Lesson 29 Section 3 To find the Base

Example 2. 14 is 66  2 3 
% of what number?  (See Lesson 16.) 
Solution 1. 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. 
We can see: 14 is 66  2 3 
% of a third more. 
A third of 14 is 7. 14 plus 7 is 21.
Solution 2. If 14 is two thirds of some number, then proportionally:
As 2 is to 3, then 14 is to ?
But 2 has been multiplied by 7. Therefore, 3 must also be multiplied by 7. (Lesson 18.)
As 2 is to 3, then 14 is to 21.
14 is two thirds of 21.
Example 3. 30 is 60%  three fifths  of what number?
Solution 1. 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.
30 is 60% of 50.
Solution 2. If 30 is three fifths of some number, then proportionally:
As 3 is to 5, so 30 is to ?
But 30 has been multiplied by 10. Therefore 5 must also be mutiplied by 10.
As 3 is to 5, so 30 is to 50.
30 is three fifths  60%  of 50.
Example 4. 27 is 75%  three fourths  of what number?
Solution. Proportionally:
As 3 is to 4, so 27 is to ?
3 has been multiplied by 9. Therefore 4 must also be multiplied by 9.
As 3 is to 4, so 27 is to 36.
27 is three fourths  75%  of 36.
*
Let us now consider a completely general method.
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 4. $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 5. 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 6. 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 7. 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 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. 
24 is 150% of  1½ times  16. (16 + 8 = 24.)
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?
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.
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