Lesson 20 Section 3
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 fact that the ratios -- the parts -- and the proper fractions have the same names
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?
The number that follows "of" -- 20 -- is the denominator.
"3 out of 20 is how many out of 100?"
Since 100 is 5 × 20, then the missing term is 5 × 3:
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:
And 12 out of 29 are boys:
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
(Lesson 10.) See
This is approximately
(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
This is approximately
(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.
Continue on to the next Lesson.
Please make a donation to keep TheMathPage online.
Copyright © 2017 Lawrence Spector
Questions or comments?