## Arithmetic Chapter 6 Multiplication And Division Of A Decimal Fraction By Whole Numbers And Decimal Fraction

**1. Multiplication of Decimal Numbers by 10, 100, 1000……**

- The effective rule of multiplication of any decimal number by 10, 100, 1000,
- Move the decimal point according to the following rule
- Move the decimal point 1 digit towards the right when a decimal number is multiplied by 10.
- Move the decimal point 2 digits towards the right when a decimal number is multiplied by 100.
- Move the decimal point 3 digits towards the right when a decimal number is multiplied by 1000.
- Move the decimal point 4 digits towards the right when a decimal number is multiplied by 10000.
- ∴ Move the decimal point n digits towards the right when a decimal number is multiplied by 10n. [n = 1, 2, 3, ]
- These imply that when a decimal number is multiplied by the multiple of 10, move the decimal point as many digits towards the right as many zeroes (0) after 1 are there in the multiple of 10.
- If the given decimal number does not contain digits after the decimal point as many zeroes are there in the multiple of 10, put the necessary number of zeroes to the right side of
- the given decimal number then put the decimal point and then put one or two more zeroes.
- For example, observe the following two examples: 2-54 x 10 = 25-4
- Here 10 contains one zero after 1, so the decimal point is moved one digit towards the right.

4.75 x 100000 = 475000.00 - Here 100000 contains 5 zeroes after 1.
- ∴ The decimal point should be moved 5 digits towards the right but in the given decimal numbers 4.75,
- There are only two digits after decimal points 7 and 5.
- That’s, why 3 zeroes are put to the right side of the given number as 4.75000 and then the decimal point.
- Is moved 5 digits towards the right and two zeroes are put after the decimal point in the final answer and we get 475000.00.

**2. Multiplication of Decimal Number by Integer :**

**Effective Rule:**

** 1. First step**: First find the number of digits after the decimal point;

** 2. Second step**: Removing the decimal point, the multiplication process is done according to the general rule.

** 3. Third step**:

- In this step, the decimal point is to be put in the product obtained (which is obtained in the 2nd step).
- Put the decimal point as many digits (towards left) before the number of digits from the end digit of the product obtained as many digits lie after the decimal point in the first step.
- If the product contains digits less than the number of digits after the decimal point in the first step.
- Then put the necessary number of zeroes to the left of the product and then put the decimal point.
- For this, observe the following example.
- 0-0002 x 12 = 0-0024.
- Here, in the given decimal number, we have 4 digits after the decimal point and after removing the decimal point, the product is 2 x 12 = 24 which contains only 2 digits.
- For this, we insert 2 zeroes to the left side of 24, and then the decimal point is put as shown above and there will be 4 digits after the decimal point in the product.

**3. Multiplication of a Decimal number by a Decimal number :**

**Effective Rule:**

**Step 1:**

- First, determine the number of digits after the decimal point of the multiplicand decimal number, and then determine the number of digits after the decimal point of the multiplier decimal number.
- Now add these two numbers of digits after the decimal point in both the multiplier and multiplicand.

Step**2:**

- Determine the product of the multiplicand and multiplier after removing the decimal point of both.
- The multiplicand and multiplier are according to the general rule.

**Step **3:

- Now put the decimal point in the product obtained in step 2 according to the following rule Place the decimal point as many digits (towards left) before the number of digits from the end digit of the product obtained in step 2 as many digits obtained in step 1.
- If the product contains digits less than the number of digits as obtained in step 1, then put the necessary number of zeroes to the left of the product obtained in step 2 and then put the decimal point.
- Observe the following examples so that you will have a clear concept of the multiplication of a decimal number by a decimal number.