A way of dividing a bunch of things into equal portions is division. It is one of the four basic arithmetic operations and produces a fair sharing result.
Division is the inverse action of multiplication. In multiplication, 4 groups of 3 equal 12; in division, 12 split into 3 equal groups equals 4 in each group.
Division with partial quotients is a deviation from the standard method. The divisor is multiplied with a number and the multiple obtained is deducted from the dividend. This multiple of the divisor is as close as possible to the dividend, that is, less than or equal to the dividend. The factor or the number with which the divisor is multiplied is the partial quotient.
So, we keep multiplying the divisor with a number and subtract that product from the dividend until the difference is less than the divisor or equal to 0. If the difference is less than the divisor then that difference is the remainder. To achieve the final quotient, all the partial quotients are combined together.
To further comprehend partial quotients, consider the following example in which 168 is divided by 14. The number 14 is multiplied by ten to get the number 140. To get 28, subtract 140 from 168.
After that, we multiply 14 by 2 to get 28. We get 0 when we subtract 28 from 28.
The partial quotients are 10 and 2, and adding them together gives us the final quotient.
10 + 2 = 12
12 is the final quotient.
Partial Quotients Using Area Model
An area model is a method of expressing what division looks like using rectangles. The partial quotients and the divisor are taken as the length and width of this rectangle. Hence, we will have as many rectangles as the number of partial quotients.
The sum of the areas of the rectangles is equal to the dividend of the division equation.
Let’s use the same example in an area model to see how it looks. The dividend is 168.
Here we have two partial quotients 10 and 2. So, there are two rectangles with both their widths being the divisor, that is, 14 units and a length of 10 units and 2 units respectively; the two partial quotients. The sum of the areas of the two rectangles is the dividend, which is 168 square units.
Solved Partial Quotient Division Examples
Example 1: Find the partial quotient of 471 when divided by 35.
Solution:
Let’s take a look at the question and then we will try to break down the dividend using the factors of the divisor. Start by multiplying 35 and ten, which equals 350. We get 121 when we subtract 471 from 350. 10 is a partial quotient. To further break down 121, we multiply 35 by 2 to get 70. We get 51 by subtracting 70 from 121.
Finally, we multiply 35 by 1 and subtract 35 from 51 to get 16. So, we used the numbers 10, 2 and 1 to multiply with 35. Your partial quotations are as follows 10,2 and 1.
By combining the partial quotients together, we get the final quotient,
10 + 2 + 1 = 13
We also see that there is a remainder here, that is, 16.
Hence, 471 divided by 35 is 13 with the remainder 16.
Example 2: Use the area model and partial quotient to find 256 ÷ 16.
Solution:
We know that the divisor is 16 and the dividend is 256.
On multiplying 16 with 10 we get,
Subtract 160 from 256
On multiplying the divisor, 16, with 6, we get
Subtract 96 from 96
Therefore, on subtracting 96 from 0, we get the remainder 0.
On adding all the partial quotients, we get the single quotient of the number = 10 + 6
= 16
The result of division of \( \frac{256}{16} \) is 16.
10 + 6 = 16, which is the answer.
Example 3: There are 75 students in a room and they were asked to distribute chocolates to the rest of the students in the school. There are a total of 6255 chocolates. Find out how many chocolates will each student get if the total number is divided equally. Also, take note of the extra chocolates that are remaining.
Solution:
To find the number of chocolates each student gets and the chocolates remaining after distribution, we need to divide the total number of chocolates by total number of students.
So the divisor is 75 and the dividend is 6255
On multiplying divisor 75 with 80, we get
Subtract 6000 from 6255
On multiplying the divisor, 75, with 3, we get
Subtract 225 from 255.
Therefore, we get 30, so the remainder is 30.
On adding all partial quotients, we get the final quotient,
= 80 + 3 = 83.
The result of the division of \( \frac{6255}{75} \) is 83 with a remainder of 30.
This means that 83 chocolates could be given to each student and there would be 30 chocolates remaining after the distribution
Example 4: 5 students were assigned for a team project in a school and there were a total of 125 students. How many teams were formed?
Solution:
To find the total number of teams we need to divide the total number of students by the total number of students in one team.
So we need to divide 125 by 5.
On multiplying divisor 5 with 20, we get
Subtract 100 from 125
On multiplying the divisor, 5, with 5, we get
Subtract 25 from 25
Therefore, we get 0 and this also means that the remainder is 0.
On adding the all partial quotients, we get the final quotient as:
= 20 + 5
= 25
So, there were a total of 25 teams for the project.
Example 5: 1335 square meters of cloth was totally available to a workshop for garment making. If each participant is given 15 square meters of cloth, how many participants were there in the workshop? Also, draw the area model for the dimensions provided.
Solution:
To find the number of participants we will divide 1335 by 15.
So the dividend is 1335 and the divisor is 15.
On multiplying the divisor, 15, with 80, we get
Subtract 1200 from 1335
On multiplying the divisor, 15, with 9, we get
Subtract 135 from 135
Therefore, we get 0, so the remainder 0.
On adding all the partial quotients, we get the final quotient
= 80 + 9
= 89
So, the total number of participants is 89.
When dividing a larger number by a smaller number using the partial quotient method, the divisor is multiplied by a number and the product obtained is subtracted from the dividend. This product should be equal to or less than the dividend. This process is continued until the difference between the dividend and the multiple of the divisor is zero or less than the divisor. The factors or the numbers with which the divisor is multiplied are known as the partial quotients. These partial quotients are added together to get the quotient.
The area model denotes the process of partial quotient division in the form of a rectangle. The area of the whole rectangle denotes the dividend. The rectangle may be further divided into smaller parts depending on the number of partial quotients. The breadth of the rectangle is the divisor and the length of the rectangle is the dividend. The sum of the partial quotients is also the length of the rectangle.