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A R I T H M E T I C

Lesson 15

PARTS
OF NATURAL NUMBERS  2

This Lesson depends on Parts of Natural Numbers  1

In this Lesson, we will answer the following:

1. What do we mean by a mixed number of times?
2. How do we take a part of a number that ends in 0's?
3. How can we find a part of a number by dividing?
4. How can we take a fifth or 20% of a number?

Section 2

5. Which numbers are the even numbers?
6. What is an odd number?
7. How much is half of 1?
8. How can we take half of an odd number?
9. How can we take a fourth or 25% of a number?
10. How can we multiply by 5?
11. How can we multiply by 15?
12. How do we increase or decrease by a given part?

 1. What do we mean by a mixed number of times? A whole number of times plus a part.

Example 1.   How much is two and a half times 8?

Answer.  "Two and a half times 8" means

Two times 8 plus half of 8.

Two times 8 is 16.  Half of 8 is 4.  16 plus 4 is 20.

Example 2.   A cheese sells for \$6 a pound, and you buy three and a half pounds.  How much do you pay?

 Answer. Three pounds cost \$18. Half a pound costs \$3. You pay \$21.

That is, "Three and a half times 6" means

Three times 6 plus half of 6.

18 + 3 = 21.

That is a mixed number of times:  A whole number of times plus a part.

Example 3.   How much is five and a quarter times 8?

 Answer. "Five times 8 is 40. "A quarter (or a fourth) of 8 is 2. "40 + 2 = 42."

 2. How do we take a part of a number that ends in 0's? A third of 120 Ignore the 0's and take that part of what remains; then, put back the 0's.

Example 4.   How much is a third of 120?

Answer.   Ignore the 0. Then

A third of 12 is 4.

A third of 120 is 40.

Similarly,

A third of 1,200 is 400.

A third of 120,000 is 40,000.

It is a 4 followed by four 0's.

Example 5.   How much is an eighth of 4,000?

Answer.  If we ignore all the 0's, then we cannot take an eighth of 4.  But if we ignore only two 0's, then

An eighth of 40 is 5.

Therefore,

An eighth of 4000 is 500.

Finding a part by dividing

We have seen in Lesson 10 that to divide a number into equal parts, we divide.  We can now restate that in the language of parts.  And everything we know about division will follow.

 3. How can we find a part of a number by dividing? Divide by the cardinal number that corresponds to the part.

To find half of a number, divide by 2; to find third, divide by 3; to find a fourth, divide by 4; and so on.  (This is a theorem whose proof is indicated in Lesson 10, Example 5.)

Example 6.   How much is half of 112?

112 = 100 + 12.

Half of 100 is 50.  Half of 12 is 6.  Therefore, half of 112 is 56.

In other words, a part can be distributed.

 If we add the same part of numbers, we will get that same part of the sum of those numbers.

(Euclid, VII. 5.)

Half of 100 + Half of 12 = Half of 112.

This is also true for adding parts:

Three fourths of 100 + Three fourths of 100 = Three fourths of 200.

Example 7.   How much is a third of 252?

Solution.   Upon decomposing 252 into 240 + 12:

 A third of 252 = A third of 240 + A third of 12 = 80 + 4 = 84.

Example 8.   Three people go to lunch and the bill is \$32.40.  How much does each one pay?

Solution.   We must find a third of \$32.40.  Now, a third of \$30 is \$10.  A third of the remainder, \$2.40, is \$.80.  Each one pays \$10.80.

Example 9.   How much is a fifth of \$37.50

Solution.   What number closest to 37 has an exact fifth part?  35. Therefore, decompose \$37.50 as

\$35 + \$2.50.

A fifth of \$35 is \$7.

A fifth of \$2.50 is \$.50.

Therefore, a fifth of \$37.50 is \$7.50.

We will see below another way to find a fifth.

Example 10.   How much is a tenth of \$62?  How much is a hundredth?

Answer.  To find a tenth of a number, divide by 10.  To divide a whole number by 10, separate one decimal place.  (Lesson 3, Question 5.) Therefore,

A tenth of \$62 is \$6.20.

To find a hundredth of a whole number, separate two decimal places:

A hundredth of \$62 is \$.62.

Now, a tenth of a number is 10% of it, and a hundredth of a number is 1%. (Lesson 3, Questions 8 and 9.)  Therefore we have found 10% and 1% of \$.62.

 4. How can we find a fifth or 20% of a number? A fifth of \$240? A fifth is twice as much as a tenth.

Example 11.   How much is a fifth or 20% of \$240?

Solution.   A tenth of \$240 is \$24.  Therefore a fifth is 2 × \$24 = \$48.

To see that a fifth is twice as much as a tenth, divide the green line into tenths, that is, into ten equal pieces.  If we divide that same line into fifths, that is, into five equal pieces, then we can see that each fifth is twice as much as a tenth.

Also, in Lesson 3, Question 9, we saw that 20% is twice as much as 10%, which is a tenth.

Example 12.   How much is 20% of \$345?

Solution.   10% or a tenth of \$345 is \$34.50.  Therefore a fifth is

 2 × \$34.50 = 2 × \$34  +  2 × \$.50 = \$68  +  \$1.00 = \$69.

Example 13.  The percent that means a third.

a)  In a recent exam, a third of the class got A.  What percent got A?

 100  3 = 99 + 1     3 =  33 + 13 =  33 13 .
 33 13 % of the class got A.
 We see, then, that 33 13 % means a third.

Again, percents are parts of 100%. (Lesson 14.)  Just as 50% means half — because 50 is half of 100 — and 25% means a quarter, because 25

 is a quarter of 100, so 33 13 % means a third.  33 13 is a third of 100.

b)  What percent means two thirds?

 Answer.  Two thirds will be 2 × 33 13 :
 2 × 33 13 = 2 × 33 +  2 × 13 .
 2 × 33 = 66 .   2 × 13 = 13 + 13 = 23 .
 2 × 33 13 = 66 23 .
 66 23 % means two thirds.

In Section 2 we will see a simple way to find a quarter or 25% of a number.

Example 14.  Calculator problem.   How much is five eighths of \$650.16?

Solution.   To find five eighths, we must first find one eighth.  Press

650.16 ÷ 8

See   81.27

Five eighths will be

5 × 81.27 = 406.35

On a simple calculator, the problem can be done in sequence by pressing

650.16 ÷ 8 × 5 =

At this point, please "turn" the page and do some Problems.

or

Continue on to the Section 2.

1st Lesson on Parts of Natural Numbers

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