Lesson 29

WHAT PERCENT?

The Method of Proportions

In Lesson 13, we saw how to use a calculator to find what percent one number is of another. In this Lesson, we will see which problems should not require a calculator. They are those problems in which the Base -- the number the follows "of" -- is either a divisor of 100 (20, 25, 50), a multiple of it (200, 300, 400), or in which the Amount is a simple part of the Base (half, a third, a quarter). With a little practice, the student should be able to do such problems easily and mentally.

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.  5 is 5 hundredths of 100.  That is the ratio of 5 to 100.

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.

If we consider this as a proportion problem, then we must find the missing term:

24 out of 200  is  ____ out of 100?

When we say "24 out of 200," the Base -- the number that follows "of" -- is 200.  But with percent, the Base must be 100.  What must we do then to 200 to make it 100?  We must take half of it, or equivalently, divide it by 2.  Therefore we must also divide 24 by 2.

24 out of 200  is  12 out of 100.

But 12 out of 100 is 12%.  Therefore, 24 out of 200 is also 12%.

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

Those questions mean the same. In each one, the base
25 follows "of."

Solution.  In this problem, the Base is 25.  To find the percent, we must make the Base 100.  Therefore must multiply both terms by 4.

8 out of 25  is equal to  32 out of 100.

That means:

8 is 32% of 25.

An all too common method these days is to make this an algebra problem.

The student is taught to cross-multiply and solve for x. This is a method for someone who do not understand percent. Whoever does understand percent will either do such a problem mentally, or, if that is too difficult, with a calculator: Percent = Amount ÷ Base.

In any percent problem, the Base will always follow "of."

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

7 is 25% of 28.

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

 Solution.   Form the proportion,

"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:

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

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

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,

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 should 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."

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

"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.

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

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.

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

20% got A.

Example 12.   21 is what percent of 75?

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.

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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