skill

S k i l l
 i n
A R I T H M E T I C

Table of Contents | Home | Introduction

Lesson 29  Section 3

To find the Base
the number that follows "of"

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.  If we

  represent 4% as the fraction    4  
100
,  then we are to evaluate
Amount ÷ Percent  =  36 ÷    4  
100
.

But we have seen that division is multiplication by the reciprocal.

36 ÷    4  
100
 =  36 ×  100
  4  
 =  9 × 100 = 900.

"4 goes into 39 nine (9) times. 9 times 100 is 900."

36 is 4% of 900.

In practice, to multipy by the reciprocal, we often just divide the Amount by the Percent, and then multiply by 100.


 
 7.   How can we find the Base when we know the
Amount and the Percent?
 
  Base = (Amount ÷ Percent) × 100
 

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 is 7.  Therefore, 6% is 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.   We can represent 9 ÷ 15 as the fraction   9 
15
.  That reduces to  3
5

or .6. (Lesson 24)

.6 × 100 = 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?

Find the Base

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.

Find the Base
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,

Find the Base

then a third of 30 is one fifth.  

Find the Base

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
. That is, we will multiply by its reciprocal.

Find the Base

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

(Lesson 26, Example 9.)

Example 9.   36 is 225% of what number?

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

Find the Base

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?

Now, 80% =  4
5
 (Lesson 24) .  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.


Please "turn" the page and do some Problems.

or

Continue on to the next Lesson.

Section 1 of this Lesson

Introduction | Home | Table of Contents


Please make a donation to keep TheMathPage online.
Even $1 will help.


Copyright © 2014 Lawrence Spector

Questions or comments?

E-mail:  themathpage@nyc.rr.com