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Division is the process of dividing a number into equal parts. It is one of the four basic operations in math. Partial quotient is a method that we use to find the result of a division operation. We perform repeated subtraction to solve division problems. Learn how to use partial quotients and the regrouping method to perform division operations. ...Read MoreRead Less

Division is one of the four basic operations of mathematics. It is an arithmetic operation where a group of objects are distributed in equal parts. In other words, we can say that in division a larger number is split into smaller numbers of the same value.

There are a number of methods to divide a large or a multi-digit number by a small or a single digit number. Here, we will discuss about the two methods of division of a multi-digit number by a single digit number:

- Partial quotient method
- Regrouping method

While dividing a multi-digit number by a single-digit number, we can use the long division method to get the results. There are other methods to divide multi-digit numbers, such as the partial quotients method and regrouping.

**Partial quotient method:**For division using the partial quotient method, we break the dividend into smaller parts to make the division simple. These smaller parts are multiples of the divisor and are subtracted from the dividend until the remainder is less than the divisor. The factors which are multiplied by the divisor are the partial quotients. The sum of these partial quotients gives the final quotient. An area model can be used to represent the partial quotients.**Regrouping method:**For the regrouping method, we use the long division method based on the place value of the digits in the dividend. Since the dividend is multi-digit, the regrouping method will include regrouping the dividend into ‘ones’, ‘tens’, ‘hundreds’, ‘thousands’ and so on.

**Example 1.** Find 8420 ÷ 5.

**Solution**: Let us use the long division method with regrouping to find the result.

_______

5 ) 8420

We will begin with the digit in the ‘thousands’ place, 8. 8 ‘thousands’ ÷ 5 will give 1 thousand with the remainder being 3 ‘thousands’.

1

_______

5 ) 8420

-5

———-

3

Now, we will bring down 4 ‘hundreds’ to the right of 3 ‘thousands’.

1

_______

5 ) 8420

-5

———-

34

By regrouping 3 ‘thousands’ and 4 ‘hundreds’, we will get 34 ‘hundreds’. We will divide 34 ‘hundreds’ by 5 to get 6 ‘hundreds’ with 4 ‘hundreds’ remaining.

16

_______

5 ) 8420

-5

———-

34

-30

———-

4

We will bring down 2 ‘tens’ to the right of 4 ‘hundreds’. By the regrouping method, 4 ‘hundreds’ and 2 ‘tens’ will give us 42 ‘tens’.

16

_______

5 ) 8420

-5

———-

34

-30

———-

42

Now, 42 ‘tens’ ÷ 5 will give us 8 ‘tens’ with 2 ‘tens’ as the remainder.

168

_______

5 ) 8420

-5

———-

34

-30

———-

42

-40

———

2

Now we will bring down 0 from the ‘ones’ place to the right of 2 ‘tens’. If we combine 2 ‘tens’ and 0 ‘ones’, then we will get 20 ‘ones’. 20 ‘ones’ divided by 5 will give us 4 ‘ones’.

1684

_______

5 ) 8420

-5

———-

34

-30

———-

42

-40

———

20

-20

————

0

Thus 8420 ÷ 5 gives us 1684 as the answer.

**Example 2**. Find 6874 ÷ 7 using the partial quotient method.

**Solution**: Let us first arrange the expression as:

_______

7 ) 6874

The partial quotient method involves subtracting a multiple of the divisor that is less than the dividend. Here, we will begin by subtracting multiples of 7 until we have 0 as the remainder. The factors that are multiplied by the divisor are called partial quotients.

So, 7 times 900 will give us 6300 which is less than the dividend, 6874. We will subtract 6300 from 6874 and that will give us 574.

900

_______

7 ) 6874

– 6300

———–

574

We are left with 574 and we will again subtract another multiple of 7. We can use the multiple of 80 as 7 times 80 gives us 560.

80

900

_______

7 ) 6874

– 6300

———–

574

– 560

————

14

We are now left with 14 and 7 times 2 gives us 14. We will subtract it from the last part of the dividend.

2

80

900

_______

7 ) 6874

– 6300

———–

574

– 560

————

14

– 14

————-

0

Now that we have reached 0 at the remainder, let us now see how we broke 6874 into 6300, 560, and 14. The corresponding partial quotients are 900, 80, and 2.

Hence, the final quotient is:

900 + 80 + 2 = 982

Thus 6874 ÷ 7 = 982.

**Example 3.** Joseph has 4750 greeting cards. He will put 5 cards in each envelope for every family in his neighborhood. How many envelopes are needed for the cards?

**Solution**: Each envelope can hold 5 cards, so we have to calculate 4750 ÷ 5.

Let’s use the long division method using regrouping.

_______

5 ) 4750

4 ‘thousands’ cannot be shared among 5 groups without regrouping. So, we will regroup 4 thousand as 47 ‘hundreds’.

9

_______

5 )4750 47 ‘hundreds’ ÷ 5

– 45

———-

2

We will bring down 5 to the right side of 2 and then divide again.

950

_______

5 )4750 47 ‘hundreds’ ÷ 5

– 45

———-

25 25 ‘tens’ ÷ 5

– 25

————

00 0 ‘ones’ ÷ 5

– 00

————–

0

Hence, 4750 ÷ 5 = 950 with 0 remainder.

The quotient is 950. This means, 950 envelopes are required in total.

**Example 4.** Nathan has 240 chocolates and he decides to give 5 chocolates to each student in his class. How many students are there in Nathan’s class? Use an area model and partial quotient method to find the answer.

**Solution**: We have to find 240 ÷ 5.

So, there are 48 students in Nathan’s class.

Frequently Asked Questions

The long division method often uses partial quotient techniques to get results. A quotient is the sum of partial quotients, hence a combination of smaller quotients leads to the final quotient value, although the techniques of arriving at them are different.

There is an arithmetic formula that builds a relationship between the dividend, divisor, quotient, and remainder.

**Dividend = Divisor × Quotient + Remainder**