6.1 percent equivalents

Business Math, Eighth EditionCleaves/Hobbs

© 2009 Pearson Education, Inc. Upper Saddle River, NJ 07458 All Rights Reserved

6.1 Percent Equivalents

Write a whole number, fraction or decimal as a percent.

Write a percent as a whole number, fraction or decimal.

1 = 100%

0.8 = 80% = 4/5



6.1.1 Write a Whole Number, Fraction or Decimal as a Percent

Percents are used to calculate markups, markdowns, discounts and many other business applications.

Hundredths and percent have the same meaning: per hundred.

100 percent is the same as 1 whole quantity, 100% = 1.

When we multiply a number by 1, the product has the same value as the original number.



Change to equivalent percents

N x 1 = N

So, if 1 = 100%, then ½ x 100% = 50%.

Also, if 1 =100%, then 0.5 x 100% = 050.% = 50%

In each case when we multiply by 1 in some form, the value of the product is equivalent to the value of the original number even though the product looks different.



Write a number as its percent equivalent

Multiply the number by 1 in the form of 100%.

The product has a % symbol.

Example:

Write 0.65 as a percent.

0.65 = 0.65 x 100% = 065.% = 65%

The decimal point moves two places to the right.



Write the decimal or whole number as a percent

0.98 = 0.98 x 100% = 098.% = 98%

1.52 = 1.52 x 100% = 152.% = 152%

0.04 = 0.04 x 100% = 004.% = 4%

5 = 5.00 x 100% = 500.% = 500%

0.003 = 0.003 x 100% = 000.3% = 0.3%



Try these examples

.48 = 48%

7.16 = 716%

0.0034 = 0.34%



Write a fraction as a percent

¼ = ¼ x 100%/1 = 25% [Reduce and multiply]

For the following, change the mixed number to an improper fraction and multiply by 100%.

3 ½ = 3 ½ x 100%/1 = 7/2 x 100%/1 =350 %

⅔ = ⅔ x 100% / 1 = 200%/3 = 66⅔%



Try these examples

⅜ = 37.5%

⅞ = 87.5%

¾ = 75%



6.1.2 Write a Percent as a Whole Number, Fraction or Decimal

When a number is divided by 1, the quotient has the same value as the original number.

N ÷ 1 = N or N/1= N

We can also use the fact that N ÷ 1 = N to change percents to numerical equivalents.

50% ÷ 100% = 50%/100% = 50/100 = ½

50%/100% = 50/100 = 0.50 = 0.5



Write the percent as a number Divide by 1 in the form of 100% or multiply by

1/100%.

The quotient does not have the % symbol.

Examples:

37% = 37% ÷ 100% = .37 = 0.37

127% = 127% ÷ 100% = 1.27

To divide by 100 mentally, move the decimal point two places to the left.



Write the percent as a fraction or mixed number

In multiplying fractions, we reduce or cancel common factors from a numerator to a denominator. Percent signs also cancel.

Division is the same as multiplying by the reciprocal of the divisor.

Similarly, % ÷ % = 1

Example:

65% = 65% ÷ 100% = 65%/1 x 1/100% = 13/20



Try these examples 150% =

1 ½

12.5% =

⅛

¼ % =

1/400



6.2 Solving Percentage Problems

Identify the rate, base and percentage in percentage problems.

Use the percentage formula to find the unknown value when two values are known.

P = R x B



6.2.1 Identify the rate, base and percentage in percentage problems

In the formula P = R x B:

“B” refers to the base which is the original number or one entire quantity.

“P” refers to percentage and represents a portion of the base.

“R” refers to rate and is a percent that tells us how the base and percentage are related.



Find the percentage The original formula is P = R x B

To find the percentage, we multiply the rate by the base.

If 60 people registered for this course and 25% are Spanish-speaking, what number of students are Spanish-speaking?

Identify the base; identify the rate.

Use the solution plan to find the answer.



1. What are you looking for?

The number of Spanish-speaking students

2. What do you know?

The base is 60 (rate); and the rate is 25% or 0.25.

3. Solution plan

P = 60 x 25% (or .25)

4. Solve

P = 15

5. Conclude

15 students are Spanish-speaking

Find the percentage



Try these problems

If 40% of the registered voters in a community of 5,600 are Democrats, how many voters are Democrats?

2,240

If 58% of the office workers prefer diet soda and there are 600 workers, how many prefer diet soda?

348



Find the base

Refer to the original formula: P = R x B.

To find “B,” we can change the formula so that it becomes: B = P/R

To find the original number, we can divide the percentage by the rate.

Example: Forty percent, or 90 diners preferred outdoor seating at the new restaurant. How many diners were interviewed in all?

Use the solution plan.



1. What are you looking for?The total number of diners surveyed.

2. What do you know?The percentage (90) and the rate (40%).

3. Solution planBase = P/R; Base = 90/.40

4. SolveB = 225

5. Conclude225 diners were interviewed in all.

Find the base



Try these examples 1700 dentists attending a convention last

month prefer fluoride treatments for preschoolers. That’s 4 out of every 5 dentists. How many dentists attended in all?

2,125

80%, or 560, of our current clients take advantage of our cash discount program for prompt payment. What is our current client base?

700



Find the rate Refer to the original formula: P = R x B.

To find “R,” we can change the formula so that it becomes: R = P/B

To find the rate, we can divide the percentage by the base.

Example: 55 insurance agents were able to meet with their clients to inform them of policy changes. If there are 220 agents in all, what percent does this represent?



Use the solution plan1. What are you looking for?

The percent or rate of agents who talked to their clients.

2. What do you know?The base or total number of agents and the percentage

who talked to their clients.

3. Solution planR = P/B ; R = 55/220

4. SolveR= .25

5. Conclusion25% of the agents talked to their clients.



Try these examples

The plant foreperson reported that 873 of the 900 items tested met the quality control specifications for production. What is the rate of acceptable items?

97%

In the new product focus group, 6,700 of the 8,375 customers rated the product as “very good” or “superior.” What was the rate?

80%



Identify what is missing

Sometimes, you will be asked to find one of the elements: rate, base or percentage when you know the other two.

Learn to “read” the problem to identify the missing element.

Example: 30% of 70 is what number? 30% is the rate. 70 is the base. You are looking for “P” or percentage. P = R x B P = 0.3 x 70 = 21



Try these problems

Identify what’s missing and then solve the problem using the correct formula.

60 is what percent of 80?

R = P/B R = 75%

35% of 350 is what?

P = R x B P = 0.35 x 350 = 122.5

25% of what number is 125?

B = P/R B = 125/.25 = 500



6.3 Increases and Decreases

Find the amount of increase or decrease in percent problems.

Find the new amount directly in percent problems.

Find the rate or the base in increase or decrease problems.



6.3.1 Find the Amount of Increase or Decrease in Percent Problems

Examples of increases in business applications include:

Sales tax

Raise in salary

Markup on a wholesale price



Decreases in percent problems

Some examples of decreases include:

Payroll deductions

Markdowns

Discounts on sale items



How to find the amount of increase

To find the amount of increase: amount of increase = new amt – beg. amt.

Example: Joe’s salary has been $400 a week. Beginning next month, it will be $450 a week. The amount of increase is $50 a week.



To find the amount of decrease:Amount of decrease = beg. amt - new amt.

Example: Roxanne’s new purse originally cost $60, but it was on sale when she bought it on Saturday for $39.99. The amount of decrease (or markdown) is $20.01.

How to find the amount of decrease



Percent of change

The amount of change is a percent of the original or beginning amount.

Find the amount (increase or decrease) from a percent of change by: Identifying the original or beginning amount

and the percent or rate of change. Multiplying the decimal equivalent of the rate

of change by the original or beginning amount.



Here’s an example

Your company has announced a 1.5% cost of living raise for all employees next month. Your monthly salary is currently $2,300. Starting next month, what will your new salary be?

You will need to find the amount of increase by multiplying the rate by the base.

To find the new amount, add the amount of increase to the original amount.



Find the new amount

Current salary = $2,300 a month Rate of change = 1.5% Amount of raise =

Percent of change x original amount.015 x $2,300 = $34.50 a month

Add $34.50 to the original amount of $2,300 to identify the new amount.

New amount = $2,334.50



6.3.2 Find the New Amount Directly in Percent Problems

Often in increase or decrease problems, we are more interested in the new amount than the amount of change.

Find the new amount by adding or subtracting percents first.

The original or beginning amount is always considered to be the base and is 100% of itself.



Find the new amount directly in a percent problem

Find the rate of the new amount.

For increase: 100% + rate of increase

For decrease: 100% - rate of decrease

Find the new amount.

P = R x B

New amount = rate of new amt. x original amt.



Here’s an example

Medical assistants are to receive a 9% increase in wages per hour. If they were making $15.25, what is the new per hour salary to the nearest cent?

Rate of new amount = 100% + rate of increase Rate of new amount = 100% + 9% = 109% Rate of new amount = $15.25 x 109% Change 109% to its decimal equivalent: 1.09 $15.25 x 1.09 = $16.6225 = $16.62



Here’s another example

A new pair of jeans that costs $49.99 is advertised at 70% off. What is the sale price to the nearest cent of the jeans?

Rate of new amount = 100% - rate of decrease = 100% - 70% = 30% New amount = rate of new amt. x original amt. New amount = 30% x $49.99 New amount = 0.3 x $49.99 = $14.997 New amount = $15.00 (nearest cent)



Try these examples The property taxes at your business office will go

up 5% next year. Currently, you pay $3,400. How much will you pay next year?

$3,570

A wholesaler is offering you a 20% discount if you purchase new inventory before the 15th of the month. If your normal invoice is $3,600, how much would you pay if you got the discount?

$2,880



6.3.3 Find the Rate or the Base in Increase or Decrease Problems

Identify or find the amount of increase or decrease.

To find the rate of increase or decrease, use the percentage formula: R = P/B.

Rate = amount of change/original amount.

To find the base or original amount, use the percentage formula: B = P/R.

Base = amount of change/rate of change.



Here’s an example

During the month of May, a graphic artist made a profit of $1,525. In June, she made a profit of $1,708. What is the percent of increase in profit?

Use the solution plan to figure out the answer.



Solution plan What are you looking for?

Percent of increase in profits.

What do you know? Original amt. = $1,525; New amt.=$1,708

Solution plan Find amt. of increase; Find percent of increase.

Solution $1,708-$1,525 = $183

$183/$1,525 = 0.12 =12%

Conclusion The rate of increase in profit is 12%.



Try these two examples

A popular detergent cost $5.99 last Saturday, but today the same detergent costs $7.50. What is the rate of increase?

25.2%

Sales in the East Region were $10,800 in January and dropped to $9,700 in February. What is the rate of decrease from January to February?

10.2%

6.1 percent equivalents

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