Division is a way of separating a group of things into equal parts. Suppose you have 18 marbles and you want to share the marbles with two friends. You want each of you to end up with the same number of marbles. To find out how many marbles each of you would get, you can count out the marbles into three equal groups. Each group has six marbles. So each of you would get six marbles as shown below. Separating a group of 18 things into three equal parts of 6 things is an example of division.
Division is one of the four basic operations in arithmetic. The others are addition, subtraction, and multiplication. You must learn how to add, subtract, and multiply before you begin to study division.
Learning to divide
Once people learned division only by memorizing. Most teachers now agree that the best way to learn division is by understanding. You can learn to understand division without much difficulty.
Writing division.
One way of separating a group into equal parts is by counting it out into equal parts. But there is a much easier way to divide. To find how many groups of 3 there are in 12, you can subtract 3 from 12 until nothing is left:
This shows that there are four 3’s in 12.
Each basic operation in arithmetic is indicated by a special symbol. The symbol for division is ÷. The statement 12 ÷ 3 = 4 means that when 12 things are separated into groups of three, there are four such groups. Or, that there are four 3’s in 12. It can also mean that when 12 things are separated into three equal groups, there are four things in each group. People who know division usually read 12 ÷ 3 = 4 as “12 divided by 3 is 4.” A problem in division also may be written this way:
The parts of a division problem have special names. The number being divided is called the dividend. The number by which the dividend is divided is the divisor. The answer, or result, of the division is the quotient.
Another way of writing a problem in division is the form used in writing fractions (see Fraction): 12/3 = 4
Division facts.
By using subtraction, you discovered that there are three equal groups of 4 things in a group of 12. Or, 12 ÷ 3 = 4. This is a division fact. You can find all the division facts by using subtraction.
It is important to learn the division facts so well that you can use them automatically. The facts are useful themselves. They are also necessary in learning how to divide larger numbers quickly and accurately.
Long division
Long division is a method that can be used to divide large numbers. In long division, you write out the work carefully.
Suppose you want to find out how many 3’s there are in 79, or 79 ÷ 3. Instead of subtracting one 3 at a time, you can shorten your work by subtracting several 3’s at once. To begin, you might subtract five 3’s, or 15, each time:
All together, you subtracted 5 + 5 + 5 + 5 + 5 or twenty-five 3’s from 79, leaving 4. You cannot subtract five more 3’s, but you can subtract one more 3, leaving a remainder of 1. Thus, there are 25 + 1 or twenty-six 3’s in 79 with 1 left over.
Subtracting five 3’s at a time shortened your work. Next, you might try subtracting ten 3’s, or 30, each time:
This time, you subtracted 10 + 10 + 5 + 1 or twenty-six 3’s from 79, and had 1 left as a remainder. A better form to use is this:
After some practice, you might subtract twenty 3’s and then six 3’s:
To gain further practice in long division, you might now try to find out how many 21’s there are in 891, or 891 ÷ 21. First, you must decide how many 21’s you will subtract at a time. Ten 21’s, or 210, might prove to be useful. Using 10’s, 100’s, or 1,000’s in multiplying the divisor makes division much easier.
When you have subtracted four 210’s or forty 21’s, you find that the remainder, 51, is too small to subtract ten more 21’s. You can, however, subtract one 21 at a time. This finally gives you 10 + 10 + 10 + 10 + 1 + 1 or forty-two 21’s in 891, with a remainder of 9.
You could have used twenty 21’s, or 420, as your first unit.
One last example will illustrate further the process of long division. Suppose you want to know how many 37’s there are in 12,526, or 12,526 ÷ 37. Once again you must decide how many 37’s to subtract at one time.
You may have to experiment on a sheet of scrap paper to find the units that you can use to solve the problem easily. You can use even larger units than 200.
Many persons use a form for long division even shorter than those outlined above. The three steps look like this:
This form does the same things that have been discussed above, but by a different method. It does not illustrate the process so well to a beginner.
When using this shorter form, it helps to notice that in all these examples you write the answer (quotient) above the proper places in the dividend. That is, when you subtract a number of 100’s, you record the number of 100’s above the 100’s place in the dividend.
Remainders in division.
There is often a remainder when you have completed a problem in division. What you do with this remainder depends on the kind of problem. If you want to know how many 3’s there are in 79, you might have had 79¢ to spend on three-cent postage stamps. You would find that you could buy 26 stamps and have 1¢ left.
If you wanted to share 79 apples among three persons, you would also find that there are twenty-six 3’s in 79 and a remainder of 1. This means that each person gets 26 apples and there is one left to share. If the sharing is to be absolutely equal, you would have to cut the remaining apple into three equal parts. Each person would receive 261/3 apples.
These examples show that what is done to a remainder depends on the problem. In some cases, further division into fractional parts is indicated. In other cases, the remainder merely tells how many are “left over.”
Division of decimal fractions.
You can also use long division to divide numbers that include decimal fractions. The statement 78.35 ÷ 3.6 is this kind of problem. In order to understand division of decimal fractions, you must learn an interesting feature of division.
You know that 15 ÷ 3 = 5 is a division fact. What would happen if both the 15 and 3 were multiplied by 10? That is, what is the result of dividing 150 by 30? Long division will show you that this quotient is also 5. Thus, 15 ÷ 3 = 5, and 150 ÷ 30 = 5. Similarly, 72 ÷ 6 = 12 and 720 ÷ 60 = 12. If the 72 and 6 are multiplied by 100, the quotient of 7,200 ÷ 600 is also 12. These examples illustrate a general rule: multiplying both the dividend and divisor by 10, 100, 1,000, or any other nonzero number, does not change the quotient.
This rule can be used to divide 78.35 by 3.6. Both 78.35 and 3.6 can be multiplied by 10. Thus, 78.35 X 10 = 783.5 and 3.6 X 10 = 36. The quotient of 783.5 ÷ 36 will be the same as the quotient of 78.35 ÷ 3.6. But the decimal points now have new positions. A useful device is to use a caret mark (ˆ) to indicate the new position of the decimal points. The decimal point in the quotient will appear directly above the caret mark in the dividend.
This shows that 78.35 and 3.6 have both been multiplied by 10. Sometimes it is necessary to multiply the dividend and divisor by 100, 1,000, or some larger multiple of 10. For example, 25.773 ÷ 17.94 should be multiplied by 100:
You should multiply the dividend and divisor by a multiple of 10 large enough to change the divisor into a whole number, or a number that does not include a decimal fraction.
For every division problem with a remainder of zero, there is a corresponding multiplication problem. The two numbers that are multiplied are the quotient and divisor in the division problem. For example:
3.25 ÷ 1.3 = 2.5
1.3 X 2.5 = 3.25
Experience with such problems has resulted in two rules. In multiplication, the number of decimal places in the product (answer to the multiplication problem) is the sum of the number of decimal places in the numbers that were multiplied. In division, the number of decimal places in the quotient is the number of decimal places in the dividend minus the number of decimal places in the divisor. If the divisor is a whole number, you can ignore the decimal point in the dividend while you are working the problem. When you get a number for the quotient, put as many decimal places in the quotient as there are in the dividend. Because the divisor has no decimal places, none must be subtracted from the number in the dividend.
In division problems, you often have to find the quotient to the nearest tenth, hundredth, and so on. You can do this easily. After you have placed the caret marks in the divisor and dividend, use just as many digits to the right of the dividend’s caret mark as the number of decimal places wanted in the answer. Sometimes it is necessary to add zeros to the dividend. For example, you must first change division problem, no carets to division problem, with carets to make the divisor a whole number. Suppose the quotient must be correct to the nearest hundredth. Then you must add a zero to the dividend, making it division problem, expanded dividend with carets.
You do not have to do anything with the remainder, because the problem asked you to be accurate only to the nearest hundredth. If the remainder is more than half of the divisor, then the digit in the divisor that is farthest to the right is increased by one. When the remainder is exactly half of the divisor, it is common to add one to the rightmost digit of the divisor if doing so will make it an even number (number that can be divided by two without a remainder).
Short division
When dividing by a one-digit number such as 7, you can do some of the work in long division without writing it down. Division of this kind, which is usually done in the mind rather than on paper, is called short division. The method is the same as in long division, but you do the work mentally.
The only difference between these two examples is that in short division you do the work mentally and indicate the remainder next to the quotient. The letter R is often used to mean Remainder. In this example, you first see that you can subtract two hundred 4’s from 849. You write the 2 in the 100’s place over the 8 in the dividend. Next, you can subtract ten 4’s from the remaining 49. You write the 1 in the 10’s place over the 4 in the dividend. Finally, you can subtract two 4’s from the remaining 9. You write the 2 in the 1’s place over the 9 in the dividend. You show the remainder to the right of the quotient.
In more difficult problems in short division, you must use a new device. The problem 415 ÷ 7 will show this.
In solving this problem, your first step is to subtract fifty 7’s or 350, which is thirty-five 10’s. Write the 5 (for 50 or five 10’s) over the 1 in the dividend. You do the subtraction mentally. Thirty-five 10’s subtracted from forty-one 10’s is six 10’s. You write a little 6 to the left of the 5 in the dividend.
Now you are dividing six 10’s and 5, or 65, by 7. You can subtract nine 7’s or 63 from 65, leaving a remainder of 2.
It is useful to see how this process is derived from long division.
Another example is 7,536 ÷ 9. As in the case of long division, you must decide how many 9’s you can subtract at one time.
First, you subtract eight hundred 9’s, or 7,200. You write the 8 (for eight 100’s or 800) over the 5 in the dividend. Mentally you subtract 72 (hundreds) from 75 (hundreds): 75 – 72 = 3. You write a little 3 to the left of the 3 in the dividend to keep the three 100’s in the work. From this new figure of 336, you can subtract thirty 9’s or 270. You write the 3 for the thirty 9’s over the 3 in the dividend. Next, 33 – 27 = 6. You write a little 6 to the left of the 6 in the dividend to keep the six 10’s in the work. From this new figure of 66, you can subtract seven 9’s, or 63. You write the 7 for the seven 9’s over the 6 in the dividend. Finally, 66 – 63 = 3. You indicate the remainder of 3 to the right of the quotient. After you have had some practice, you will be able to leave out the little numbers as reminders of figures that must be included in the work. You will soon be able to remember these numbers in your head.
How to check division
You will be wise to check the answer to a division problem to be sure you have solved it correctly.
Rounding off.
One way to check is to see whether or not the quotient is a sensible answer. You can estimate a quotient by rounding off the dividend and divisor. To estimate the quotient of 158 ÷ 76, you can round off 158 to 160 and 76 to 80. Because 160 ÷ 80 = 2, the quotient of 158 ÷ 76 should be about 2. To estimate the quotient of 5,124 ÷ 36, you can round off 5,124 to 5,000 and 36 to 50. You can see that 5,000 ÷ 50 = 100, and 5,000 ÷ 25 = 200. Thus, the quotient of 5,124 ÷ 36 should be somewhere between 100 and 200. Estimating the quotient will help you decide whether your answer is sensible.
Checking by multiplication.
Another way of checking a quotient is to multiply the quotient by the divisor to see if the product is the dividend. If you have multiplied correctly, this method will catch any error. This is because multiplication is the opposite of division.
The next example shows how to use the remainder in checking by multiplication:
The quotient is multiplied by the divisor, and the remainder is added to the product.
Four key division ideas
Here are four important rules to remember for solving division problems.
1. Remember that division means breaking up a number into smaller equal groups. The divisor can show the size of these groups or the number of groups.
2. Learn the division facts so well that you do not have to stop and figure them out each time. You will use the division facts constantly in everyday arithmetic, and will need to know them to divide larger numbers.
3. Remember the method for dividing larger numbers used in long division. In long division, subtract the divisor from the dividend as many times as possible in a single step. In this way, you can reduce the number of steps in long division.
4. Always check the answer after finishing a division problem. You can do this by estimating or by multiplying the quotient by the divisor and adding any remainder.
Fun with division
Space
is a game played with cards much like those used in bingo. Each card has a square drawn on it. The square is subdivided into 25 smaller squares. The letters S P A C E are written across the top of the card. The squares are filled in with any arrangement of the numerals from 1 to 9. Each square has one number, except the one in the center which is marked F for “free.” Each card should have a different pattern of numerals on it. Each player has a card and a set of small markers. The leader of the game calls out questions on the division facts, for example, “Under A, the 4’s in 20.” There are five 4’s in 20. If the players have the number 5 under A on their card, they cover the number. The first player to completely cover all numbers in a row, a column, or a diagonal calls out “Space!” and wins the game. The leader keeps a record of the division facts called and uses this record to check the winner’s card. For a new game, exchange the cards among the players.
Divide-down
is an arithmetic version of a spelldown. The players are divided into two teams. Each player is asked one of the division facts, such as “how many 6’s in 42?” If the players answer correctly, they stay in the game. If they miss, they must leave the game. When all the members of one team have missed, the other team is declared the winner.
