FRACTIONS

Lesson 19  Section 2

the whole, which is represented by 1.

*

3 out of 5 signifies the ratio three fifths. For, when we say that three fifths responded yes, we are not measuring anything. "Three fifths" is the ratio 3 to 5 -- the ratio of those who responded yes to the whole. (Lesson 17.)

Yet we can calculate ratio problems with fractions, and that is one of the reasons that the word fraction has become confused with the word ratio. Also, the English names of the fractions are the same as the names of the ratios. To make matters worse, the mathematical philosophy of the last century sought to explain arithmetic in terms of algebra, rather than verbally. And algebra depends only on the written appearance of symbols.

Now, when 3 out of 5 responded 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 vulgarism. You will never see it in any newspaper or journal. When we say it, of course, there is no problem.

Fractions, nevertheless, afford a written way of doing ratio problems. The fraction 3/5 represents, or signifies, a ratio, in the same sense that the letter t in physics signifies time. The symbol is never the thing.

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?

were present.

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

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

Since 100 = 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 17, 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

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

See

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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