S k i l l
i n
A R I T H M E T I C

PARTS
OF NATURAL NUMBERS  1

Lesson 14  Section 2

1 of the 3 rectangles has been shaded -- 1 out of 3.  That is, the third part -- or one third -- of the entire figure has been shaded.  1 is one third of 3.

2 of the 3 rectangles have now been shaded.  2 out of 3.  But each rectangle is a third part of the whole.  Therefore those two rectangles together are two third parts of the whole.  Or simply two thirds.

Those words, "two thirds," are to be taken literally, like two apples or two chairs.  Two thirds of a number are twice as much as one third.  Count them:

One Third, two Thirds.  1,  2.

Now, 2 is not a part of 3, because 3 is not a multiple of 2.  We say it is parts of 3, plural.

2 is two third parts of 3.

Notice how each number -- 2 out of 3 -- says its name.  "Two thirds."

In Spanish, they say dos terceras partes -- two third parts -- which is literally true. But in English, unfortunately, we are not used to saying that.

 4. What does it mean to say that a smaller number is parts (plural) of a larger number? It means it is more than one of a given part.

If the entire figure now represents 15, then 5 is the third part of 15, and 10 is two third parts of 15.

If the entire figure represents 18, then 6 is the third part of 18, and 12 is two third parts of 18.

If the entire figure represents 21, then 7 is the third part of 21, and 14 is two third parts of 21.

And so on.  Two thirds of a number are twice as much as one third.

Example of fifths.   Here is 5 divided into fifths, that is, into five equal pieces:

Each 1 is a fifth part of 5.

2 is two fifth parts of 5.

3 is three fifth parts of 5.

4 is four fifth parts of 5.

And 5 is all five of its fifth parts.

Notice how each number says its name:

1 is one fifth of 5.

2 is two fifths of 5.

3 is three fifths of 5.

The first number says its cardinal name: one, two, three.   5 says its ordinal name, fifth.

 5. How can we calculate parts of a number? Two thirds of 12 First name the part, "One." Then multiply.

Example 1.   How much is two thirds of 12?

Answer.  To name two thirds, we must first name one third.  Say, "One third of 12 is 4Two thirds are 2 × 4 = 8 ."

Two thirds of 12, then, are equivalent to 8 out of 12.

Example 2.   How much is three fourths, or three quarters, of 28?

Answer.  Three fourths are three times more than one fourth.  Begin:

"One fourth of 28 is 7.  So, three fourths are three 7's:  21.

We can illustrate this with any number that has a fourth part, namely, any multiple of 4.  For example, 12, 40, 100:

If 12 is divided into fourths -- that is, into 3's -- then each 3 is a fourth part of 12.   6 is two fourth parts of 12.   9 is three fourth parts of 12.  Count those Fourths!

If 40 is divided into its fourths, then 10 is the fourth part of 40.   20 is two fourths (also half) of 40.  30 is three fourths of 40.

Finally, if 100 is divided into its fourths, then 25 is one fourth of 100.  50 is two fourths or half of 100.  And 75 is three fourths of 100.

Example 3.   How much is four fifths of 10?

Answer.  To name four fifths, we must first name one fifth.  One fifth of 10 is 2.

Each 2 is a fifth part of 10.

 2 + 2 = 4 is two fifth parts of 10. 2 + 2 + 2 = 6 is three fifth parts of 10. 2 + 2 + 2 + 2 = 8 is four fifth parts, or simply four fifths, of 10.

Percent: Parts of 100%

Percent is another way of expressing a part.  Since 100% is the whole (Lesson 3, Question 7), and since 50% is half of 100%, then 50% means half.

50% of 12 -- half of 12 -- is 6.

Since 25% is a quarter of 100%, then 25% is another way of saying a quarter.  A quarter of 40 -- 25% of 40 -- is 10.

Since 75% is three quarters of 100%, then 75% means three quarters.  30 is 75% of 40.

Whatever part or parts the percent is of 100%, that is the part or parts we mean.

Fifths

Let the circle represent 100%, and let us divide it into fifths, that is, into 20's.  Each 20% is a fifth of the circle.  40%, then, is two fifths of the circle.  60% is three fifths.  80% is four fifths.

That is what those percentages mean:

20% means one fifth.

40% means two fifths.

60% means three fifths.

80% means four fifths.

100% is the whole; it is all five fifths.

How much, then, is 60% of 45?

Answer.  60% means three fifths.  To name three fifths, we must first name one fifth.  One fifth of 45 is 9.  Therefore three fifths are three 9's:  27.

In general, since percent is how many for each 100, then percents are hundredths.

32 of those 100 squares have been shaded.  That is, 32%, or 32 hundredths, of them have been shaded.

*

Finally, we will state this theorem:

 Each number is either a part of a larger number or parts of it.

(Euclid, VII.4.)

We will illustrate this with each number less than 9.  We will see that each number less than 9 is either a part of 9 or parts of 9.

Now, 9 units can be divided either into Ninths or Thirds:

(If 9 is divided into Ninths, then it is divided into 1's.  If 9 is divided into Thirds, then it is divided into 3's:  3, 6, 9.)

Let us now see how to relate each number to 9.

1 is the ninth part -- or one ninth -- of 9.

2 is two ninth parts of 9.

(The point again is that each 1 is a ninth part of 9.)

3 is three ninths of 9 -- and also the third part of 9.

4 is four ninths of 9. Count them!

5 is five ninths of 9.

6 is six ninths -- and also two thirds -- of 9.   3 + 3.

7 is seven ninths of 9.

8 is eight ninths of 9.

Notice again how each number says its name:

1 is one ninth of 9.

2 is two ninths of 9.

3 is three ninths of 9.

The first number says its cardinal name.  9 says its ordinal name.

Each number less than 9, then, is either a part of 9 or parts of it. We can therefore express in words how each number is related to 9. We can say that 5, for example, is "five ninths" of 9.

Example 4.   What relationship has 9 to 10?

Answer.  If we divide 10 into 1's, then each 1 is a tenth part of 10.

1 is one tenth of 10.

2 is two tenths of 10.

3 is three tenths of 10.  And so on, until we come to 9:

9 is nine tenths of 10.

Again, the numbers 9 and 10 say their names.  9 says its cardinal name "nine."  10 says its ordinal name "tenth."

or

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