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A R I T H M E T I C

Lesson 11

THE MEANING OF DIVISION

In this Lesson, we will address the following:

What is the problem of "division"?
If we divide a number into equal parts, then how can we know how many there are in each part?
What is a rate?
Which numbers will be exactly divisible by a given number? Exact versus inexact division.
The division bar.

Section 2. Decomposing the dividend

How do we divide by decomposing the dividend?

Section 3. Three properties of division

DIVISION, we are about to see, is the inverse of multiplication. Therefore skill in division depends on skill in multiplication. It depends on knowing the multiplication table!

Now in multiplication we are given two numbers, the multiplier and the multiplicand, and we have to name their product.

4 × 15 = ?

But in what we call the inverse of multiplication, we are given the product and the multiplicand --

? × 15 = 60

-- and we have to name the multiplier.

"What number times 15 equals 60?"

Equivalently,

"How many times do we have to add 15 to get 60?"

We call that division, because it is the number of times that we could divide 60 -- break it up -- into groups of 15.

We write:

60 ÷ 15 = 4.

"60 divided by 15 equals 4."

That means

4 × 15 = 60.

"4 times 15 equals 60."

Equivalently, we could subtract 15 from 60 four times. Multiplication is repeated addition. Therefore we can think of division as repeated subtraction.

When we write:

60 ÷ 15 = 4,

then 15, the number on the right of the division sign ÷ , is called the divisor. It is into which numbers 60 will be divided. 60 is called the dividend; it is the number being divided. 4 is called the quotient. (It corresponds to the multiplier in multiplication.)

1.	What is the problem of "division"?

Dividend ÷ Divisor = Quotient

	We are to name the number of times one number, called the Divisor, is contained in another number, called the Dividend. That number of times is called the Quotient.
	Equivalently, we are to say what number times the Divisor will equal the Dividend. That is, how many times do we have to add the Divisor so that it will equal the Dividend?
	Or, how many times we could subtract the Divisor from the Dividend?

Example 1. A bottle of juice contains 18 ounces. How many times could you fill a 6 ounce glass?

Answer. How many times is 6 ounces contained in 18? This is a division problem.

18 ÷ 6 = 3.

"18 divided by 6 equals 3."

6 -- the number to the right of the division sign ÷ -- is the divisor. 18 is the dividend. 3 is the quotient.

Here is the picture of 18 ÷ 6 :

6 "goes into" 18 three times. That is,

3 × 6 = 18.

Equivalently, we could subtract 6 from 18 three times.

Here, on the other hand, is the picture of 18 ÷ 3:

18 can be divided into six 3's.

18 ÷ 3 = 6.

That is,

6 × 3 = 18.

Dividend ÷ Divisor	=	Quotient

Quotient × Divisor	=	Dividend

Note: A divisor may not be 0 -- 6 ÷ 0 -- because any number times 0 will still be 0. Therefore division by 0 is an excluded operation.

As for 0 ÷ 0, that is ambiguous because it could be any number. Any number times 0 equals 0!

Example 2. What number times 10 will equal 72?

Answer. Any question that asks "What number times. . .?" is a division problem. The number that follows the word "times" is the divisor. We have to divide 72 by 10. On separating one decimal digit:

72 ÷ 10 = 7.2

(Lesson 4.) That is,

7.2 × 10 = 72.

Example 3. If it takes 3 yards of material to make a suit, how many suits could be made from a piece of material that is 15 yards?

Answer. We have to cut 3 yards from 15 yards as many times as we can. That number of times is 15 ÷ 3.

15 ÷ 3 = 5.

That is,

15 yd ÷ 3 yd = 5,

because

5 × 3 yd = 15 yd.

You could make 5 suits.

This problem illustrates the following: The dividend and divisor must be units of the same kind. We can only divide yards by yards, dollars by dollars, hours by hours. We cannot divide 8 apples by 2 oranges --

8 apples ÷ 2 oranges = ?

-- because there is no number times 2 oranges that will equal 8 apples

What is more, we see that the quotient is always a pure number.

Dividend ÷ Divisor = Quotient.

It is the number that multiplies the divisor to produce the dividend.

Example 4. A bus is scheduled to arrive every 12 minutes. In the course of 2 hours, how many buses will arrive?

Answer. How many times is 12 minutes contained in 2 hours? But the units must be the same. Since 1 hour = 60 minutes, then 2 hours = 2 × 60 = 120 minutes.

Therefore,

120 minutes ÷ 12 minutes = 10.

10 times 12 minutes = 120 minutes. (Lesson 4.)

In the course of 2 hours, 10 buses will arrive.

(See Problem 6 at the end of the Lesson.)

Division into equal parts

2.	If we divide a number into equal parts, then how can we know how many there are in each part?

	Divide by the number of parts.

If we divide a number into 2 equal parts,

then to know how many there are in each part, divide by 2; if into 3 equal

parts, divide by 3; and so on.

That is why to divide the whole of something, which is 100%, into 100 equal parts – that is, to find 1% of a number – we divide by 100. (Lesson 4, Question 6.)

Example 5. If we divide 28 people into four equal parts, then how many will be in each part?

In Lesson 15 we will see that we are dividing 28 people into "quarters" or "fourths."

Solution. Divide by 4. 28 ÷ 4 = 7.

There will be 7 people in each part.

But that is the picture of 28 ÷ 7 Why does 28 ÷ 4 give the right answer?

Because of the order property of multiplication. 28 ÷ 4 = 7 means

7 × 4 = 28.

But that implies

4 × 7 = 28.

That means 28 is made up of four 7's.

Example 6. Christopher bought 3 pens for a total of $18. Each pen cost the same. How much did each pen cost?

Solution. If we divide $18 into 3 equal parts, we will know the answer.

18 ÷ 3 = 6.

Each pen cost $6.

It seems that we are dividing units of different kinds: 18 dollars by 3 pens. Although it may seem a bit strange, we must say that we are dividing $18 by $3.

$18 � $3 = 6.

That is,

6 × $3 = $18.

But again, according to the order property, that implies

3 × $6 = $18

-- which is the answer we want! The order property is always implied whenever we divide a quantity into equal parts.

A problem in which we appear to be dividing units of different kinds is called a rate problem, as we are about to see.

Rates

3.	What is a rate?

	A rate is a relationship between units of different kinds. Dollars per person. Miles per hour. And so on.

A rate is typically indicated by per, which means for each or in each.

In a calculation, per always indicates division.

Example 7. In a certain country, the unit of currency is the corona. With $11 Ana was able to buy 55 coronas. What was the rate of exchange? That is, how many coronas per dollar?

Solution. Follow the sequence: coronas per dollar: 55 ÷ 11 = 5.

The rate of exchange was 5 coronas per dollar.

Any rate problem -- dollars per person, miles per hour -- is equivalent to dividing a number into equal parts. (In this example, we divided 55 coronas into 11 equal parts; each part being worth 1 dollar.) Rate problems can therefore can be analyzed in the same manner as the Example above. To preserve the meaning of division, we must divide units of the same kind, even though that is not how it appears.

Exact versus inexact division

4.	Which numbers will be exactly divisible by a given number?

	The multiples of that number.

The numbers exactly divisible by 3 are the multiples of 3:

3, 6, 9, 12, and so on.

And since they are divisible by 3, so are

30, 60, 90, 120, . . .

300, 600, 900, 1200, . . .

The numbers exactly divisible by 8 are the multiples of 8:

8, 16, 24, 32, . . .

80, 160, 240, 320, . . .

800, 1600, 2400, 3200, . . .

Example 8. A bottle holds 35 ounces. A glass holds 8 ounces. How many glasses can you fill from that bottle?

Solution. We must calculate 35 ÷ 8. Now, 8 goes into 32 exactly, but 8 does not go into 35 exactly:

There is a remainder of 3.

35 = 4 × 8 + 3.

Therefore you could fill 4 glasses, and 3 ounces will remain in the bottle.

As division, we write the following:

35 ÷ 8 = 4 R 3

The remainder 3 is what we have to add to 4 × 8 to get 35.

Possible remainders

Say that there is a large group of people, and we want to divide them into groups of 5.

But say we discover that there is not an exact number of 5's. Then how many people might we not be able to group? How many people might remain?

Answer: Either 1, or 2, or 3, or 4. Because if more than 4 remained, we could make another group of 5

The point is:

The remainder is always less than the divisor.

If we divide by 5, then the possible remainders are 1, 2, 3, or 4.

Example 9.

a) If 7 is the divisor, what are the possible remainders?

Answer. 1, 2, 3, 4, 5, 6.

b) How many 7's are there in 61?

Answer. 8. 8 × 7 is 56 -- plus 5 is 61.

61 ÷ 7 = 8 R 5.

That is,

61 = 8 × 7 + 5.

The remainder 5 is what we must add to 56 to get 61.

Example 10. Prove: 47 ÷ 9 = 5 R 2.

Proof. 5 × 9 + 2 = 45 + 2 = 47.

Example 11. Divide 53 by 8. Write the whole number quotient and the remainder. Do not write the division box

Answer. 53 ÷ 8 = 6 R 5.

To arrive at that aswer, say:

"8 goes into 53 six (6) times: 48." The remainder will be the number we must add to 48 to get 53. Now, 8 plus what number ends in 3? 8 plus 5 ends in 3. (13.) 5 is the remainder.

Or:

48 plus 2 is 50, plus 3 is 53.

Lesson 7.

48 + 5 = 53.

See Problems 7 - 11.

The division bar

In what follows, we will signify division in this way:

16
8

= 2

"16 divided by 8 is 2."

Dividend
Divisor

= Quotient