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Lesson 29  Section 3

PERCENT OF A NUMBER

To find the Base

Back to Section 1

Example 1.   7 is 25% of what number?

Here, the Base is missing, the number that follows of.  This is an elementary problem in finding the Base. It's Example 11 in Lesson 28.

Now we have seen that to find the Amount, we multiply.

Amount = Base × Percent

Therefore, according to the relationship between multiplication and division, to find the Base, we divide.

Base = Amount ÷ Percent

Example 2.   $36 is 4% of how much?

Answer.  The Base -- the number that follows "of" -- is missing.  Let us represent 4% as the decimal .04.  And let us signify division by the division bar.

Then

Amount
 Percent		  =   36
.04
 
   =  3600
  4		   (Lesson 11)
 
   =  900.

36 is 4% of 900.

In practice, we may divide the Amount by the Percent without changing to a decimal. But then we must multiply by 100.
 
 5.   How can we find the Base when we know the
Amount and the Percent?
 
 

Example 3.   42 is 6% of what number?  Check the answer.

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

42 is 6% of 700.

Here is the check:

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

Example 4.   8% of what number is 20?

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

Check:  1% of 250 = 2.5

Therefore, 8% of 250 = 8 × 2.5 =  2.5 × 8  (Two and a half times 8)
 
  (Lesson 27, Examples 10 and 11)
 
   =  20.

Example 5.   9 is 15% of what number?

  Answer.     9 
15		  =  3
5		  = .6 (Lesson 24); times 100 is 60.

9 is 15% of 60.

(10% of 60 is 6.  5% is 3.  6 + 3 = 9.)

  Example 6.   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 7.   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 8.   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 26, Example 9.)

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

Example 9.   36 is 225% of what number?

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

Example 10.   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.

Example 11.   Michelle paid $82.68 for a pair of shoes -- but that included a tax of 6%.  What was the actual price of the shoes before the tax?

Solution.  This is Example 7:  Percent with a Calculator.


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