A normal year has 365 days. A leap year has 366 days. We multiply normal years by 365.

$\text{Days from normal years} = 75 \times 365$

$= \mathbf{27375}$

We multiply leap years by 366.

$\text{Days from leap years} = 25 \times 366$

$= \mathbf{9150}$

Now, we add these days together. This gives us the total number of days.

$\text{Total days} = 27375 + 9150$

$\boxed{\mathbf{36525} \text{ days}}$

Step 3 — Write the Expression

Let $Y$ be the total number of years. A normal year has 365 days. A leap year has 366 days. The number of leap years is $Y$ divided by 4. The number of normal years is $Y$ minus the leap years. We can write this as $Y - \frac{Y}{4}$ . We multiply normal years by 365. We multiply leap years by 366. Then we add these two amounts.

$\text{Total days} = \left(Y - \frac{Y}{4}\right) \times 365 + \left(\frac{Y}{4}\right) \times 366$

We can simplify this expression. We can write $366$ as $365 + 1$ .

$\text{Total days} = \left(Y - \frac{Y}{4}\right) \times 365 + \left(\frac{Y}{4}\right) \times (365 + 1)$

We distribute the multiplication.

$\text{Total days} = Y \times 365 - \frac{Y}{4} \times 365 + \frac{Y}{4} \times 365 + \frac{Y}{4} \times 1$

The terms $-\frac{Y}{4} \times 365$ and $+\frac{Y}{4} \times 365$ cancel out.

$\text{Total days} = Y \times 365 + \frac{Y}{4} \times 1$

$\text{Total days} = \mathbf{365Y + \frac{Y}{4}}$

Answer

(i) The number of days in 100 calendar years is 36525 days. (ii) An expression to calculate the number of days for $Y$ years is $\mathbf{365Y + \frac{Y}{4}}$ .

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 g of Cinnamon, 100 g of Cumin seeds, 25 g of Cardamom and 250 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, $\frac{123}{10}$ , $\frac{24}{100}$ or $\frac{678}{1000}$ ? 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 × 248 = 147808, can you immediately write down the product of 5.96 × 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 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?