# Quotient

A quotient is the result of a division problem. It is one of three main parts of a division problem, with the other two being the dividend and divisor.

## What is the quotient

A quotient is the solution to the question "how many times does a number (the divisor) go into another (the dividend)?"

### Parts of a division problem

The three main parts of a division problem are the quotient, divisor, and dividend.

- Dividend - the dividend is the number that is being divided.
- Divisor - the divisor is the number that the dividend is being divided by.
- Quotient - the quotient is the result of dividing the dividend by the divisor; it is the solution to the division problem.

#### Quotient and remainder

Depending on the divisor and dividend, and whether or not we are using decimals, we may end up with a remainder. A remainder is the number left over when we cannot evenly divide the divisor into the dividend such that the quotient is not an integer. For example, if we divide 10 by 4 we have a remainder of 2 because 4 goes into 10 twice (4 × 2 = 8), leaving 2.

10 ÷ 4 = 2 R2

### How to write a division problem

A division problem can be structured in a number of different ways. The figure below shows three different ways to write a division problem, as well as the parts of the division problem. The quotient is shown in blue, the dividend in orange, and the divisor in blue.

There are other symbols used to indicate division as well, such as 12 / 3 = 4. In all the cases, the problem is the same, and the quotient is 4.

## How to find the quotient

There are a few different ways to find the quotient. One of the most straightforward (but not very efficient) ways to do so is through repeated subtraction. Another more consistent but potentially tedious way is long division.

### Repeated subtraction

To use repeated subtraction to find the quotient, use the following steps.

- Identify the dividend and divisor.
- Repeatedly subtract the divisor from the dividend until doing so results in 0, or would result in a negative number.
- If the result is 0, the dividend is evenly divided by the divisor, and the quotient is the number of times that the divisor was subtracted until this result was reached.
- If subtracting the divisor from the dividend will result in a negative number, stop subtracting. The number of times the divisor was subtracted is the quotient; the final non-zero result is the remainder.

Below is a repeated subtraction example.

Example

Find the quotient and remainder (if any) of 40 ÷ 12.

The dividend is 40 and the divisor is 12, so we subtract 12 from 40:

40 - 12 = 28

28 - 12 = 16

16 - 12 = 4

At this point, subtracting 12 from 4 would result in a negative number. Since we subtracted 12 from 40 three times, the quotient is 3, leaving 4 as the remainder. We can write our final solution as:

40 ÷ 12 = 3 R4

#### What is a partial quotient

A partial quotient is a division method similar to that of repeated subtraction. To use partial quotients to find the quotient of a division problem, use the following steps.

- Multiply the divisor by as large a factor as possible such that you know the product will still be less than the dividend.
- Subtract the product from step 1 from the dividend.
- Repeat steps 1 and 2 until the result is 0 or you have a remainder.
- If the division problem can be carried out such that the remainder is 0, the quotient is the sum of the factors by which we multiplied as well as the number of times (if any) we repeatedly subtracted. The quotient is the same even if the problem cannot be carried through to 0, but the non-zero result is the remainder.

Below is a partial quotient example.

Example

Use the partial quotient method to find the quotient and remainder (if any) of 43 ÷ 2.

2 goes into 43 21 times such that the product of 2 × 21 = 42. Subtracting 42 from 43 gives us:

43 - 42 = 1

Since we cannot subtract 2 from 1 without getting a negative result, the quotient is 21 and the remainder is 1. In this example, we immediately knew that 21 was the largest factor we could multiply by. Even if this is not the case, we can still use partial quotients. For example, say that we know 2 × 10 = 20. We then subtract 20 from 43:

43 - 20 = 23

From here, we may notice that we can instead multiply by 11 to get 22, which we then subtract from 23:

23 - 22 = 1

Now we have arrived at the same point, where we cannot subtract 2 from 1. The quotient is the sum of the factors we multiplied by as well as any repeated subtraction (which we did not do in this case):

10 + 11 = 21

This gives us the same result as above: 43 ÷ 2 = 21 R1.

### Long division

Long division is a process used to find the quotient in a division problem. Long division is written as follows:

The quotient is written above the dividend and radical symbol (the symbol around which a division problem is written).

To find the quotient using long division, we will use the above example as reference, as well as the following steps:

- Reading from left to right, we first want to find the smallest sequence of digits (in the dividend, 564) that the divisor, 6, can go into at least once. 6 cannot go into the 5 in 564. Since it can't, move on to the next number formed, which is 56. 6 can go into 56 a total of 9 times to equal 54.
- Write the first value of the quotient above the dividend. In this case, write 9 above the 6 in 56 to indicate that 6 goes into 56 a total of 9 times. The position that the number 9 is written in is important. When doing long division, make sure that the numbers align. The starting point of the quotient should be above the last digit (reading from left to right) in the smallest sequence of numbers that the divisor can go into; in this case above the 6 in 56. The next number that forms the quotient should be written directly to the right of the first.
- Write the product, in this case 6 × 9 = 54, below 56, and perform subtraction; in this case there is a remainder of 2.
- Bring the 4 in 564 down next to the remainder to form 24, keeping in mind that alignment is important.
- Repeat the process starting from step 1, treating 24 as the new dividend. Continue this process as long as necessary until the remainder is 0, or we find a repeating pattern. In this case, since 6 × 4 = 24, subtract 24 from 24 to get 0, and the long division is complete. If, on a different problem, the remainder is not 0, continue to step 6.
- If there is still a remainder and there are no more new numbers from the dividend to bring down, add a decimal point and a 0, then bring the 0 down to the remainder and continue the process above (including adding 0's) until there is no longer a remainder, or until a repeating pattern is found.

Since the remainder is 0, the quotient is 94. If, as an example, the remainder were 2, we could write the solution as 94 R2. If we wanted to continue the long division process to get a decimal quotient, we continue the division process, adding a decimal point, and bringing down a 0 for each subsequent place value in the dividend until we arrive at a solution, or find that the decimal repeats. Refer to the figure below:

## Concept of division

It is important to understand the various parts of a division sentence (and equations in general) to be able to understand and communicate mathematics. However, it is also important to understand division as a concept. One way to conceptualize division is to think of it in terms of a given number of objects being broken up into a number of groups containing an equal number of objects. For example, in 8 ÷ 2 = 4, there are a total of 8 objects that are broken into 4 groups. In order for each group to have an equal number of objects, each group would have to have 2 objects, as in the figure below.

The dividend in the figure above is represented by the 8 red circles.