Multiplication of Multi-Digit Numbers

Introduction

Multiplication is one of the four basic operations in mathematics with the other three being addition, subtraction and division. Before we move to learn how to multiply multi-digit numbers, let us recall what we mean by multiplication.

How do we define Multiplication?

Multiplication is defined as the process of finding out the product between two or more numbers. The result thus obtained is called the product. Suppose you bought 6 pens on one day and 6 pens on the next day. Total pens you bought are now 2 times 6 or 6 + 6 = 12.

This can also be written as 2 x 6 = 12

Not the symbol used for multiplication. The symbol (x) is generally used to represent multiplication. Other common symbols that are used for multiplication are the asterisk (*) and dot (.)

Symbol for Multiplication

Note the symbol used in the example above for multiplication. The symbol (x) is generally used to represent multiplication. Other common symbols that are used for multiplication are the asterisk (*) and dot (.)

Now, let us have a look at some important terms that are used when two numbers are multiplied.

Important terms in the multiplication

Some important terms used in multiplication are –

Multiplicand – The number to be multiplied is called the multiplicand.

Multiplier – The number with which we multiply is called the multiplier.

Product – The result obtained after multiplying the multiplier and the multiplicand is called the product.

The relation between the multiplier, multiplicand and the product can be expressed as –

Multiplier × Multiplicand = Product

Let us understand this using an example.

Suppose we have two numbers 9 and 5. We wish to multiply 9 by 5.

So, we express it as 9 x 5 which gives us 45.

Therefore, 9 x 5 = 45

Here, 9 is the multiplicand, 5 is the multiplier and 45 is the product.

Now, that we have understood what we mean by multiplication and the terms associated with it, let us move to learn multiplication of 1 – digit numbers.

Now, let us understand how to perform multiplication when we have multi-digit numbers.

How to multiply Multi-Digit Numbers?

Before we come to understand the multiplication of multi-digit numbers, it is important to recall what is meant by multi-digit numbers?

Recall that every digit of a number has a place value. For instance, the number 5 is a single digit number where 5 is at the one’s place. Similarly, in number 27, the digit 2 is at the ten’s place while the digit 7 is at the one’s place. So, how do we define multi-digit numbers? Multi-digit numbers are numbers that consist of more than 1 digit. For example, the numbers 535 and 678 are both multi-digit numbers.

Now, let us move to learning the multiplication of multi-digit numbers. When it comes to the multiplication of multi-digit numbers, there are two methods of multiplying the numbers. These methods are the expanded notation method and the column method. Let us understand both the methods.

Expanded Notation Method

In the expanded notation method we expand the multiplicand as per the place values and then multiply each number by the multiplier. We then sum up all the results obtained to get our final answer. Let us understand it through an example.

For example, Multiply 1235 by 40

Solution

We will solve this step by step.

Step 1 – Write the number (multiplicand) in the expanded form. We get,

1235 = 1000 + 200 + 30 + 5

Step 2 – Multiply each number by the given number (multiplier) one by one. We get,

1000 x 40 + 200 x 40 + 30 x 40 + 5 x 40

= 40000 + 8000 + 1200 + 200

Step 3 – Add the results obtained. We get,

40000 + 8000 + 1200 + 200 = 49400

Hence, 1235 x 40 = 49400

This method, though simple, may not be suited for larger numbers. But it is used to understand the basic concepts of multiplication.

Column Method

In this method, we split the numbers into columns and multiply the numbers by the multiplicand one by one. There are two scenarios when using this method.

Let us understand them one by one

Multiplication without Regrouping

This method comes into force when we have smaller numbers that do not involve carrying forward any numbers to the digit at the next place value. Let us understand it through an example.

For example, Multiply 1021 by 32

Solution

We will use the following steps to obtain our result.

Step 1 – First we write the multiplicand and the multiplier in columns. Here we have 1021 as the multiplicand and 32 as the multiplier.

Step 2 – Now, we multiply the number at the one’s place of the multiplicand, i.e. 1 by the number at the one’s place of the multiplier, which in this case is 2. We get

Step 3 – Now, we multiply the number at the ten’s place of the multiplicand by 2. We get

Step 4 – Next, we multiply the number at the hundred’s place of the multiplicand by 2. We get

Step 5 – Last, we multiply the number at the thousand’s place of the multiplicand by 2. We get

Step 6 – Now, we need to place 0 at the ones’ place in the next line as a placeholder. We will get

Step 7 Since, we are complete with the multiplication of the multiplicand with the first digit of the multiplier, we perform the same steps as above for the multiplication of the multiplicand with the next number of the multiplier and then write the result in the line against the 0 that we placed as a placeholder in the previous step. We will get –

Step 8 Now that we have multiplied all the digits of the multiplier with the multiplicand, we will add the digits obtained in a vertical manner. We will get –

The result thus obtained is our answer. Hence 1021 x 32 = 32672

Multiplication with Regrouping

In the above case, we have small multiplications that did not involve two-digit results at any step. But in the case of larger numbers, there will be a need to carry forward the number to the number at the next place value. This is called Multiplication with Regrouping. Let us understand it through an example.

For example, Multiply 1025 by 34

Solution

We will use the following steps to obtain our result.

Step 1 – First we write the multiplicand and the multiplier in columns.

Step 2- Multiply the one’s digit of the multiplicand by 4. We have 4 x 5 = 20. Write 0 in the one’s column and carry over 2 to the ten’s column.

Step 3 – Multiply ten’s digit of the multiplicand by 4. We get 2 x 4 = 8. Add 2 that was carried over to it to get 8 + 2 = 10 Now, write 0 in the ten’s column and carry over 1 to the hundred’s column.

Step 4 – Multiply the hundred’s digit of the multiplicand by 4. We get 0 x 4 = 0. Add 1 that was carried over to it to get 0 + 1 = 1. Now, write 5 in the hundred’s column.

Step 5 – Multiply the digits at the thousand’s place of the multiplicand with 4. We get 4 x 1 = 4. Write 4 in the thousand’s column to get –

Step 6 Now, we need to place 0 at the ones’ place in the next line as a placeholder. We will get

Step 7 – Next, we repeat the above steps to multiply all digits of the multiplicand by the digit at the ten’s place of the multiplier. We will get

Step 8 Now that we have multiplied all the digits of the multiplier with the multiplicand, we will add the digits obtained in a vertical manner. We will get –

Hence, 1025 x 34 = 34850

The above steps can be generalised to define multiplication, which is commonly known as long multiplication. Let us define these steps.

Long Multiplication

Long multiplication is similar to the column method except for the fact that here we multiply the larger numbers. This method is used when the multiplicand is greater than 9, i.e. the multiplicand is more than a one-digit number. This method involves the following steps –

First, we write the multiplicand and the multiplier in columns.
First, multiply the number at the one’s place of the multiplier with all the numbers of the multiplicand and write them horizontally.
Make sure you write numbers from right to left and each number below the corresponding place value of the multiplicand.
Now, move to the next line.
Place a 0 at the one’s place of this line.
Now, look for the digit at the ten’s place of the multiplier. Multiply the number at the ten’s place of the multiplier with all the numbers of the multiplicand and write them horizontally in the line where you had marked 0.
Again move to the next line.
Place a 0 at the one’s as well as ten’s place of this line.
Now, look for the digit at the hundred’s place of the multiplier. Multiply the number at the hundred’s place of the multiplier with all the numbers of the multiplicand and write them horizontally in the line where you had marked the two zeros.
Continue in this manner by adding an extra zero in each line until you have reached the end of the multiplier
Add the numbers vertically according to their place values.
The number so obtained is your result.

Let us understand this by an example

For example, Multiply 132 by 13

Solution

First, we write the multiplicand and the multiplier in columns.

2. First multiply the number at the one’s place of the multiplier with all the numbers of the multiplicand and write them horizontally.

3. Place a 0 at the one’s place of the next line

4. Now, look for the digit at the ten’s place of the multiplier. Multiply the number at the ten’s place of the multiplier with all the numbers of the multiplicand and write them horizontally in the line where you had marked 0.

5. There is no more number in the multiplicand. Now, add the numbers vertically according to their place values.

6. The final answer is 1716. Hence 132 x 13 = 1716

Let us see another example where we 3 digits in the multiplicand.

For example, Multiply 364 by 123

Solution

1. First we write the multiplicand and the multiplier in columns

2. First multiply the number at the one’s place of the multiplier with all the numbers of the multiplicand and write them horizontally.

3. Place a 0 at the one’s place of the next line

5. Place a 0 at the one’s as well as ten’s place of the next line.

6. Now, look for the digit at the hundred’s place of the multiplier. Multiply the number at the hundred’s place of the multiplier with all the numbers of the multiplicand and write them horizontally in the line where you had marked the two zeros.

7. There is no more number in the multiplicand. Now, add the numbers vertically according to their place values.

8. Hence the final product is 44,772. We can say that 364 x 123 = 44772

Multiplication by 100 and 1000

To multiply a number by 100, put two zeros to the right of the number. For example, 435 x 100 = 43500
To multiply a number by 1000, put three zeros to the right of the number. For example, 435 x 1000 = 435000
To multiply a number by the product of 100 and 1000 with the counting numbers, multiply the numbers with the non-zero numbers and then add the number of zeros to the result. For example, multiply 45 by 200. To solve this, we will first multiply 45 by 200. We get 45 x 2 = 90. Now we add 2 zeros ( as 200 had 2 zeros) to the right of 90. We get 9000. Hence, 45 x 200 = 9000

Solved Examples

Example 1 A tyre factory produces 6348 tyres a day. How many tyres will the factory produce in 460 days?

Solution We have been given that a tyre factory produces 6348 tyres a day. We need to find how many tyres the factory will produce in 460 days. Let us summarise this information as –

Tyres produced by a factory in one day = 6348

The number of tyres the factory will produce in 460 days = ?

To obtain this value, we will have to multiply 6348 by 460. We will use the long multiplication steps defined above to find the value. We will get –

Hence, the number of tyres that will be produced in 460 days will be 2920080.

Example 2 World Tour Travels charges £ 80563 for a 7 day trip to the USA per passenger. If 790 passengers take that trip in a year, how much did the travel agency earned during that year ?

Solution We have been given that World Tour Travels charges £ 80563 for a 7 day trip to the USA per passenger. Also, 790 passengers take that trip in a year. We are required to find the earnings of the World Tour Travels in that year. Let us summarise the information available to us.

The total amount charged by World Tour Travels for a 7 day trip to the USA per passenger = £ 80563

Number of passengers taking that trip in a year = 790

Total earnings of the World Tour Travels in that year = ?

To find the total earnings of the World Tour Travels in that year we will have to find multiply 80563 by 790. We will get

Hence, the earnings of the World Tour Travels in a year when it charged £ 80563 for a 7 day trip to the USA per passenger and 790 passengers take that trip in a year was £ 63644770.

Key Facts and Summary

Multiplication is defined as the process of finding out the product between two or more numbers.
The number to be multiplied is called the multiplicand.
The number with which we multiply is called the multiplier.
The result obtained after multiplying the multiplier and the multiplicand is called the product.
Multi-digit numbers are numbers that consist of more than 1 digit.
In the expanded notation method we expand the multiplicand as per the place values and then multiply each number by the multiplier. We then sum up all the results obtained to get our final answer.
In the column method, we split the numbers into columns and multiply the numbers by the multiplicand one by one.

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