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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. How do we take a part of a number that ends in 0's?
  2. How do we increase or decrease a number by a given part?
  3. What do we mean by a mixed number of times?
  4. How can we find a part of a number by dividing?
  5. How can we find a fifth of a number?
  6. What percent means a third?

    Section 2

  7. Which natural numbers are called the even numbers?
  8. How much is half of 1?
  9. How can we take half of an odd number?
  10. How can we take a fourth or 25% of a number?
  11. How can we multiply by 5?
  12. How can we take 5% of a number?
  13. How can we multiply by 15?
  14. How can we take 15% of a number?

 1.   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 1.   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 2.   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.

 2.   How do we increase or decrease a number by a given part?
  To increase by a given part, add that given part; to decrease, subtract.

Example 3.   A jacket originally sold for $150.

a)  It now sells for a third more.  What is the present price?

Answer.   A third of $150 is $50.  It now sells for $50 more.  It sells for $200.

b)  It now sells for a third less.  What is the present price?

Answer.   $150 − $50 = $100.

Example 4.   A photograph measures 8 inches by 12 inches.

a)  It will be enlarged by one quarter.  What are the new dimensions?

Answer.   A quarter of 8 is 2.  A quarter of 12 is 3.  The new dimensions are:

(8 + 2) inches by (12 + 3) inches = 10 inches by 15 inches.

b)  It will be reduced by one quarter. What are the new dimensions?

Answer.  (8 − 2) inches by (12 − 3) inches = 6 inches by 9 inches.

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

Example 5.   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 6.   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 7.   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."

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.

 4.   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.)

In every case, we will divide by decomposing the dividend (Lesson 10).


Example 8.   How much is half of 112?

Solution.   We must divide 112 by 2.  But to do that, we can easily decompose 112 into two numbers that are obviously divisible by 2 -- that is, into two numbers whose half we know:

112 = 100 + 12.

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

In other words, we may "distribute" a part to each term of a sum.
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, plural:

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

Example 9.   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 10.   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 11.   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 12.   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 is 1%. (Lesson 3, Questions 7 and 8.)  Therefore we have found 10% and 1% of $62.

 5.   How can we find a fifth of a number?
 
  A fifth is twice as much as a tenth.

Example 13.   How much is a fifth 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 fifths, that is, into five equal pieces.  If we now divide each fifth in half, then we see that the whole line has been divided into ten equal pieces, that is, into tenths.  Each fifth is twice as much as a tenth.

Also, a fifth is the same as 20%. (Lesson 14.)  And in Lesson 3, Example 9, we saw that 20% is twice as much as 10%, which is a tenth.

Example 14.   How much is a fifth 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.


 6.   What percent means a third?
 
 
  For the explanation why, see the following Example.

Example 15.  The percent that means a third. 

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

Answer.  Since the whole class is 100%, then a third of the class will be a third of 100%.  We must divide 100 by 3.  It will not be a whole number. (Lesson 10.)

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

b)  What percent means two thirds?

  Answer.  Two thirds will be 2 × 33 1
3		  :
2 × 33 1
3		   =   2 × 33   +  2 ×  1
3		 .
2 × 33  = 66 .   2 ×  1
3		  =  1
3		  +  1
3		  =  2
3		 .
2 × 33 1
3		   =   66 2
3		 .
66 2
3		 % means two thirds.

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

Example 16.  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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