Example 5. Tenths, hundredths. How much is a tenth of $275? How much is a hundredth?
Answer. To find a tenth, divide by 10.
275 ÷ 10 = 27.5
(Lesson 3, Question 4.)
Since this is money, we report the answer as $27.50. (Lesson 2.)
As for a hundredth, we will separate two decimal digits:
$275 ÷ 100 = $2.75
Now, in Lesson 3 we saw that when we divide by 10, we have taken 10% of the number. And when we divide by 100, we have taken 1%. Therefore, 10% of $275 is $27.50; 1% is $2.75.
Note: Whenever we divide by any power of 10 -- the digits do not change.
275 ÷ 100 = 2.75
Conversely, then, if two numbers have the same digits, they differ by a power of 10.
Example 6. $85 is which part of $850?
Answer. Apart from the 0 at the end of $850, those numbers have the same digits. Therefore, they differ by a power of 10. 850 is in fact 10 times 85. (Lesson 3, Question 1.) Therefore, $85 is the tenth part of $850. To say the same thing, $85 is 10% of $850.
Example 7. $.98 is which part of $98?
Answer. They have the same digits. They differ by a power of 10.
$.98 is the hundredth part of $98. It is 1% of it.
Repeated division in half
If we start with the whole of something, and divide it into two equal parts --

-- then we have divided the whole in Half.
If we now divide each Half in half --

-- the whole will now be in four equal parts, or Quarters.
If we divide each Quarter in half --

-- the whole will then be in twice as many, that is, eight equal parts, or Eighths.
Each time we take half, the number of equal parts doubles. Half of a Quarter is an Eighth. Half of an Eighth is a Sixteenth. Half of a Sixteenth is a Thirty-second. And so on.
Now, here are the cardinal forms of those ordinal numbers.
2, 4, 8, 16, 32, 64, and so on.
Each cardinal number is the number of equal parts, starting with 2.
Those numbers are called the powers of 2. Repeated division in half is very common. The student should know the names of the sequence of those parts:
Halves, Quarters, Eighths, Sixteenths, Thirty-seconds, and so on.
Divisors and parts
The divisors of a number will go into the number exactly. 3 is a divisor of 12. 5 is not. The divisors of a number (except for the number itself) are the only parts that a number has. 3 is the fourth part of 12. 5 is not called a part of 12.
Example 8. Find all the divisors of 30 in pairs. Each divisor (except 30) is which part of 30?
Answer. Here are all the divisors of 30 in pairs:
1 and 30. (Because 1 × 30 = 30.)
2 and 15. (Because 2 × 15 = 30.)
3 and 10. (Because 3 × 10 = 30.)
5 and 6. (Because 5 × 6 = 30.)
On naming which part of 30, each divisor will say the ordinal name of its partner:
1 is the thirtieth part of 30.
2 is the fifteenth part of 30. 15 is half of 30.
3 is the tenth part of 30. 10 is the third part of 30.
5 is the sixth part of 30. 6 is the fifth part of 30.
Thus divisors always come in pairs. And that implies the following:
Theorem. For every divisor (except 1) that a number has, it will have a part with the ordinal name of that divisor.
(Euclid, VII. 37.)
Since 18, for example, has a divisor 3, then 18 has a third part. Since 18 has a divisor 6, then 18 has a sixth part. And so on.
Here is an illustration that 18 has a divisor 3:

18 = 6 × 3.
But according to the order property of multiplication:
18 = 3 × 6.

This shows that 6 -- the partner of 3 -- is the third part of 18.
In other words, since 18 has a divisor 3, then 18 has a third part.
Example 9. Into which parts could 12 people be divided?
Answer. The divisors of 12 are
1, 2, 3, 4, 6, and 12.
Corresponding to each divisor (except 1), there will be a part with the ordinal name of the divisor. 12 people, therefore, could be divided into
Halves, thirds, fourths, sixths, and twelfths.
You cannot take a fifth of 12 people. 12 does not have a divisor 5.
Percent: Parts of 100%
A percent is another way of expressing a part. For whatever part the percent is of 100%, that is the part we mean.
Since 50% is half of 100%, then 50% means half. 50% of 12 -- half of 12 -- is 6.
Since 25% is a quarter, or a fourth, of 100% --

-- then 25% is another way of saying a quarter. 25% of 40 -- the fourth part of 40 -- is 10.
In the next Lesson, Question 10, we will see how to take 25% by taking half of half
Since 20% is the fifth part of 100% --

(100 is made up of five 20's) -- then 20% is another way of saying a fifth. 20% of 15 -- the fifth part of 15 -- is 3.
See Problems 13 and 14.
*
When we say "5 is the third part of 15," we do not imply a sequence: the first part, the second part, the third part, and so on. When we speak of the third part, that is a different meaning for the word "third." It means each one of three equal parts that together make up the whole.

We say that we have divided 15 into thirds.
Yet "third" still retains an ordinal character. Because to the question, "Which part of 15 is 5?", we answer,
"5 is the third part of 15." We use that ordinal number because 15 is the third multiple of 5.
At this point, please "turn" the page and do some Problems.
or
Continue on to the next Section: Parts, plural
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