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Lesson 29

# The Method of Proportions

In this Lesson, we will answer the following:

1. Any number is what percent of 100?
2. How do we find the Percent by the method of proportions?

Section 2

3. What is a general method for finding the Percent?

 1. Any number is what percent of 100? 5 is  ? % of 100. Any number is that same percent of 100.

5 is 5% of 100.  12 is 12% of 100.  250 is 250% of 100.

For, percent is how many for each 100, which is to say, percents are hundredths (Lesson 14).  5 is 5 hundredths of 100.  That is the ratio of 5 to 100.
 5  100 = 5%, 12 100 = 12%, 250100 = 250%.

When the percent is less than or equal to 100%, then we can say "out of" 100.  5% is 5 out of 100.  12% is 12 out of 100.  But 250% cannot mean 250 out of 100. That makes no sense.  It means 250 for each 100, which is two and a half times (Lesson 15).

Example 1.   \$42.10 is what percent of \$42.10?

Answer.   100%  100% is the whole thing.

The method of proportions

Example 2.  24 out of 100 is 24%.  But what percent is 24 out of 200?

If there are 24 for each 200, then for each 100 there are 12.
(12 + 12 = 24.)

24 is 12% of 200.

Again, percent is how much for each 100.

 24 200 = ?  100

"24 out of 200 is how many out of 100?"

To change 200 into 100, we must take half.  Therefore the missing term is half of 24, which is 12.

Example 3.   8 out of 25 is what percent?  That is, 8 is what percent
of 25?

Solution.  Now, percent is not out of 25, it is out of 100.

But since there are 8 out of each 25,  and in 100 there are four 25's, then in 100 there are four 8's:  32.

8 is 32% of 25.

Again, if we had to do this problem formally, then we could write the proportion,

 8 25 = ?  100

"8 out of 25 is how many out of 100?"

Since four times 25 is 100, then the missing term is four times 8, which is 32.

2.   How do we find the Percent by the method of
proportions?
Solve the proportion in which 100 is the fourth term, and the Amount and Base are the first and second.
 Amount  Base = ?  100
"The Amount is to the Base as what number is to 100?"

See the vocabulary of Lesson 27:  "Amount," "Base."

Example 4.   7 out of 28 students got A.  What percent got A?

Solution.   Form this proportion,

 7 28 = ?  100

Now, 7 is one quarter of 28.  And 25 is one quarter of 100.  Therefore,

 7 28 = 25 100 = 25%.

7 is 25% of 28.

Example 5.   11 out of 50 students studied French.  What percent studied French?

Solution.   Form the proportion,

 1150 = ?  100

"11 out of 50 is how many out of 100?"

In this case, looking directly at the ratio of 11 to 50 does not help.  We must look alternately (across).  To make 50 into 100, we have to multiply by 2.  Therefore, we must multiply 11 by 2, also:

 1150 = 22 100 = 22%.

Example 6.   11 out of 200 studied French.  What percent studied French?

Solution.   In this case, to make 200 into 100, we must divide by 2, or take half.  Therefore we must also take half of 11:

 11 200 = 5½100 = 5½%.

We see that to deal with fractions (ratios):

We must divide both terms by the same number,
or we must multiply both terms by the same number.

Example 7.   3 is what percent of 25?

Solution.  This question means the same as,

3 out of 25 is what percent?

In every case, the Base follows "of."

Proportionally,

 3 25 = ?  100

25 has been multiplied by 4.  Therefore we must multiply 3 by 4, also:

 3 25 = 12  4 × 3 100 4 × 25 3 25 = 12%.

3 is 12% of 25.

With practice, this need not be a written calculation.  By simply looking at the numbers, and realizing that the Base must be 100, the student can know the Percent.

Example 8.   9 out of 20 students were able to stop smoking.  What percent were able to stop smoking?

Answer.  45%.  For, to make 20 into 100, we must multiply by 5.  Therefore we must multiply 9 by 5, also:

9 out of 20   is equal to  45 out of 100.

To find the Percent, the Base must be 100.

Example 9.   What percent of 400 is 33?

Solution.  400 is the Base; it follows "of."

 33  400 = ?   100

To go from 400 to 100, we must divide by 4.  Therefore we must divide 33 by 4, also.

 33  400 = 8¼  100 = 8¼%.

"4 goes into 33 eight (8 ) times (32) with 1 left over."

Example 10.   1000 people voted in the recent election, and 763 voted for Jones.  What percent voted for Jones?

Solution.  763 out of 1000 voted for Jones.

 763  1000 = ?   100

1000 has been divided by 10.  Therefore 763 also must be divided by 10.  We will separate one decimal place:   (Lesson 3, Question 5)

 763  1000 = 76.31

76.3% voted for Jones.

Example 11.   In a class of 45 students, there were 9 A's.  What percent got A?

Solution.  9 out of 45 got A.

 9 45 = ?  100

Here, we must look directly (down).  9 is a fifth of 45.  And 20 is a fifth of 100.

 9 45 = 20 100 .

20% got A.

Example 12.   21 is what percent of 75?

 Solution. 2175 = ?   100

Now, 100 is a third more than 75, because the difference between them is 25, and 25 is a third of 75.  Therefore, the missing term is a third more than 21:

21 + 7 = 28.

 2175 = 28 100 .

21 is 28% of 75.

But it will not alway be clear how to make the 4th term 100.  We will consider that in the next Section.

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

or

Continue on to the next Section.

1st Lesson on Percent

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