__maths__>__Whole Numbers__>__Whole Numbers: Multiplication and Division__### Multiplication : Simplified Procedure for Large Numbers

This page extends the multiplication in first principles into a simplified procedure for multiplication of large numbers, which is called *multiplication by place-value with regrouping*.

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Consider multiplication of `12` and `3`. Which of the following step helps in the multiplication?

- Combine `12` repeatedly `3` times and count the combined quantity
- Combine `12` repeatedly `3` times and count the combined quantity
- two digit number multiplication is not possible

The answer is 'Combine `12` repeatedly `3` times and count the combined quantity'.

Considering the multiplication `12xx3`. Combining `12` repeatedly `3` times is shown in the figure.

What is the count shown in the combined quantity?

- `121212`
- `36`
- `36`

The answer is '`36`'. There are `3` tens and `6` units which together form value `36`.

Considering the multiplication `12xx3`. The number `12` is given in the place value format in the figure. The repeated addition is visualized and the multiplication is performed as illustrated in the figure.

What is the result of the multiplication?

- `123`
- `36`
- `36`

The answer is '`36`'. The units place is multiplied as `2xx3=6` and the tens place is multiplied as `1xx3 = 3`.

Consider the multiplication `26xx2`. That is, `26` is repeated twice as shown in the figure. Counting the two quantities together, there are `4` tens and `12` units.

What is the combined value of the quantities?

- `412`
- `52`
- `52`

The answer is '`52`'. This is explained in the next page

Considering the multiplication `26xx2`. That is, `26` is repeated twice and combined as shown in the figure. Note that `10` units are regrouped together to form the `1` ten. So the result of the multiplication is `52`.

Considering the multiplication of `26xx2`. The number `26` is given in the place value format in the figure. A simplified procedure **Multiplication by Place-value with regrouping** is given in the figure.

In the units place, `6` and `2` are multiplied to `12`. The `10` unit is converted to `1` ten and moved to the tens position. This regrouping `10` of lower place-value to `1` of the immediate higher place-value is carry over. The remaining units `2` is retained.

In the tens place, `2` and `2` are multiplied to `4` and the carry over is added to that. So, the tens place of result is `4+1=5`.

The result is `52`.

Consider the multiplication: `34xx4`. One `34` is shown in the figure. How to perform the multiplication?

- repeat the quantity `4` times and count
- use the simplified procedure to multiply in place-value form
- either one of the above
- either one of the above

The answer is 'either one of the above'.

Considering the multiplication: `34xx4`. The value `34` is repeated `4` times and combined as shown in the figure. The combined quantity makes `12` tens and `16` units.

What is the value of the combined quantity?

- `1216`
- `136`
- `136`

The answer is '`136`'. The `10` tens are regrouped into `1` hundred. The `10` units are regrouped into `1` ten. Then the number of hundreds, tens, and units are counted to the result `136`.

Considering the multiplication: `34xx4`. The numbers are given in the place value in the figure. A simplified procedure, **Multiplication by Place-value** is given in the figure.

The digit in the units place is multiplied by the multiplier. The tens formed is carried over to the tens position,

Then the digit in the tens place is multiplied by the multiplier. The hundreds formed is carried over to the hundreds position.

What is the product as given in the figure?

- `136`
- `136`
- `346`

The answer is '`136`'.

Consider the multiplication of `47` and `23`. The numbers are shown in the place value format in the figure. Which of the following is correct?

`47xx23 = 141` or

`47xx23=1081`.

- `141`
- `1081`
- `1081`

The answer is '`1081`'.

**Multiplication by Place-value with Regrouping** - Simplified Procedure : Two numbers are multiplied as illustrated in the figure. Note: The procedure combines `10` units into `1` ten as carry over and so on for higher place-values.

*Solved Exercise Problem: *

What is `284 xx 20`

- `568`
- `5680`
- `5680`

The answer is "`5680`"

*Solved Exercise Problem: *

What is `11xx123`?

- `253`
- `1353`
- `1353`

The answer is "`1353`"

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