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FRACTIONS INTO DECIMALS

Lesson 23  Section 2

In the previous section, we saw the most frequent
and therefore the most important decimal and percent equivalents. Nevertheless, we now ask the following:
 3.   What is a general method for changing a fraction to a decimal?
 4 
11
 
  Divide the numerator by the denominator.

  Example 1.   Write   4 
11		  as a decimal. 
  Solution.     4 
11		  = 4 ÷ 11.  As they said in the Little Red School House,

"Let the 4 fall into the house"

11 does not go into 4.  Write 0 in the quotient, place a decimal point, and add a 0 onto the dividend. (Lesson 11)

"11 goes into 40 three (3) times (33) with 7 left over.

"11 goes into 70 six (6) times (66) with 4 left over."

Since we are dividing 11 into 40 again, we see that this division will never be exact.  We will have 36 repeated as a pattern:

 4 
11		 = 0.363636. . .
The three dots called ellipsis mean, "It is not possible to express   4 
11

exactly as a decimal.  However we can approximate it to as many decimal places as we please according to the indicated pattern; and the more

  decimal places we take, the closer we will come to   4 
11		 ."
  Thus we cannot express   4 
11		  exactly as a decimal.  Therefore if we want

to use that number as a decimal, we must approximate it.  Let us approximate it to three decimal places (Lesson 10):

 4 
11		 0.364
  Example 2.   Write 5  4 
11		  as a decimal. 

Answer.  According to the previous problem,

5  4 
11		 5.364  

*

We say that any decimal for   4 
11		  is incomplete.  But the decimal for ¼,

for example, which is .25, is complete.

The decimal  .363636  in and of itself is complete. But
as a value for  4 
11		 , it is incomplete.

Fractions, then, when expressed as decimals, will be either complete or incomplete.

Which fractions will have complete decimals? Only those in which the factors of the denominator are made up of 2's or 5's.  They are fractions with these denominators:  2, 4, 5, 8, 10, 16, 20, 25, 40, 50, and so on.

Example 3.

  a)   Show the decimal pattern that  1
9		  generates.

9 goes into 1 zero (0).

9 goes into 10 one (1) time with 1 left over.

Again, 9 goes into 10 one (1) time with 1 left over.

And so on. This division will never be exact -- we will keep getting 1's in the quotient.  

1
9		  = 0.111111. . .
  b)   Use that value for  1
9		  to find the value of   8
9		 .
  Solution.     8
9		  = 8 ×  1
9		   =  8 × 0.111111. . . =  0.8888888. . .
  c)   Round off   8
9		  to three decimal places.
8
9		 0.889
  Example 4.  Calculator problem.   Write as a decimal:   73
96		 .

Answer.  Divide 73 by 96.  Press

73÷ 96 =

Displayed is

 0.7604166 

Therefore, to three decimal places,

73
96		 .760

Example 5.   In a class of 52 students, 29 were women.

a)  What fraction were women?

  Answer.  Since 29 out of  52 were women, then  29
52		  were women.

(Lesson 20, Question 4.)

b)  Use a calculator to express that fraction as a decimal.

Answer.  Press

2 9 ÷ 5 2 =

See

 0.5576923 

This is approximately  .558.

c)  What percent were women?

Answer.   To change a number to a percent, multiply it by 100.

.558 = 55.8%

(Lesson 3, Question 11.)

In summary, look at what we have done:

29 out of  52  =  29
52		  =  29 ÷ 52 .558  =  55.8%

"Out of," with a calculator, always signifies division.  Division of a smaller number by a larger.

We will return to this in Lesson 30.



Please "turn" the page and do some Problems.

or

Continue on to the next Lesson.

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