Unit 10: The Percentage Formula


Performance Objectives

After completing Unit 10, you will be able to:



THE PERCENTAGE FORMULA
The principle Percentage = Rate x Base is used in many business problems. The formula is P = RB.
 
Letter Explanation
P
Percentage (portion, part, or share), always without a percent sign
R
Rate or percent, usually with a percent sign
B
Base, the original amount

Example: What is the sales tax (P) on a purchase of $25 (B) if the rate of sales tax (R) is 6 1/2%?

Solution:       P =  R     x    B

Sales tax = 6 1/2% x $25
Sales tax = $1.63
THE DIFFERENCE BETWEEN PERCENT AND PERCENTAGE
Sometimes the words "percent" and "percentage" are used interchangeably. In business mathe-matics, percentage, P, is always a number without a percent sign. Rate, R, (percent) is usually a number followed by percent sign.

1. What is the percentage formula? __________

2. Can equations or formulas be turned around--that is, can the left side be switched with the right side? __________

3. Is RB = P the same as P = RB? __________

4. If P and B are known, what is unknown? ______(a)______

Is P an amount or is it a number followed by a %? ______(b)______

Is R an amount or is it a number followed by a %? _______(c)_______

COMPUTING THE PERCENTAGE
The percentage formula can be pictured using a triangle as shown at the top of the next page. The triangle can be used as a memory aid for all the variations of the percentage formula.

The following figure shows a way to remember the formula for computing the percentage.
Example: A discount of 23% is offered on a desk costing $374. How much is the discount? The two knowns are the rate, R, which is 23%, and the base, B, $374. The unknown is the percentage, P. The triangle shows the following:
P = RB

P = 23% x $374

P = $86.02

5. Compute the percentage in the following problems. Remember to convert the percent to a fraction or a decimal. If necessary, use the Aliquot Parts Table in Unit 7.
(a)  What is 33 1/3% of $1,710? _________

(b)  What is 4% of $1,620? _________

(c)  7% of $25,000 is _________.

(d)  What is 12 1/2% of $48,016? _________

(e) 1/2% of $100 is _________.

ANALYZING WORD PROBLEMS ASKING FOR A PERCENTAGE
Problems in which the percentage, P, is unknown can be analyzed as follows.

Problem: A sale allows 10% off all marked prices. What is the allowance on an item marked $195?


6. An $80 article is being sold at a 15% discount.

(a)  What part of the percentage formula is unknown? ___________

(b)  What is the amount of the discount? ___________

7. During a 40%-off sale, how much is taken off the price of a snowmobile listed at $795? ____________

8. A salary increase of 6 1/2% will take effect next month.  How much will an employee's raise be if her present salary if $1,345 per month? _____________

9. How much is the sales tax on a new car selling for $6,340 if the sales tax rate is 6 1/2%? ___________

10. The power commission has authorized an extra charge of 15.5% for electricity to cover increases in fuel costs. How much of a monthly increase is this for a customer whose electricity bills average $140.00 per month? ____________

11. In the problem "What is 10% of $560?", what is the base? ___________

FINDING THE RATE OF PERCENT
Note that the percentage formula P  = RB can be rewritten as R = P
B
.
The following illustrates a way to remember the rate formula.

12. Rewrite the formula P = RB to compute the rate (R).
(a)  R = ___________

(b)  R is a number followed by ________________.

13. Using the formula R =  P
B
, compute the rate (R) is P is $0.50 and B is $4.50. _______

14. A loss of $34 was sustained on the sale of a desk calculator that cost $520. What was the percent of loss? ____________

ANALYZING PERCENT PROBLEMS STATED IN WORDS
Problems in which the percent R is unknown can be analyzed as follows.

What percent of 75 is 25?

"What percent" is the unknown, R.
"Of" means multiply.
The number following "of" is the base, B, in this case 75.
"Is" means equals.
The number following "is" is the percentage, in this case 25.

Example: The sales ticket on a sport coat lists the original price of $75. The sale price is $50, or $25 off. What is the percent of reduction based on the original price?

(a)  Because R is unknown, use the formula ___________.
(b)  R = ___________
(a)  Because R is unknown, use the formula ___________.

(b)  R = ___________

17. Analyze the following problem. If a customer obtained a $15 discount on an article marked $50, what was the percent of discount?
Analysis:
(a)  First, restate the problem. ________________________________

(b)  Second, write the equation.

(c)  Third, solve the equation. Because R is unknown, which percentage formula should be used? ____________

(d)  What is the amount of R? ___________________

(e)  Fourth, prove. _______________________________

18. Solve the following problems.
(a) What percent of 150 is 50? __________

(b) What percent of 6 is 1 1/2? ____________

(c) 28 is what percent of 40? ____________

In problems 19-20, go through the same analysis process presented in problem 17, above.

19. Analyze and solve. Kim Borka received a raise in wages of $30 a week. If her wages were $200 a week, what was the percent of increase? _____________________________________________

20. Analyze and solve. A direct mailing company sent out 5,000 sale circulars and received a return of 40. What was the percent of return? ______________________________________________

21. $85 is what percent of $135?

(a)  The formula is ______________.

(b)  The answer, to the nearest whole percent, is ______________.

22. In analyzing percentage problems, what word does the base amount always follow? ____________

23. What percent of $735 is $110.25?

(a)  Formula: ____________

(b)  Answer: ____________________

24. What is 20% of 125? Include in your answer the formula used. ____________(a)__________

What percent of 16 is 5? Include in your answer the formula used. _____________(b)____________

FINDING THE BASE
Note that the formula P = RB can be rewritten as B = P/R. The following illustrates a way of remembering the formula.

25. Rewrite the formula P = RB to compute the amount of B. __________________

ANALYZING BASE PROBLEMS STATED IN WORDS
Assume that May White received a $150 increase in her monthly salary, which was a 12% increase. Compute her original monthly salary.

26. Which part of the formula in the above problem is an unknown quantity? _______(a)______

Which formula should be used? ________(b)________

What was May White's salary before the raise? ________(c)________

27. Jon Estro received a salary increase of $165, which was a 15% raise. What was his original salary? __________

28. 8% of what amount is 216?

(a)  Restate as an equation. ___________

(b)  What formula should be used? ___________

(c)  Solve the equation. ___________

(d)  The answer is ____________________

29. 1,045 is 22% of what amount?
(a)  Restate as an equation. ___________

(b)  Solve the equation. ___________

(c)  What is the answer? ___________

30. To prove the answer to this type of problem, multiply the rate times the answer, which should equal the percentage.

Solve and prove the answers to the following problems.

(a)  6 is 3% of what amount? _______________ Prove. __________

(b)  15% of what amount is $186? ___________ Prove. ___________

(c)  $15 is 200% of what amount? ___________ Prove. ___________

(d)  60% of what amount is $2,160? ___________ Prove. ___________

(e)  110% of what amount is $167.20? ___________ Prove. ___________

(f)  2.50 is 1/2% of what amount? ___________ Prove. ___________

31. $176 is 200% of what amount? (Convert 200% to 2.00.) ______________________

32. Solve the following problems.

(a)  $20 is 1/2% of what amount? (Convert ~21% to 0.005.) ___________

(b) 500 is 80% of what amount? ____________

33. $1,800 is 300% of what amount? _____(a)_____

$2,100 is 150% of what amount? _____(b)_____

Business Applications

Analyze and solve the following problems.

34. The sales tax on an article was $4.83 and the tax rate is 7%. What did the article cost?

(a)  Restate the problem. ______________________

(b)  Which of P, R, or B is unknown? ___________

(c)  Write an equation. ____________________

(d)  How much did the article cost? ____________

35. Jack Bader paid $5.82 sales tax on a ski outfit in a city where the local and state sales tax was 6%.
(a)  Restate the problem. ______________________

(b)  Which of P, R, or B is unknown? ___________

(c)  Write an equation. __________________

(d)  How much did the ski outfit cost? ___________

36. Lorri received an additional $54.08 in her salary check, which represented a 10.3% raise.
(a)  Restate the problem.

(b)  Which of P, R, or B is unknown? ___________

(c)  Write an equation. ____________________

(d)  What was the amount of her original salary? __________

37. $1,600 was received on a $7,200 investment. What was the percent of return?
(a)  Restate the problem. ______________________

(b)  Which of P, R, or B is unknown? ___________

(c)  Write an equation. ____________

(d)  What was the percent of return on the investment? ___________

38. Assume that you were to receive an 8% increase in your hourly wage of $12. What would be the amount of increase?
 (a)  Restate the problem. ______________________

(b)  Which of P, R, or B is unknown? ___________

(c)  Write an equation. ____________

(d)  What would be the amount of the salary increase? ___________

39. Assume that an experienced worker receives 225% of the base weekly wage. As an experienced worker you receive $675 a week. What is the base wage?
(a)  Restate the problem. ______________________

(b)  Which of P, R, or B is unknown? ___________

(c)  Write an equation. ____________________

(d)  What is the base wage? ___________



You have finished Unit 10. Please select one of the choices below.