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Lesson 12

SHORT DIVISION

Division is the inverse of multiplication; therefore it depends on knowing the multiplication table.

The problem of division is to find what number times the divisor will equal the dividend. (Lesson 11.) To find that number, there is a method called short division.

In this Lesson, we will answer the following:

When we write the division box, where do we write the divisor?
How do we do short division?
When the dividend is a decimal, where does the point go in the quotient?
Section 2
How do we round off, or approximate, a decimal to a given number of decimal digits?
How do we express the quotient as a decimal?
How do we proceed when the dividend is less than the divisor?
Calculator problems.

Dividend ÷ Divisor = Quotient

Example. Using the division box, write 1 ÷ 5. (There is nothing to calculate.)

Answer. 5 is the divisor. It goes outside the box.

Before going on to short division, the student should be clear about division with remainder, and how to name the remainder: Example 11 of that Lesson.

2.	How do we do short division?


	From the left of the dividend (252), take as many digits as necessary to form a number (25) that will contain the divisor (7) at least once but less than ten times. Divide that partial dividend by the divisor, and obtain the first digit of the quotient (3). Write it over the last digit (5) of the partial dividend -- and write the remainder (4) beside the next digit of the dividend. Continue the operation, until all the digits of the dividend have been exhausted.

Speak

"7 goes into 25 three (3) times (21) with 4 left over."

Write the remainder 4 beside the 2. Continue:

"7 goes into 42 six (6) times exactly."

Compare the simplicity of that with long division:

In long division, we bring down the 2 and write it next to the remainder 4. In short division, we simply write the remainder next to 2. Long division, moreover, de-emphasizes the spoken nature of arithmetic, and gives the false impression that, like algebra, it is a written skill.

Long division now properly belongs to the history of mathematics.

For no other reason than tradition, long division is still taught. Since both long and short division are nothing but methods, neither one requires a true understanding of division. That comes only with decomposing the dividend -- which in any case is the principle on which all methods are based.

Begin, "5 goes into 17 three (3) times (15) with 2 left over."

Write 3 over the 7 (not over the 1), and write the remainder 2 next to the 9.

Continue: "5 goes into 29 five (5) times (25) with 4 left over.

Write 5 over the 9, and write the remainder 4 next to the 8.

Finally, "5 goes into 48 nine (9) times (45) with 3 left over."

Write 9 over the 8. The final remainder is 3.

This problem will illustrate the following point:

Over every digit in the dividend
we must write a digit in the quotient.
We proceed one digit at a time.

We will write a digit over the 1, then over the 6, then over the 0, and so on, until finally we write a digit over the 3.

Begin,

"4 goes into 21 five (5) times (20) with remainder 1 ."

Next, "4 goes into 16 four (4) times exactly."

Next, "4 goes into 0 zero (0)."

Whenever the partial dividend is less than the divisor
-- 0 is less than 4 -- write 0 in the quotient.

Next, we must write a digit over the 2: "4 goes into 2 zero (0)."

Now the 2 remains. It is the remainder.

Whenever the quotient is 0, that digit beneath it
in the dividend is the remainder.

"4 goes into 24 six (6) times exactly."

Finally, "4 goes into 3 zero (0)."

3 is the final remainder.

Again, whenever the quotient is 0, the digit beneath it in the dividend is the remainder.

"3 goes into 15 five (5) times. 3 goes into 2 zero (0)."

2 is the remainder.

That is, 152 = 50 × 3 + 2.

We use short division whenever it is easy to multiply the divisor.

Example 4. Harold has a debt of $3,164. He is able to pay $25 a week. How many weeks will it take him to pay the debt?

Solution. How many 25's will equal 3,164. To find out, we must divide:

"25 goes into 31 one (1) time (25) with 6 left over."

"25 goes into 66 two (2) times (50) with 16 left over."

"25 goes into 164 six (6) times (150) with 14 left over."

At the end of 126 weeks, then, the debt will be almost paid. $14 will remain. Therefore, it will take Harold 127 weeks.

Problem. The multiples of 8 that are in the numbers 1 to 100, are what percent of the numbers from 1 to 100?

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Do the problem yourself first!

We will now consider the case in which the dividend is a decimal. We have already seen (Lesson 11) that the divisor must always be a whole number.

3.	When the dividend is a decimal, and the divisor a whole number, how do we proceed?


	Divide in the usual way. But on coming to the decimal point, place a decimal point above it in the quotient, and continue the operation.