Another Peek Beyond the Point | IT

Question 24

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

Solution

Understand the Question

The Earth completes one full revolution around the Sun in $364.2422\text{ days}$ .
The decimal portion ( $0.2422\text{ days}$ ) can be converted into standard units of time (hours, minutes, and seconds) using the conversion factors $1\text{ day} = 24\text{ hours}$ , $1\text{ hour} = 60\text{ minutes}$ , and $1\text{ minute} = 60\text{ seconds}$ .
We can also round the period to standard decimal places and calculate the accumulated time drift over multiple years.

Step 1 · Round the Revolution Period

To round $364.2422\text{ days}$ to the nearest hundredth, observe the digit in the thousandths place ( $2$ ). Since $2 < 5$ , the digit at the hundredths place remains unchanged. Diagram 1

$364.2422 \approx 364.24\text{ days}$

Step 2 · Convert the Decimal Part to Hours, Minutes, and Seconds

The decimal part of the revolution is $0.2422\text{ days}$ .

Convert days to hours ( $1\text{ day} = 24\text{ hours}$ ): $0.2422 \times 24 = 5.8128\text{ hours}$

Convert decimal hours to minutes ( $1\text{ hour} = 60\text{ minutes}$ ): $0.8128 \times 60 = 48.768\text{ minutes}$

Convert decimal minutes to seconds ( $1\text{ minute} = 60\text{ seconds}$ ): $0.768 \times 60 = 46.08\text{ seconds}$

Therefore, $0.2422\text{ days} = 5\text{ hours}, 48\text{ minutes}, 46.08\text{ seconds}$ .

Step 3 · Calculate the Calendar Difference over 10 Years

Difference between the actual revolution period and a $364\text{-day}$ calendar year: $364.2422 - 364 = 0.2422\text{ days}$

Total difference after $10\text{ years}$ : $0.2422 \times 10 = 2.422\text{ days}$

Answer

(i) Rounded value: $364.24\text{ days}$ (ii) Decimal part in time units: $5\text{ hours}, 48\text{ minutes}, 46.08\text{ seconds}$ (iii) $10\text{-year}$ difference: $2.422\text{ days}$

Common Mistakes

Decimal Time Conversion: Incorrectly multiplying the decimal fraction by $100$ or $10$ instead of the time base factors ( $24$ for hours and $60$ for minutes/seconds).
Rounding Rules: Checking the wrong decimal place when rounding to the nearest hundredth (the thousandths digit must be checked).

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.

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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?