RATIO AND PROPORTION  2

Lesson 17  Section 2

This illustrates 3 out of 5:  the ratio of 3, the part, to the whole 5.
3 is three fifths of 5;  Lesson 16.  (Here, "part" refers to whatever is less than the whole.)

"Out of" is often how fractions are introduced -- as ratios but with fractional symbols. A fraction, however, is a number we need for measuring. "Out of" does nothing to explain where the number 3/5 belongs on the number line. See Lesson 19.

Example 1.   In a class of 40 students, 3 out of 5 got B.  How many students got B?

Solution 1.  To say that 3 out of 5 got B, is to say that three fifths of the students got B.  One fifth of 40 is 8.  Therefore, three fifths are 3 × 8 = 24.

Solution 2.  Proportionally,

40 is the whole number of students.  Therefore, it corresponds to 5 (not 3).

To solve the proportion, we see that 5 has been multiplied by 8.  Therefore, 3 also must be multiplied by 8:

3 5

=

24 8 × 3  40 8 × 5

24 students got B.

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

"3 out of 5" means that for every 5,

there are 3.   If there are two 5's,

there will be two 3's.  If there are three 5's, there will be three 3's:

And so on.  Therefore, if there are eight 5's, there will be eight 3's.  This is the Theorem of the Same Multiple (Section 1).

Example 2.   In a recent survey, 7 out of 10 people responded Yes.  If 280 people responded Yes, then

a)  how many people were surveyed?      b)  how many responded No?

Solution.  In this Example, 280 is the part that responded Yes.  It corresponds to 7.

Now, 280 is what number times 7?  280 is 40 × 7.  Therefore the missing term is 40 × 10.

7  10

=

280 40 × 7 400 40 × 10

400 is the whole number of people surveyed.

b)   Since 280 is the number that responded Yes, then the difference,

400280,

b)    will be the number that responded No.

400280 = 120.

Percent: Ratio to 100

6 is 6% of 100.  35 is 35% of 100.  98.2 is 98.2% of 100.

For, percent is how many for each 100, which is to say, percents are hundredths. (Lesson 14.)  6 is 6 hundredths of 100.  That is the ratio of 6 to 100.

Example 3.   8 out of 100 is equal to what percent?

 Answer.   8 out of 100 is the ratio of 8 to 100, which is 8%:

Note:  When we consider the ratio of a smaller number to 100, then we may say "out of" 100.  But when we have the ratio of a larger number to 100 -- 200 to 100 -- then 200 "out of" 100 make no sense.  In that case, we must say "for each."

Example 4.   200 for each 100 is equal to what percent?

Answer.   200%.

Example 5.   8 out of 25 is equal to what percent?

 Answer.   Now percent is not out of 25, it is out of 100.  Therefore, let us complete this proportion:

"8 out of 25 is equal to how many out of 100."

Alternately, since 100 = 4 × 25, then the third term will be 4 × 8:

8  25

=

32 4 × 8 100 4 × 25

But 32 out of 100 is 32%.  And since that is equal to 8 out of 25, then 8 out of 25 is also 32%!

Note that the following questions mean the same:

8 out of 25 is what percent?

8 is what percent of 25?

We can state the following principle:

Finding a percent by making the fourth term 100 is called the method of proportions.  We will go into it more in Lesson 29.

Example 6.   18 is what percent of 200?

Solution.  Proportionally,

"18 is to 200  as  what number is to 100?"

Here, 200 has been divided by 2.  Therefore, 18 also will be divided by 2:

18 200

=

9     18 ÷ 2 100 200 ÷ 2

"18 is the same percent of 200  that  9 is of 100."

18 is 9% of 200.

At this point, please "turn" the page and do some Problems.

or

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