Another Peek Beyond the Point | IT

Question 9

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?

Solution

Understand the Question

When multiplying decimal numbers, the multiplication can initially be performed by treating them as whole numbers (ignoring decimal points).
The position of the decimal point in the final product is determined by the sum of decimal places in the multiplicand and multiplier.
Looking at the fraction representation, each decimal place corresponds to a power of $10$ in the denominator, so the zeroes in the denominator add up directly.

Step 1 · Count Decimal Places in Factors

Count the number of decimal digits in the numbers being multiplied:

For $9.5$ and $5$ : $1 + 0 = 1 \text{ decimal place}$
For $12.5$ and $7.5$ : $1 + 1 = 2 \text{ decimal places}$
For $1.64$ and $6$ : $2 + 0 = 2 \text{ decimal places}$
For $5.7$ and $13.35$ : $1 + 2 = 3 \text{ decimal places}$

Step 2 · Compare with Decimal Places in Products

Observe the decimal places in each calculated product:

$9.5 \times 5 = 47.5$ ( $1$ decimal digit)
$12.5 \times 7.5 = 93.75$ ( $2$ decimal digits)
$1.64 \times 6 = 9.84$ ( $2$ decimal digits)
$5.7 \times 13.35 = 76.095$ ( $3$ decimal digits)

In every case, the number of decimal places in the product matches the sum of decimal places in the given factors.

Step 3 · Formulate the Multiplication Rule

Based on the pattern:

Step 1: Multiply the two numbers as whole numbers, ignoring the decimal points.
Step 2: Count the total number of decimal digits in both original numbers.
Step 3: Starting from the rightmost digit of the product, count towards the left by that total number of places and place the decimal point.

Answer

Rule to multiply two decimals:

Multiply the numbers as whole numbers (ignoring decimal points).
Count the total number of decimal digits in both factors.
Starting from the right end of the product, move left by that total number of places to insert the decimal point.

Common Mistakes

Counting from Left to Right: Counting decimal places starting from the left of the product instead of starting from the rightmost digit.
Premature Zero Dropping: Dropping trailing zeroes before placing the decimal point (e.g. $0.4 \times 0.5 = 0.20$ ; dropping $0$ first to get $2$ results in an incorrect answer like $0.02$ or $2$ ). Always place the decimal point before simplifying trailing zeroes.

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?