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PERCENT OF A NUMBER

Lesson 28  Section 2

To find the Base

We have seen that to find the Amount we multiply.  To find

3% of $600,

where $600 is the Base, first name 1%:

1% of $600 is $6.00.

Therefore, 3% is

3 × $6.00 = $18.00.

That is,

$18.00 is 3% of $600.

But say that we are given $18.00, and we ask:

$18.00 is 3% of how much?

In other words,

3 × ? = $18.00?

According to the relationship between multiplication and division, then to find the Base, we have to divide.

$18 ÷ 3 = $6.

But $6 is 1% of the Base.  The Base itself is 100 times that.

$18 is 3% of $600.

And we can check it.  1% of $600 is $6.00.  So 3% is 3 × $6.00 = $18.00.
 4.   How can we find the Base when we know the Amount and the Percent?
 
$36 is 4% of _?_
 
  Base = Amount ÷ Percent × 100

Example 1.   $36 is 4% of how much?

Answer.  The Base -- the number that follows "of" -- is missing.  And

$36 ÷ 4 = $9.

But $9 is 1% of the Base.  The Base itself is 100 times that.

$36 is 4% of $900.

Let's check it.  1% of $900 is $9.  Therefore, 4% is 4 × $9 = $36.

If we wrote 4% as    4 
100		 , then the Base is equal to
36 ÷ 4% = 36 ÷   4 
100		  = 36 × 100
  4		  = 36 ÷ 4 × 100.

Base = Amount ÷ Percent × 100.

Example 2.   42 is 6% of what number?  Check the answer.

Answer.  42 ÷ 6 = 7; times 100 is 700.

Here is the check:

1% of 700 = 7.  Therefore, 6% = 6 × 7 = 42.  

Example 3.   8% of what number is 20?

   Answer.   20 ÷ 8 = 2 4
8		  = 2 1
2		 ; times 100 is 250.

20 is 8% of 250.

Check:  1% of 250 = 2.5

Therefore, 8% = 8 × 2.5  =  2.5 × 8   (Two and a half times 8)
  (Lesson 26, Examples 10 and 11)
   =  20.  
  Example 4.   14 is 66 2
3		 % of what number?

Answer.  The problem asks:

14 is two thirds of what number?

But if 14 is two thirds, then half of 14 is one third.  Half of 14 is 7.  And 7 is one third -- of 21.

14 is 66 2
3		 % of 21.

Example 5.   30 is 60% -- three fifths -- of what number?

Answer.  If 30 is three fifths of some number,

then a third of 30 is one fifth.  

A third of 30 is 10.  And 10 is one fifth -- of 50.

Example 6.   24 is 150% of what number?

  Answer.  150% is represented by the number 1½, or  3
2		 .  We must divide
  24 by  3
2		 .

24 is 150% of -- 1½ times -- 16.   (16 + 8 = 24.)

(Lesson 25, Example 8.)

We see then that to find the Base, we may multiply the Amount by the reciprocal of the fractional form of the percent.

Example 7.   36 is 225% of what number?

  Solution.   225% = 2¼ =  9
4		 .  Multiply 36 by  4
9		 .

Example 8.   Maria is retired and withdraws money from her retirement account.  But a tax of 20% is automatically withheld.  If she needs $1200, how much must she actually request?

Solution.   Since 20% will be withheld, Maria will receive 80% of her request.  So the question is:  $1200 is 80% of how much?

To calculate that, we must divide 1200 by .8 or, equivalently,  4
5		 .  But
  that will result in multiplying by  5
4		  or 1¼.  In other words, she must

request one and a quarter times, or one quarter more, than what she actually needs.

One quarter of $1200 is $300.  Therefore she must withdraw $1500.

It is then a simple matter to see that 20%, or one fifth, of $1500 is $300, so that her net amount will in fact be $1200.


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