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Lesson 20  Section 3

"Out of"

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The word 'fraction,' in everyday speech, has come to mean a part -- half, a third, a fourth, a fifth -- as in the phrase "a fraction of the students," or "3 out of 5 students." 3 out of 5, however, is strictly a ratio. 3 is three fifths of 5. We have gone into that in Lesson 18.

In this Lesson, however, we defer to the everyday use of a fraction to express "out of." Thus when 3 out of 5 respond yes, we ask, "What fraction responded yes?" It would be very wordy to ask, "What is the ratio of those who responded yes to the total number surveyed?" Yet that is what the former question means. To write a fraction -- "3/5 responded yes" -- is a stylistically unacceptable. You will never see it in any newspaper or journal. When we say it, of course, there is no problem. (That avoids for the moment the confusion that arises from the ratios -- the parts -- and the proper fractions having the same names.)


 12.   What kind of fraction signifies "out of"?
3 out of 5
  A proper fraction. The numerator will be smaller than the denominator.
The fraction represents the ratio of a part to the whole.

  3 out of 5 is represented by the proper fraction  3
5
.  3 out of 5 is less

than the whole, which is represented by 1.

Note that the number that follows "of" must be the denominator. For that number signifies the whole; the numerator is the part of the whole.

Also, "3 out of 5" -- a smaller number out of a larger -- makes sense.  It would make no sense to say "5 out of 3."

Example 1.   In a class of 20 students, 3 were absent.  What fraction were absent?  What fraction were present?

What percent were absent?  What percent were present?

  Answer.  3 out of 20 students were absent:     Part  
Whole
 =   3 
20
.

The number that follows "of" -- 20 -- is the denominator.

Now, if 3 out of 20 were absent, then the rest, 17 out of 20 --  17
20
 --

were present.

As for the percent, it is so many out of 100.  Proportionally,

 3 
20
 =    ? 
100

"3 out of 20  is  how many out of 100?"

Since 100 is 5 × 20, then the missing term is 5 × 3:

 3 
20
 =   15 
100
 = 15%.

15% were absent.  The rest, 100% − 15% = 85%, were present.

Example 2.  The whole is the sum of the parts.   In a class, there are 17 girls and 12 boys.  What fraction of the class are girls, and what fraction are boys?

Answer.  In this problem, we are not given the whole number of students.  But the whole is the sum of the two parts:

Girls + Boys = 17 + 12 = 29.

Therefore, 17 out of 29 are girls:

  Part  
Whole
 =  17
29
.

And 12 out of 29 are boys:

  Part  
Whole
 =  12
29
.

Compare Lesson 18, Example 12.

Example 3.  Calculator problem.   In a class election, 135 students voted for candidate A, and 212 voted for candidate B.  What percent voted for A, and what percent voted for B?

Solution.  Again, the whole is the sum of the two parts:

135 + 212 = 347

Therefore, 135 out of 347 voted for A, while 212 out of 347 voted for B.

To find the percent that voted for A, press

135 ÷  347 %

(Lesson 10.)  See

 38.90489  

This is approximately

38.9%.

(We could anticipate that this would be less than 50% because 135 is less than half of 347.)

For the percent that voted for B, press

 212  ÷  347

See

 61.0951 

This is approximately

61.1%.

(We could anticipate that this would be more than 50%, because 212 is more than half of 347.)

Or, since 38.9% voted for A, then the number that voted for B is

100% − 38.9%

The student should easily find this to be 61.1%.  (Lesson 6, Question 6.)


Please "turn" the page and do some Problems.

or

Continue on to the next Lesson.

Section 1 on Fractions

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