chapter 6: percents
CHAPTER 6: PERCENTS
CHAPTER 6 CONTENTS 6.1 Introduction to Percents 6.2 Solve Percent Problems 6.3 Application Problems 6.4 Financial Literacy 6.5 Circle Graphs
6.1 Introduction to Percents and Conversions to Fractions and Decimals Introduction Fractions are used to describe a part of a whole, no matter what makes up the whole. For example, describes 2 parts of a whole that is divided into 5 equal pieces. And as you have just learned decimals, too, describe parts of a whole, but in terms of base 10. Thus, 0.2 represents 2 parts of a whole divided into 10 equal parts, .34 represents 34 parts of a whole divided into 100 parts, and .502 represents 502 parts of a whole divided into 1000 equal parts. Although fractions and decimals are two very common ways of identifying parts of a whole, they are not the only ways. Percents can also be used. By definition, a percent is specifically a representation of a part of a whole that is divided into exactly 100 equal pieces. The term is a form of the latin phrase, per centum, meaning for every hundred. Thus we have come to accept a percent as meaning part of one hundred. Therefore, to say 85 percent is to signify 85 parts of one hundred. However, when identifying a percent it would be too much to always have to write the word percent after the part identified. To avoid having to do this we use a symbol to represent the term percent. That symbol is % and is called, none other than, a percent symbol. We can see this demonstrated in the following two examples:
Example 1: What percent would be used to represent 34 parts of a whole divided into 100 equal parts? Solution: Since we are interested in parts of a whole divided into 100 equal parts we are already dealing with a percent. Therefore, if we use the percent symbol, 34 parts of 100 equal parts will be represented by 34%.
Example 2: Write 57% by identifying the parts of a whole. Solution: Remembering that the percent symbol, %, represents parts of 100, we see that 57% must represent 57 parts of a whole divided into 100 equal parts.
Now that we know that percents are yet another way to represent a number of parts of a whole divided into equal parts we can demonstrate how fractions, decimals, and percents
are related to one another. In fact, they are simply different representations of the same value. Thus, it is possible to convert one representation in terms of one of the other two. Let’s examine each translation one at a time.
Convert Between Percents and Fractions 1(a) - Percent to Fraction: When translating a percent to a fraction we can simply remember the meaning of a percent in general and the percent symbol in particular. The procedure requires that we multiply by the unit ratio of
. Of course, we should also remember to reduce the
resulting fraction to lowest terms. This is demonstrated in the following examples.
Example 3: Convert 63% into a fraction. Solution: By definition 63% represents the value of 63 parts of a whole divided into 100 equal parts. If we multiply the percent by 1/100% we get
Since there are no common factors other than 1 remaining in the numerator and denominator this fraction is already in reduced form.
Example 4: Convert 45% into a fraction. Solution: 45% represents the value of 45 parts of a whole divided into 100 equal parts. Again, multiplying by 1/100% we have,
Using the common factor of 5 in the numerator and denominator, we can reduce the fraction
Example 5: Convert 120% into a fraction. Solution: The translation of percents greater than 100% follow the same pattern as any other percent to fraction translation. Therefore 120% represents 120 parts of a whole divided into 100 equal parts. Following the pattern, we have
Recall that this is an improper fraction and can be written as should then be reduced to
. The fraction
.
Example 6: Convert 6 2/3% into a fraction. Solution: To convert percents that are mixed numbers (or contain fractions,) it should first be converted into an improper fraction. Once this is completed we then should follow the procedure we have for the rest of the examples. Thus,
Which reduces to
1(b) - Fraction to Percent: To convert a fraction into a percent we can use a different form of the same unit ratio by multiplying by
. The following example demonstrates what must be done.
Example 7: Convert into a percent. Solution:
Example 8: Convert Solution:
into a percent.
Convert Between Percents and Decimals 2(a) - Percent to Decimal: To convert a percent into a decimal we need to again multiply the percent by the unit ratio 1/100%. Notice this is the same procedure we used to convert a percent to a fraction. However, we do not want a fraction so we must continue the process by simply carrying out the division as presented in the resulting fraction. The following examples demonstrate this process. Example 9: Convert 82% into a decimal. Solution:
82% x 1/100% = 82%/100% = 82/100 82/100 = 0.82
Example 10: Convert 17% into a decimal Solution:
17% x 1/100% = 17% / 100% = 17/100 = 0.17
2(b) - Decimal to Percent To convert a decimal to a percent we will reverse the percent to decimal process and multiply by 100% instead of multiplying by 1/100%. The following examples demonstrate how this works. (include patter of moving decimal to right when multiplying by powers of 10) Example 11: Convert .94 as a percent. Solution: .94 x 100% = 94% Example 12: Convert .058 as a percent. Solution: .058 x 100% = 5.8%
6.1 Introduction Exercises: Solve each problem. 1. Convert 25% into a fraction 2. Convert 56% into a fraction 3. Convert 13% into a fraction.
4. Convert 71 1/2% into a fraction. 5. Convert 90% into a fraction. 6. Convert 1/3 into a percent. 7. Convert 5/6 into a percent. 8. Convert 21/25 into a percent. 9. Convert 2 7/8 into a percent 10. Convert 37/50 into a percent. 11. Convert 29% into a decimal. 12. Convert 24.6% into a decimal. 13. Convert 5.75% into a decimal. 14. Convert 1 2/3% into a decimal. 15. Convert 0.78 into a percent. 16. Convert 0.5 into a percent. 17. Convert 0.025 into a percent. 18. Convert 2.07 into a percent. 19. Convert 0.0038 into a percent. 20. Complete the table by finding the missing conversions. Percent Fraction Decimal 16% 27/50 13/10 82.8% .40 .056
6.1 Introduction Exercises Answers: 1. ¼ 2. 14/25 3. 13/100
4.
143/200
5. 9/10 6.
33 1/3%
7.
83 1/3%
8.
84%
9.
287 1/2%
10. 74% 11. 0.29 12.
0.246
13.
0.0575
14.
0.01667
15.
78%
16.
50%
17.
2.5%
18.
207%
19.
0.38%
20. Percent 16% 54% 130% 82.8% 40 5.6%
Fraction 4/25 27/50 13/10 207/250 2/5 7/125
Decimal 0.16 0.54 1.30 0.828 .40 .056
6.2 Solve Percent Problems Percentages are used to express how large one quantity is, relative to another quantity. The first quantity usually represents a part of, or a change in, the second quantity, which should be greater than zero. Also when we translate a statement in English to Math the word “of” is interpreted as multiplication. So here we can use the pattern in which all percentage problems can be set up. Proportion Method We can set up the percentage problem as a proportion. We know that Percent is always out of 100, we can write the proportion as:
=
from Amount is Percent% of Base
We then can substitute given values and solve for the unknown value. Example 1: 90 is what percent of 150? Here 90 is amount and 150 I base. We need to find the percent by substituting these values in the proportion as follows: =
This then can be written as P =
= 60%
Example 2: What number is 55% of 40? Here 55 is percent and 40 is base. We need to find the amount by setting up the proportion as follows: =
This then can be written as A =
= 22
Example 3: 19 is 20% of what number? Here 19 is amount and 20 is percent. We need to find the base by setting up the proportion as follows: =
This then can be written as B =
Formula Method We can also set up a percentage problem as a formula as follows: Percent * Base = Amount OR _____% of ______ = ________. When reading a problem we will have to decide which value is the base and which is the amount. Or in other words which number goes in which place. For example “15% of 200 is 30” can be written as 0.15*200 = 30. Do not forget to change 15% to decimal form so we can use it in multiplication. 200 is known as the base and 30 as the amount. Example 4: What percent of 64 is 16? In this question value of percent is to be found (calculated) while base is 64 and amount is 16. We translate this as percent × 64 = 16 n × 64 = 16 n = 16/64 = 0.25 n = 25% Example 5: 36 is what percent of 180? In this question 36 is amount and 180 is base. We need to calculate the value of percentage. We translate this as
36 = P × 180 36/180 = P 1/5 = 0.2 = P P = 20%
Example 6: What percent of 40 is 100? Again this problem 100 is amount and 40 is base. We need to calculate the value of percentage. We translate this as
P * 40 = 100 P = 100/40
P = 5/2 = 2.5 P = 250%
Examples to find Base: Example 7: 30% of what number is 45? Here we can see that 30 is percent, 45 is amount and the value of base needs to be calculated. Change the 30% to a decimal (0.3). So we can substitute these values in the formula as follow: 0.30 × B = 45 B = 45/0.30 B = 150
Example 8: 30 is 15% of what number? Here again we can identify 30 as Amount and 15 as Percent. We now substitute these values in the formula as follows: 0.15 × B = 30 B = 30/0.15 B = 200
Example 9: 180% of what number is 81? Here 180 is the Percent, 81 is the Amount and we need to calculate the value of base. We substitute the value and calculate base as follows: 1.80 × B = 81 B = 81/1.80 B = 45
Examples to find Amount: Example 10: What is 20% of 35? Here 20 is Percent and 35 is Base. By substituting these values in the formula we can calculate Amount as follows: A = 0.20 × 35 A=7 Example 11: 30% of 17 is what number? Using the formula we can write: 0.30 × 17 = 5.1 Example 12: What is 110% of 80? Substituting appropriate numbers in the formula we get A = 1.10 × 80 A = 88
6.2 Solve Percent Problem Exercises 1) What number is 30% of 300? 2) What number is 8.9% of 10? 3) 120% of 4000 is what number? 4) 15% of 90 is what number? 5) 15 is 25% of what number? 6) 32 is 80% of what number? 7) What percent of 125 is 25? 8) What percent of 100 is 150?
9) 40 is what percent of 400? 10) 24 is what percent of 60?
6.2 Solve Percent Problem Exercise Answers 1) 90 2) .89 3) 4800 4) 13.5 5) 60 6) 40 7) 20% 8) 150% 9) 10% 10) 40%
6.3 Application Problems Now that you have practiced working with percents, we will move on to applying these skills to problems seen in the real world. Calculating sales tax, commission earnings, and percent increase / decrease are all important skills to have. In the previous section, we saw three patterns. Each of the problems in this section will be solved using one of those three patterns: What number is x% of y? x is what percent of y? y is x% of what number?
Sales Tax When something is purchased, there is almost always sales tax to pay. The sales tax rate is given by a percent. That percent depends usually on the state, but sometimes varies by cities or
counties within a state. To find the amount of tax for an item, multiply the purchase price by the tax rate. Use the “What number is x% of y?” pattern. To find the total cost, add the purchase price and the amount of tax. When completing the word problems, be careful to answer the question that is asked. Sometimes just the amount of tax is asked for, sometimes the total price, and sometimes the tax rate itself. Let’s look at a few examples: Example 1: Julia is going to buy a shirt for $12.00. How much tax will she pay for the item if the tax rate is 6%? Use the pattern: What number is x% of y? “What number” is the amount of tax to be paid. It is the unknown, so we will assign it the variable t. Be sure to change the percent to a decimal and remember the “of” means to multiply. What number is x% of y? t = 6% of 12.00 t = 0.06 * 12.00 t = 0.72 The amount of tax is $0.72. Example 2: The television that Miguel wants to purchase costs $449. a.
How much will he pay if the sales tax is 8%?
b.
How much is the total cost?
As in example 1, use the pattern “what number is x% of y?” t = 8% of 449 t = 0.08 * 449 t = 35.92 The amount of tax is $35.92. This is the answer to part a. Part b asks for the total cost. This is found by adding the original cost and the tax amount. $449 + $35.92 = $484.92 The total cost is $484.92. This is the answer to part b.
Example 3: Nathanael wants to buy a science kit with a sticker price of $37.50. After a 6% sales tax, what is the final cost? Again, use the pattern “what number is x% of y?” t = 6% of 37.50 t = 0.06 * 37.50 t = 2.25 The amount of tax is $2.25. The total cost is needed, not the amount of tax. Add the original cost and the amount of tax. $37.50 + $2.25 = $39.75 The total cost is $39.75.
Example 4: Jada found a pair of jeans that she would like to purchase. She has $65 to spend. If the price is $62.75 and the tax rate is 8%, does she have enough money? Explain your answer. In this case, we need to first find out how much the jeans cost. What number is x% of y? t = 8% of 62.75 t = 0.08 * 62.75 t = 5.02 The amount of tax is $5.02. The total cost is $62.75 + $5.02 = $67.77 The question asked if she would enough money. The total cost is $67.77, but Jada only has $65. Therefore, No, Jada does not have enough money. The question asks for an explanation, a good explanation is: “Since the total cost is $67.77 and Jada only has $65, she does not have enough to buy the jeans.”
Example 5: Samuel bought a computer for $1450 and paid $101.50 in sales tax. What is the tax rate for the state he bought the computer? In this question, what we are looking for is the tax rate, so we need to look at a different pattern. The pattern with the variable as the percent is: x is what percent of y? Since the tax rate or percent is always multiplied by the cost of the item, the cost of 1450 will replace y in the equation. Since product gives the amount of tax, the 101.50 will replace x. We will use the variable “p” to represent the tax rate as a decimal. 101.50 = p * 1450 To solve the equation for x, divide both sides by 1450. 101.50 = p * 1450 1450 1450 0.07 = p The tax rate is 7%
Commission Many jobs include a base salary plus bonuses called commissions. Some employees only receive commission as their pay. This is popular in many sales jobs. Many salespeople do a better job by selling more of the product or service when their income depends on how much they sell. Commission is typically calculated using a percent. The percent of the amount sold is the commission. Example 6: Temperance is a real estate agent who earns a 3% commission on each house sold. If she sells a house for $225,000, what will her commission be? Use the pattern: what number is x% of y? “What number” represents the commission, We’ll use “c” for the variable. Don’t forget to change the percent to a decimal. c = 0.03 * 225,000 c = 6,750 The commission of the sale of the house is $6,750.
Example 7: Seeley gets a commission of 2% and received a commission check of $7,000 in a certain month. How much did he sell in that month? Use the pattern: y is x% of what number? This time, “what number” represents the amount that was sold, we’ll assign “a” as the variable. 7,000 = 0.02 * a To solve the equation, divide both sides of the equation by 0.02. 7,000 = 0.02 * a 0.02 0.02 350,000 = a The house sold for $350,000 if the commission was $7,000.
Example 8: Angela sells cars for a living. In one week, she sold $37,500 worth of cars and earned a commission of $1875. What is her commission rate? Since we are looking for the percent we’ll use the pattern: “x is what percent of y” and assign “p” as the variable to represent to percent. x is the amount of commission and y is the amount of sales. 1,875 = p * 37,500 1,875 = p * 37,500 37,500 37,500 . 0.05 = p The percent is 5% and represents the commission rate.
Percent Increase and Percent Decrease When measuring growth and decline in business or the economy, many companies report percent increase or percent decrease. The difference between the original amount and the final amount will be the actual increase (or decrease). Then, the increase (or decrease) is a percent of the original amount. Let’s look at a few examples:
Example 9: Gas prices increased from $3.75 to 3.90. What is the percent increase? First, find the difference.
$3.90 - $3.75 = $0.15
Then, use the pattern: “x is what percent of y.” The difference replaces x and the original price replaces the y, we’ll use “p” for the variable. The solution to the equation will be a decimal and must be changed to a percent. 0.15 = p * 3.75 0.15 = p * 3.75 3.75 3.75 . 0.04 = p The percent increase is 4%.
Example 10: A certain water bill was $134. The next month, there was an 8% increase. If the same amount of water is used what can be expected the next bill will be? Use the pattern: What number is x% of y? “what number” represents the amount of increase, we’ll use “a” for the variable. x represents the percent and y represents the original amount. a = 0.08 * 134 a = 10.72 Remember, “a” represent the amount the bill will increase. The question asks for the expected amount of the new bill. To find this, add the original bill amount with the amount of increase. $134 + $10.72 = $144.72 With an increase of 8%, the next bill is expected to be $144.72 Example 11: The average attendance of a high school basketball game decreased from 1600 to 1400 from one year to the next. What is the percent decrease in attendance at the basketball games? Find the difference first.
1600 – 1400 = 200
Use the pattern: “x is what percent of y.” x represents the difference, y represents the original amount of 1600. We’ll use “p” as the variable. 200 = p * 1600
200 = p * 1600 1600 1600 0.125 = p The percent decrease is 12.5%.
6.3 Application Exercises: Solve each problem. 1. Alberto wants to buy a remote control car with a sale price of $14.50. If the tax rate is 6%, how much will he pay in tax? 2. Benjamin bought camping supplies for an upcoming trip and spent $312. If the tax rate was 6%, what was the total cost of the camping items?
3. Casper bought new furniture with a price tag of $1,250.00, but actually paid $1,312.50. What was the amount of tax? What was the tax rate?
4. Dave sells $56,000 worth of advertising in one month. If his commission rate is 8%, how much is his commission check? 5. Eliza hopes to earn a commission bonus of $4500. If her commission rate is 5%, how much does she have to sell to earn that bonus?
6. Fran sold $4,258,000 worth of real estate last year. Her commission earnings were $63,870. What was her commission rate?
7. Garth weighed 200 pounds a year ago, he now weighs 160 pounds. What is the percent decrease? 8. The value of Helen’s house increased by $24,000 in 5 years. What is the percent increase if the original value was $400,000?
9. Isaiah is selling his car that he bought a few years ago for $14,500. He figures that the percent decrease is 22%. How much value was lost in the car? 10. The amount of tax paid on a new bed was $37.10 in a state where the tax rate is 7%. What was the price of the bed? 11. What is the commission made on a sale if the rate is 3% and the amount of sales is $52,345? 12. The value of an autographed base ball increased from $25 when purchased to $75 two years later. What is the percent increase? 13. The sale price of a dining room set is $1250. If the amount of tax paid was $93.75, what was the tax rate? 14. An antique increased in value by 33% while Jermaine owned it. If he bought the antique for $150, how much is it worth now? 15. Katherine sold $178,000 last month. His commission check was $7,120. What is the percent that he earns for commission? 16. For a hotel stay, the tax rate is much higher than a regular sales tax, often around 12%. If the total hotel stay costs $525, how much is paid in tax?
17. A piece of property in Nicaragua increased in value from $60,000 to $84,000 in 5 years. What was the percent increase? 18. With a budget of $95, LaToya wants to purchase a pair of jeans and a shirt. The jeans cost $45 and the shirt costs $40. If the tax rate is 7%, does he have enough money to purchase the outfit? 19. Michelle needs $1650 to landscape her back yard. She sells computers and wants to earn extra money in commissions to pay for the landscaping. Her employer pays a commission of 11%. If she wants to earn that $1650 in one month, what is the amount of sales she much have?
20. Nancy has a record collection that was worth $450, but due to damage to some of the albums, they are now only worth $382.50. What is the percent decrease?
6.3 Application Exercise Answers: 1. $0.87 2. $330.72 3. $62.50, 5% 4.
$4,480
5.
$90,000
6.
1.5%
7.
20%
8.
6%
9.
$3,190
10. $530 11. $1,570.35 12. 200% 13. 7.5% 14. $199.50 15. 4% 16. $63 17.
40%
18.
Yes
19. $15,000 20.
15%
6.4 Financial Applications of Percents Now that you have reviewed how percents work and the basic method of solving percents problems, we can explore some real world problems in the financial area that contain application of percents. Just as completing the percent problems had a pattern in terms of set-up and solving, the same pattern will be used to complete application problems involving percent. The basic format of ___ % of ____ is ____ will be in the problem, but you will need to determine which value goes where. Remember that if the two given values are on the same side of the equal side, you multiply them and if they are on different sides, then you need to divide. Interest Interest is the cost of borrowing money. An interest rate is the cost stated as a percent of the
amount borrowed per period of time, usually one year. Interest can be divided in two categories Simple Interest and Compound Interest Simple interest is calculated on the original principal only. Accumulated interest from prior periods is not used in calculations for the following periods. Simple interest is normally used for a single period of less than a year, such as 30 or 60 days. Compound interest arises when interest is added to the principal, so that from that moment on, the interest that has been added also earns interest. We need complicated formula to calculate compound interest. We will skip problems from Compound Interest in this section. How do we calculate simple interest on a loan? The solution is in the formula I = Prt, where I is the interest, P is the principle (amount loaned), r is the interest rate, and t is the time measured in years. Example 1: What is the simple interest on a loan of $6,000 for one year if the interest rate is 8.25%? I = Prt I = (6000)(0.0825)(1 year) I = 495 Thus the interest is $495.
Example 2: If one borrows $2000 for 3 years at a 13% annual simple interest rate, what is the interest? I = Prt I = ($2000)(.13)(3) I = $780 Thus the interest is $780. For the loan, one would pay the $2000 borrowed plus an additional $780 for a total amount owed of $2780.
Example 3: What is the interest gained with a $1500 simple interest investment at 8% annual rate for 9 months? Notice that the time is given in months, and since we have an annual interest rate, we must convert the time to years. I = Prt I = ($1500)(.08)(9 12) I = $90 Thus the interest is $90. For the investment, one would get the $1500 invested plus an additional $90 for a total amount received at the end of 9 months of $1590. In the simple interest formula, you can solve for any of the four quantities (I, p, r, and t) if any three of them are known. Example 4: One needs $500 in 2 years. What annual rate does a simple interest investment need if the initial investment is $200? In this case, we want the interest rate (r). We will have $500 in 2 years, but starting with $200. Therefore, the principal is $200, the interest is $500 $200 = $300. $300 is the extra money we made on top of the $200 initial amount. Thus: I = Prt $300 = ($200) r (2) r = $300 ($200×2) r = .75 or 75% Thus the interest rate is 75%
Example 5: How much time does it take $3000 to double if the annual simple interest rate is 20%? In this case, we are solving for time (t). Since we are doubling $3000 (to $6000), we made an additional $3000 on this investment, which is the interest. Therefore: I = Prt $3000 = $3000 (.20) t t = $3000 ($3000 .20) t=5 The amount of time is 5 years. Total Amount If interest is charged, the total amount is the principal and interest added together. Recall that we used these variables in calculating simple interest: P = principal I = interest r = rate t = time Let’s use A for the total amount. So…A = P + I The formula for simple interest that we have used is: I = Prt Putting the two together we get a formula we can used to find the total amount: A = P + Prt Example 6: Ray buys a $900 television. The store gives him credit calculated with simple interest at 30% over 2 years. How much does Ray pay for his television? Using the formula A = P + Prt, we want to know A, the total amount of money Ray paid for the television. The television costs $900 (so P = 900); the interest rate is 30%(so r = .30); the time is 2 years (so t = 2). Substituting into our formula we get: A = P +Prt A = 900 + (900)(.3)(2) A = 900 + 540 A = 1440, So the total amount Ray would pay for his television is $1440.
Credit Cards and Simple Interest Lessons Credit Cards are used throughout the world to purchase goods and services instead of cash money. This material is to discuss some of the concepts of the credit card. Annual Percentage Rate (APR): The annual interest rate associated with the credit card. Credit Limit: The maximum amount allowed charged on a credit card. Transaction Fee: Fees charged to the customer for late payments, exceeding the credit limit, or other non-interest fees. Periodic Rate: The Annual Percentage Rate divided by the period used to calculate the balance (usually 365). Purchases: Charges on a credit card from purchasing goods and services. Cash Advances: Charges on a credit card from cash received. Balance Transfers: Charges on a credit card from transfers from other accounts. Finance Charge: The total amount to be paid in transaction fees and interest. Grace Period: The period of time in which if the new balance of a credit card is paid off, there is no finance charge. How is the interest on a credit card calculated? Most widely used credit cards (Visa, MasterCard, etc.) use the Average Daily Balance approach to finding the amount of interest on purchases during a period of time. In this section we will explain how to calculate the Average Daily Balance.
Example 9: Let's say we have this credit card statement:
Tingling National Bank Credit Card Covering the Period April 15 through May 14 APR = 22.9% Beginning Balance: $240.00
DATE
TRANSACTION
AMOUNT
4/18
Peasant Pastures Gas
$10.00
4/23
Blue Mollusk Seafood Restaurant
$35.00
5/01
Payment Received
$100.00
5/10
New Army Clothing Store
$65.00
What is the amount of interest for this period? The first thing we must find is the Average Daily Balance. This credit card statement is for 30 days. From April 15 through April 17, the balance is $240.00 (3 days) From April 18 through April 22, the balance is $240.00 + $10.00 = $250.00 (5 days) From April 23 through April 30, the balance is $250.00 + $35.00 = $285.00 (8 days) From May 1 through May 9, the balance is $285.00 - $100.00 = $185.00 (9 days) From May 10 through May 14, the balance is $185.00 + $65.00 = $250.00 (5 days) To find the Average Daily Balance, we must average the above: ($240 3 + $250
5 + $285
8 + $185
9 + $250
5) ÷ 30
= (720 + 1250 + 2280 + 1665 + 1250) ÷ 30 = (7165) ÷ 30 = $238.83 The amount of interest is obtained using the Simple Interest Equation: I = Prt. The principal is the average daily balance. The rate is the annual percentage rate. The time is in years, thus the time is the number of days divided by 365. I = Prt I = 238.83× (0.229) × (30/365)
I = 4.50 Therefore the interest for this account for this period is $4.50.
Example 10: Let's look at another statement: Weripoff National Bank Credit Card Covering the Period July 23 through August 24 APR = 19.8% Beginning Balance: $174.76 DATE
TRANSACTION
AMOUNT
7/25
McReynold's Restaurant
$11.45
7/28
Washington Merlins Basketball Tickets $212.00
7/31
Payment Received
$200.00
8/12
High's Black Swamp Theatres
$31.75
8/20
WEW Cooked Wrestling Tickets
$150.00
What is the amount of interest for this period? From July 23 through July 24, the balance is $174.76 (2 days) From July 25 through July 27, the balance is $174.76 + $11.45 = $186.21 (3 days) From July 28 through July 30, the balance is $186.21 + $212.00 = $398.21 (3 days) From July 31 through August 11, the balance is $398.21 - $200.00 = $198.21 (12 days) From August 12 through August 19, the balance is $198.21 + $31.75 = $229.96 (8 days) From August 20 through August 24, the balance is $229.96 + $150.00 = $379.96 (5 days) The Average Daily Balance is: ($174.76×2 + $186.21×3 + $398.21×3 + $198.21×12 + $229.96×8 + $379.96×5) ÷ 33 = $249.11
The amount of interest is: I = Prt I = 249.11*(0.198)*(33/365) I = 4.46 Therefore the interest for this account for this period is $4.46. These two examples show the amount of interest for one month, I would say, reasonable amounts. Of course this is only one month's worth of interest and many of us have higher balances than the ones above. Remember, the Average Daily Balance approach is used to determine the interest on purchases. Interest obtained from the use of cash advances and balance transfers may be obtained another way, usually with an entirely different Annual Percentage Rate. In fact, most cash advances use the Compound Interest Formula compounded daily.
6.4 Financial Literacy Exercises: Read each problem carefully. Write your answer to the point and in complete sentences. 1. A bank charges 6% simple interest. How much interest must you pay on a loan of $2000 for 3 years? 2. A bank charges 5% simple interest. How much interest must you pay on a loan of $700 for 2 years? 3. A bank charges 8.75% simple interest. How much interest must you pay on a loan of $4000 for one year? 4. A bank charges 7.50% simple interest. How much interest must you pay on a loan of $8000 for one year? 5. What is the interest on a $650 simple interest loan with a 10% rate for 18 months? 6. What is the total amount made on a $1000 simple interest investment if the rate is 8% and the term is 3 years? 7. How much would Jerry totally pay if he borrows $2000 at 15% simple interest for 5 years? 8. A computer costs $1100. If a company is charging 30% simple interest for 3 years, how much is totally paid for the computer?
9. Find the average daily balance and the amount of interest from purchases for the credit card account below: Tingling National Bank Credit Card Covering the Period January 11 through February 10 APR = 15.5% Beginning Balance: $200.00 DATE
TRANSACTION
AMOUNT
1/17
Payment
$80.00
1/20
Purchase
$35.00
2/01
Purchase
$100.00
2/04
Purchase
$45.00
10. Find the average daily balance and the amount of interest from purchases for the credit card account below: Tingling National Bank Credit Card Covering the Period November 17 through December 17 APR = 16.9% Beginning Balance: $1231.24 DATE
TRANSACTION
AMOUNT
11/25
Purchase
$65.21
11/30
Purchase
$11.48
12/6
Purchase
$163.95
12/13
Payment
$300.00
11. Find the average daily balance and the amount of interest from purchases for the credit card account below: Tingling National Bank Credit Card Covering the Period October 10 through November 9 APR = 9.9% Beginning Balance: $569.18 DATE
TRANSACTION
AMOUNT
10/19
Purchase
$23.99
10/22
Purchase
$100.00
10/31
Purchase
$65.50
11/2
Payment
$20.00
11/6
Purchase
$78.06
12. Find the average daily balance and the amount of interest from purchases for the credit card account below: Tingling National Bank Credit Card Covering the Period February 11 through March 7 APR = 12.0% Beginning Balance: $0.00 DATE
TRANSACTION
AMOUNT
2/11
Purchase
$100.00
2/14
Purchase
$100.00
3/02
Credit
$25.00
13. Find the average daily balance and the amount of interest from purchases for the credit card account below: Tingling National Bank Credit Card Covering the Period January 1 through February 1 APR = 9.9% Beginning Balance: $1,987.65 DATE
TRANSACTION
AMOUNT
01/01
Beginning Balance
$ 1,987.65
01/02
Purchase
$256.78
01/04
Payment
$497.15
01/05
Purchase
$78.99
01/07
Purchase
$22.21
01/20
Payment
$800.00
6.4 Applications of Percents Exercise Answers: 1. The interest is $360. 2. The interest is $70. 3. The interest is $350. 4. The interest is $600. 5. The interest is $97.50 6. The interest is $240.00. So the total amount is $1240.00 7. Total amount paid is $3500 8. The total amount paid for the computer was $2090 9. The average daily balance is $202.74; The interest for the period is $2.67 10. The average daily balance is $1301.36; The interest for the period is $18.68 11. The average daily balance is $673.54; The interest for the period is $5.66 12. The average daily balance is $183.00; The interest for the period is $1.50 13. The average daily balance is $1930.73; The interest for the period is $16.76
6.5 Circle Graphs Statistical information is often presented in a table form or in the form of a graph such as, line graph, bar graph, or circle graph. These graphs are a way of presenting or visualizing statistical data for easy comparisons. For circle graphs, also known as pie charts, the information is presented in the form of a circle divided into sectors, each sector representing a part of a whole as percent. Example 1: A class of 25 students took a mathematics test. The results are presented by the circle graph below.
Test Results
A, 20% B, 32%
B F C
C, 40%
F, 8%
A
Use the circle graph above to answer the following questions. a. How many students obtained a C grade? Answer: The largest sector of the circle is 40% of the whole circle. Therefore, we calculate 40% of the 25 students who took the test. 40% of 25 = .4×25 = 10 students. b. How many students received an A or a B on the test? Answer: 20% of the students obtained an A, and 32% obtained a B. 20% + 32% = 52% 52% of 25 = .52×25 = 13 students. c. What percent of the students failed the test? How many were they? Answer: The smallest sector of the graph indicates that 8% of the students failed test. 8% of 25 = .08×25 = 2. Two students failed the test. d. How many more students obtained a C grade than a B grade? Answer: 40% of 25 = .4×25 = 10 C students 32% of 25 = .32× 25 = 8 B students 10 – 8 = 2 more students obtained C than B.
e. What percent of the entire class passed the test? Answer: Either, Subtract the 8% who failed from the 100% 100% - 8% = 92% Or, 20% + 32% + 40% = 92% passed
6.5 Circle Graph Exercises 1 – 4 Ms. Blossom is a school teacher. She earns $3,500 per month for her hard work as a teacher. Her monthly expenses are represented by the circle graph below.
Budget Savings, 10% Utilities, 12%
Food, 25%
Food Rent Transportation
Transportation, 8%
Utilities Savings Rent, 45%
Use Ms. Blossom’s circle graph to answer the following questions about her budget. 1. How much does Ms. Blossom spend on rent? 2. How much does she save every month? 3. What percent of her monthly income does Ms. Blossom spend on utilities and transportation? What is the amount? 4. What is Ms. Blossom’s total expenditure other than rent?
5-9. A company named Compassion employs people of all ages, from twenty-year-olds to seventy-year-olds. The circle graph below shows the number of employees in each age group. Use Compassion’s circle graph to answer questions 5- 9.
Ages 70 yr olds, 6 20 yr olds, 12
20 yr olds 30 yr olds
60 yr olds, 15
50 yr olds, 10
40 yr olds 30 yr olds, 24
50 yr olds 60 yr olds 70 yr olds
40 yr olds, 8
5.
Find the total number of employees of this company.
6. What age group constitutes the largest group of employees? 7. What percent of employees are in their sixties and seventies? How many are they? 8. How many employees are under fifty years of age? 9. What age group makes up 16% of the total number of employees/
10-16. Assuming there is a very small country with population of 24 million. This country is divided into six ethnic groups, Akan, European, Ewe, Ga, Gurma, and Moshi. This information is presented in the circle graph below. Use the graph to answer questions 10 –16
Ethnic Groups
Moshi, 16%
Moshi Ga
Akan, 44%
Ga, 8% Ewe, 13%
Ewe European Gurma Akan
European, 16% Gurma, 3%
10. Which is the predominant ethnic group in this population? 11. What percent is represented by the largest ethnic group? 12. What is the population of the largest group of people? 13. Which ethnic groups have the same percentage of people? 14. How many people are in each group of people with the same percent? 15. The Ewe group is larger than the Ga group by what percent? 16. How many more people belong to the Ewe group than the Ga group?
17-20. A survey asked 160 mathematics students what they thought were the most pertinent contributory factors to their success in mathematics education. Among the factors considered were, the role of the instructor, taking responsibility for their homework assignments, study skills and time management, being tutored on one-on-one basis, and study groups. The results of the survey are presented in the circle graph below Use the graph to answer questions 17 – 20.
Math Students Tutoring, 5%
Study Groups, Study Skills, 10% 15%
Study Skills Instructor
Doing Homework, 45%
Instructor, 25%
Doing Homework Tutoring Study Groups
17. What was the most predominant thought of the students surveyed on how to be successful in learning mathematics? 18. How many students thought instructors play a major role in getting them to be successful in mathematics? 19. What percent of the students believed that study skills and time management are very important in their mathematics education? 20. How many students believed in study skills and time management?
6.5 Circle Graph Exercise Answers 1. $1,575 2. $350 3. 20%, $700 4. $1,925 5. 75 6. 30 year-olds 7. 28%, 21 8. 44 9. 20 year-olds 10. Akan 11. 44% 12. 10,560,000 13. European and Moshi 14. 3,840,000 15. 5% 16. 1,200,000 17. Taking responsibility in doing homework assignments 18. 40 19. 15% 20. 24
CHAPTER 6 CONTENTS 6.1 Introduction to Percents 6.2 Solve Percent Problems 6.3 Application Problems 6.4 Financial Literacy 6.5 Circle Graphs
6.1 Introduction to Percents and Conversions to Fractions and Decimals Introduction Fractions are used to describe a part of a whole, no matter what makes up the whole. For example, describes 2 parts of a whole that is divided into 5 equal pieces. And as you have just learned decimals, too, describe parts of a whole, but in terms of base 10. Thus, 0.2 represents 2 parts of a whole divided into 10 equal parts, .34 represents 34 parts of a whole divided into 100 parts, and .502 represents 502 parts of a whole divided into 1000 equal parts. Although fractions and decimals are two very common ways of identifying parts of a whole, they are not the only ways. Percents can also be used. By definition, a percent is specifically a representation of a part of a whole that is divided into exactly 100 equal pieces. The term is a form of the latin phrase, per centum, meaning for every hundred. Thus we have come to accept a percent as meaning part of one hundred. Therefore, to say 85 percent is to signify 85 parts of one hundred. However, when identifying a percent it would be too much to always have to write the word percent after the part identified. To avoid having to do this we use a symbol to represent the term percent. That symbol is % and is called, none other than, a percent symbol. We can see this demonstrated in the following two examples:
Example 1: What percent would be used to represent 34 parts of a whole divided into 100 equal parts? Solution: Since we are interested in parts of a whole divided into 100 equal parts we are already dealing with a percent. Therefore, if we use the percent symbol, 34 parts of 100 equal parts will be represented by 34%.
Example 2: Write 57% by identifying the parts of a whole. Solution: Remembering that the percent symbol, %, represents parts of 100, we see that 57% must represent 57 parts of a whole divided into 100 equal parts.
Now that we know that percents are yet another way to represent a number of parts of a whole divided into equal parts we can demonstrate how fractions, decimals, and percents
are related to one another. In fact, they are simply different representations of the same value. Thus, it is possible to convert one representation in terms of one of the other two. Let’s examine each translation one at a time.
Convert Between Percents and Fractions 1(a) - Percent to Fraction: When translating a percent to a fraction we can simply remember the meaning of a percent in general and the percent symbol in particular. The procedure requires that we multiply by the unit ratio of
. Of course, we should also remember to reduce the
resulting fraction to lowest terms. This is demonstrated in the following examples.
Example 3: Convert 63% into a fraction. Solution: By definition 63% represents the value of 63 parts of a whole divided into 100 equal parts. If we multiply the percent by 1/100% we get
Since there are no common factors other than 1 remaining in the numerator and denominator this fraction is already in reduced form.
Example 4: Convert 45% into a fraction. Solution: 45% represents the value of 45 parts of a whole divided into 100 equal parts. Again, multiplying by 1/100% we have,
Using the common factor of 5 in the numerator and denominator, we can reduce the fraction
Example 5: Convert 120% into a fraction. Solution: The translation of percents greater than 100% follow the same pattern as any other percent to fraction translation. Therefore 120% represents 120 parts of a whole divided into 100 equal parts. Following the pattern, we have
Recall that this is an improper fraction and can be written as should then be reduced to
. The fraction
.
Example 6: Convert 6 2/3% into a fraction. Solution: To convert percents that are mixed numbers (or contain fractions,) it should first be converted into an improper fraction. Once this is completed we then should follow the procedure we have for the rest of the examples. Thus,
Which reduces to
1(b) - Fraction to Percent: To convert a fraction into a percent we can use a different form of the same unit ratio by multiplying by
. The following example demonstrates what must be done.
Example 7: Convert into a percent. Solution:
Example 8: Convert Solution:
into a percent.
Convert Between Percents and Decimals 2(a) - Percent to Decimal: To convert a percent into a decimal we need to again multiply the percent by the unit ratio 1/100%. Notice this is the same procedure we used to convert a percent to a fraction. However, we do not want a fraction so we must continue the process by simply carrying out the division as presented in the resulting fraction. The following examples demonstrate this process. Example 9: Convert 82% into a decimal. Solution:
82% x 1/100% = 82%/100% = 82/100 82/100 = 0.82
Example 10: Convert 17% into a decimal Solution:
17% x 1/100% = 17% / 100% = 17/100 = 0.17
2(b) - Decimal to Percent To convert a decimal to a percent we will reverse the percent to decimal process and multiply by 100% instead of multiplying by 1/100%. The following examples demonstrate how this works. (include patter of moving decimal to right when multiplying by powers of 10) Example 11: Convert .94 as a percent. Solution: .94 x 100% = 94% Example 12: Convert .058 as a percent. Solution: .058 x 100% = 5.8%
6.1 Introduction Exercises: Solve each problem. 1. Convert 25% into a fraction 2. Convert 56% into a fraction 3. Convert 13% into a fraction.
4. Convert 71 1/2% into a fraction. 5. Convert 90% into a fraction. 6. Convert 1/3 into a percent. 7. Convert 5/6 into a percent. 8. Convert 21/25 into a percent. 9. Convert 2 7/8 into a percent 10. Convert 37/50 into a percent. 11. Convert 29% into a decimal. 12. Convert 24.6% into a decimal. 13. Convert 5.75% into a decimal. 14. Convert 1 2/3% into a decimal. 15. Convert 0.78 into a percent. 16. Convert 0.5 into a percent. 17. Convert 0.025 into a percent. 18. Convert 2.07 into a percent. 19. Convert 0.0038 into a percent. 20. Complete the table by finding the missing conversions. Percent Fraction Decimal 16% 27/50 13/10 82.8% .40 .056
6.1 Introduction Exercises Answers: 1. ¼ 2. 14/25 3. 13/100
4.
143/200
5. 9/10 6.
33 1/3%
7.
83 1/3%
8.
84%
9.
287 1/2%
10. 74% 11. 0.29 12.
0.246
13.
0.0575
14.
0.01667
15.
78%
16.
50%
17.
2.5%
18.
207%
19.
0.38%
20. Percent 16% 54% 130% 82.8% 40 5.6%
Fraction 4/25 27/50 13/10 207/250 2/5 7/125
Decimal 0.16 0.54 1.30 0.828 .40 .056
6.2 Solve Percent Problems Percentages are used to express how large one quantity is, relative to another quantity. The first quantity usually represents a part of, or a change in, the second quantity, which should be greater than zero. Also when we translate a statement in English to Math the word “of” is interpreted as multiplication. So here we can use the pattern in which all percentage problems can be set up. Proportion Method We can set up the percentage problem as a proportion. We know that Percent is always out of 100, we can write the proportion as:
=
from Amount is Percent% of Base
We then can substitute given values and solve for the unknown value. Example 1: 90 is what percent of 150? Here 90 is amount and 150 I base. We need to find the percent by substituting these values in the proportion as follows: =
This then can be written as P =
= 60%
Example 2: What number is 55% of 40? Here 55 is percent and 40 is base. We need to find the amount by setting up the proportion as follows: =
This then can be written as A =
= 22
Example 3: 19 is 20% of what number? Here 19 is amount and 20 is percent. We need to find the base by setting up the proportion as follows: =
This then can be written as B =
Formula Method We can also set up a percentage problem as a formula as follows: Percent * Base = Amount OR _____% of ______ = ________. When reading a problem we will have to decide which value is the base and which is the amount. Or in other words which number goes in which place. For example “15% of 200 is 30” can be written as 0.15*200 = 30. Do not forget to change 15% to decimal form so we can use it in multiplication. 200 is known as the base and 30 as the amount. Example 4: What percent of 64 is 16? In this question value of percent is to be found (calculated) while base is 64 and amount is 16. We translate this as percent × 64 = 16 n × 64 = 16 n = 16/64 = 0.25 n = 25% Example 5: 36 is what percent of 180? In this question 36 is amount and 180 is base. We need to calculate the value of percentage. We translate this as
36 = P × 180 36/180 = P 1/5 = 0.2 = P P = 20%
Example 6: What percent of 40 is 100? Again this problem 100 is amount and 40 is base. We need to calculate the value of percentage. We translate this as
P * 40 = 100 P = 100/40
P = 5/2 = 2.5 P = 250%
Examples to find Base: Example 7: 30% of what number is 45? Here we can see that 30 is percent, 45 is amount and the value of base needs to be calculated. Change the 30% to a decimal (0.3). So we can substitute these values in the formula as follow: 0.30 × B = 45 B = 45/0.30 B = 150
Example 8: 30 is 15% of what number? Here again we can identify 30 as Amount and 15 as Percent. We now substitute these values in the formula as follows: 0.15 × B = 30 B = 30/0.15 B = 200
Example 9: 180% of what number is 81? Here 180 is the Percent, 81 is the Amount and we need to calculate the value of base. We substitute the value and calculate base as follows: 1.80 × B = 81 B = 81/1.80 B = 45
Examples to find Amount: Example 10: What is 20% of 35? Here 20 is Percent and 35 is Base. By substituting these values in the formula we can calculate Amount as follows: A = 0.20 × 35 A=7 Example 11: 30% of 17 is what number? Using the formula we can write: 0.30 × 17 = 5.1 Example 12: What is 110% of 80? Substituting appropriate numbers in the formula we get A = 1.10 × 80 A = 88
6.2 Solve Percent Problem Exercises 1) What number is 30% of 300? 2) What number is 8.9% of 10? 3) 120% of 4000 is what number? 4) 15% of 90 is what number? 5) 15 is 25% of what number? 6) 32 is 80% of what number? 7) What percent of 125 is 25? 8) What percent of 100 is 150?
9) 40 is what percent of 400? 10) 24 is what percent of 60?
6.2 Solve Percent Problem Exercise Answers 1) 90 2) .89 3) 4800 4) 13.5 5) 60 6) 40 7) 20% 8) 150% 9) 10% 10) 40%
6.3 Application Problems Now that you have practiced working with percents, we will move on to applying these skills to problems seen in the real world. Calculating sales tax, commission earnings, and percent increase / decrease are all important skills to have. In the previous section, we saw three patterns. Each of the problems in this section will be solved using one of those three patterns: What number is x% of y? x is what percent of y? y is x% of what number?
Sales Tax When something is purchased, there is almost always sales tax to pay. The sales tax rate is given by a percent. That percent depends usually on the state, but sometimes varies by cities or
counties within a state. To find the amount of tax for an item, multiply the purchase price by the tax rate. Use the “What number is x% of y?” pattern. To find the total cost, add the purchase price and the amount of tax. When completing the word problems, be careful to answer the question that is asked. Sometimes just the amount of tax is asked for, sometimes the total price, and sometimes the tax rate itself. Let’s look at a few examples: Example 1: Julia is going to buy a shirt for $12.00. How much tax will she pay for the item if the tax rate is 6%? Use the pattern: What number is x% of y? “What number” is the amount of tax to be paid. It is the unknown, so we will assign it the variable t. Be sure to change the percent to a decimal and remember the “of” means to multiply. What number is x% of y? t = 6% of 12.00 t = 0.06 * 12.00 t = 0.72 The amount of tax is $0.72. Example 2: The television that Miguel wants to purchase costs $449. a.
How much will he pay if the sales tax is 8%?
b.
How much is the total cost?
As in example 1, use the pattern “what number is x% of y?” t = 8% of 449 t = 0.08 * 449 t = 35.92 The amount of tax is $35.92. This is the answer to part a. Part b asks for the total cost. This is found by adding the original cost and the tax amount. $449 + $35.92 = $484.92 The total cost is $484.92. This is the answer to part b.
Example 3: Nathanael wants to buy a science kit with a sticker price of $37.50. After a 6% sales tax, what is the final cost? Again, use the pattern “what number is x% of y?” t = 6% of 37.50 t = 0.06 * 37.50 t = 2.25 The amount of tax is $2.25. The total cost is needed, not the amount of tax. Add the original cost and the amount of tax. $37.50 + $2.25 = $39.75 The total cost is $39.75.
Example 4: Jada found a pair of jeans that she would like to purchase. She has $65 to spend. If the price is $62.75 and the tax rate is 8%, does she have enough money? Explain your answer. In this case, we need to first find out how much the jeans cost. What number is x% of y? t = 8% of 62.75 t = 0.08 * 62.75 t = 5.02 The amount of tax is $5.02. The total cost is $62.75 + $5.02 = $67.77 The question asked if she would enough money. The total cost is $67.77, but Jada only has $65. Therefore, No, Jada does not have enough money. The question asks for an explanation, a good explanation is: “Since the total cost is $67.77 and Jada only has $65, she does not have enough to buy the jeans.”
Example 5: Samuel bought a computer for $1450 and paid $101.50 in sales tax. What is the tax rate for the state he bought the computer? In this question, what we are looking for is the tax rate, so we need to look at a different pattern. The pattern with the variable as the percent is: x is what percent of y? Since the tax rate or percent is always multiplied by the cost of the item, the cost of 1450 will replace y in the equation. Since product gives the amount of tax, the 101.50 will replace x. We will use the variable “p” to represent the tax rate as a decimal. 101.50 = p * 1450 To solve the equation for x, divide both sides by 1450. 101.50 = p * 1450 1450 1450 0.07 = p The tax rate is 7%
Commission Many jobs include a base salary plus bonuses called commissions. Some employees only receive commission as their pay. This is popular in many sales jobs. Many salespeople do a better job by selling more of the product or service when their income depends on how much they sell. Commission is typically calculated using a percent. The percent of the amount sold is the commission. Example 6: Temperance is a real estate agent who earns a 3% commission on each house sold. If she sells a house for $225,000, what will her commission be? Use the pattern: what number is x% of y? “What number” represents the commission, We’ll use “c” for the variable. Don’t forget to change the percent to a decimal. c = 0.03 * 225,000 c = 6,750 The commission of the sale of the house is $6,750.
Example 7: Seeley gets a commission of 2% and received a commission check of $7,000 in a certain month. How much did he sell in that month? Use the pattern: y is x% of what number? This time, “what number” represents the amount that was sold, we’ll assign “a” as the variable. 7,000 = 0.02 * a To solve the equation, divide both sides of the equation by 0.02. 7,000 = 0.02 * a 0.02 0.02 350,000 = a The house sold for $350,000 if the commission was $7,000.
Example 8: Angela sells cars for a living. In one week, she sold $37,500 worth of cars and earned a commission of $1875. What is her commission rate? Since we are looking for the percent we’ll use the pattern: “x is what percent of y” and assign “p” as the variable to represent to percent. x is the amount of commission and y is the amount of sales. 1,875 = p * 37,500 1,875 = p * 37,500 37,500 37,500 . 0.05 = p The percent is 5% and represents the commission rate.
Percent Increase and Percent Decrease When measuring growth and decline in business or the economy, many companies report percent increase or percent decrease. The difference between the original amount and the final amount will be the actual increase (or decrease). Then, the increase (or decrease) is a percent of the original amount. Let’s look at a few examples:
Example 9: Gas prices increased from $3.75 to 3.90. What is the percent increase? First, find the difference.
$3.90 - $3.75 = $0.15
Then, use the pattern: “x is what percent of y.” The difference replaces x and the original price replaces the y, we’ll use “p” for the variable. The solution to the equation will be a decimal and must be changed to a percent. 0.15 = p * 3.75 0.15 = p * 3.75 3.75 3.75 . 0.04 = p The percent increase is 4%.
Example 10: A certain water bill was $134. The next month, there was an 8% increase. If the same amount of water is used what can be expected the next bill will be? Use the pattern: What number is x% of y? “what number” represents the amount of increase, we’ll use “a” for the variable. x represents the percent and y represents the original amount. a = 0.08 * 134 a = 10.72 Remember, “a” represent the amount the bill will increase. The question asks for the expected amount of the new bill. To find this, add the original bill amount with the amount of increase. $134 + $10.72 = $144.72 With an increase of 8%, the next bill is expected to be $144.72 Example 11: The average attendance of a high school basketball game decreased from 1600 to 1400 from one year to the next. What is the percent decrease in attendance at the basketball games? Find the difference first.
1600 – 1400 = 200
Use the pattern: “x is what percent of y.” x represents the difference, y represents the original amount of 1600. We’ll use “p” as the variable. 200 = p * 1600
200 = p * 1600 1600 1600 0.125 = p The percent decrease is 12.5%.
6.3 Application Exercises: Solve each problem. 1. Alberto wants to buy a remote control car with a sale price of $14.50. If the tax rate is 6%, how much will he pay in tax? 2. Benjamin bought camping supplies for an upcoming trip and spent $312. If the tax rate was 6%, what was the total cost of the camping items?
3. Casper bought new furniture with a price tag of $1,250.00, but actually paid $1,312.50. What was the amount of tax? What was the tax rate?
4. Dave sells $56,000 worth of advertising in one month. If his commission rate is 8%, how much is his commission check? 5. Eliza hopes to earn a commission bonus of $4500. If her commission rate is 5%, how much does she have to sell to earn that bonus?
6. Fran sold $4,258,000 worth of real estate last year. Her commission earnings were $63,870. What was her commission rate?
7. Garth weighed 200 pounds a year ago, he now weighs 160 pounds. What is the percent decrease? 8. The value of Helen’s house increased by $24,000 in 5 years. What is the percent increase if the original value was $400,000?
9. Isaiah is selling his car that he bought a few years ago for $14,500. He figures that the percent decrease is 22%. How much value was lost in the car? 10. The amount of tax paid on a new bed was $37.10 in a state where the tax rate is 7%. What was the price of the bed? 11. What is the commission made on a sale if the rate is 3% and the amount of sales is $52,345? 12. The value of an autographed base ball increased from $25 when purchased to $75 two years later. What is the percent increase? 13. The sale price of a dining room set is $1250. If the amount of tax paid was $93.75, what was the tax rate? 14. An antique increased in value by 33% while Jermaine owned it. If he bought the antique for $150, how much is it worth now? 15. Katherine sold $178,000 last month. His commission check was $7,120. What is the percent that he earns for commission? 16. For a hotel stay, the tax rate is much higher than a regular sales tax, often around 12%. If the total hotel stay costs $525, how much is paid in tax?
17. A piece of property in Nicaragua increased in value from $60,000 to $84,000 in 5 years. What was the percent increase? 18. With a budget of $95, LaToya wants to purchase a pair of jeans and a shirt. The jeans cost $45 and the shirt costs $40. If the tax rate is 7%, does he have enough money to purchase the outfit? 19. Michelle needs $1650 to landscape her back yard. She sells computers and wants to earn extra money in commissions to pay for the landscaping. Her employer pays a commission of 11%. If she wants to earn that $1650 in one month, what is the amount of sales she much have?
20. Nancy has a record collection that was worth $450, but due to damage to some of the albums, they are now only worth $382.50. What is the percent decrease?
6.3 Application Exercise Answers: 1. $0.87 2. $330.72 3. $62.50, 5% 4.
$4,480
5.
$90,000
6.
1.5%
7.
20%
8.
6%
9.
$3,190
10. $530 11. $1,570.35 12. 200% 13. 7.5% 14. $199.50 15. 4% 16. $63 17.
40%
18.
Yes
19. $15,000 20.
15%
6.4 Financial Applications of Percents Now that you have reviewed how percents work and the basic method of solving percents problems, we can explore some real world problems in the financial area that contain application of percents. Just as completing the percent problems had a pattern in terms of set-up and solving, the same pattern will be used to complete application problems involving percent. The basic format of ___ % of ____ is ____ will be in the problem, but you will need to determine which value goes where. Remember that if the two given values are on the same side of the equal side, you multiply them and if they are on different sides, then you need to divide. Interest Interest is the cost of borrowing money. An interest rate is the cost stated as a percent of the
amount borrowed per period of time, usually one year. Interest can be divided in two categories Simple Interest and Compound Interest Simple interest is calculated on the original principal only. Accumulated interest from prior periods is not used in calculations for the following periods. Simple interest is normally used for a single period of less than a year, such as 30 or 60 days. Compound interest arises when interest is added to the principal, so that from that moment on, the interest that has been added also earns interest. We need complicated formula to calculate compound interest. We will skip problems from Compound Interest in this section. How do we calculate simple interest on a loan? The solution is in the formula I = Prt, where I is the interest, P is the principle (amount loaned), r is the interest rate, and t is the time measured in years. Example 1: What is the simple interest on a loan of $6,000 for one year if the interest rate is 8.25%? I = Prt I = (6000)(0.0825)(1 year) I = 495 Thus the interest is $495.
Example 2: If one borrows $2000 for 3 years at a 13% annual simple interest rate, what is the interest? I = Prt I = ($2000)(.13)(3) I = $780 Thus the interest is $780. For the loan, one would pay the $2000 borrowed plus an additional $780 for a total amount owed of $2780.
Example 3: What is the interest gained with a $1500 simple interest investment at 8% annual rate for 9 months? Notice that the time is given in months, and since we have an annual interest rate, we must convert the time to years. I = Prt I = ($1500)(.08)(9 12) I = $90 Thus the interest is $90. For the investment, one would get the $1500 invested plus an additional $90 for a total amount received at the end of 9 months of $1590. In the simple interest formula, you can solve for any of the four quantities (I, p, r, and t) if any three of them are known. Example 4: One needs $500 in 2 years. What annual rate does a simple interest investment need if the initial investment is $200? In this case, we want the interest rate (r). We will have $500 in 2 years, but starting with $200. Therefore, the principal is $200, the interest is $500 $200 = $300. $300 is the extra money we made on top of the $200 initial amount. Thus: I = Prt $300 = ($200) r (2) r = $300 ($200×2) r = .75 or 75% Thus the interest rate is 75%
Example 5: How much time does it take $3000 to double if the annual simple interest rate is 20%? In this case, we are solving for time (t). Since we are doubling $3000 (to $6000), we made an additional $3000 on this investment, which is the interest. Therefore: I = Prt $3000 = $3000 (.20) t t = $3000 ($3000 .20) t=5 The amount of time is 5 years. Total Amount If interest is charged, the total amount is the principal and interest added together. Recall that we used these variables in calculating simple interest: P = principal I = interest r = rate t = time Let’s use A for the total amount. So…A = P + I The formula for simple interest that we have used is: I = Prt Putting the two together we get a formula we can used to find the total amount: A = P + Prt Example 6: Ray buys a $900 television. The store gives him credit calculated with simple interest at 30% over 2 years. How much does Ray pay for his television? Using the formula A = P + Prt, we want to know A, the total amount of money Ray paid for the television. The television costs $900 (so P = 900); the interest rate is 30%(so r = .30); the time is 2 years (so t = 2). Substituting into our formula we get: A = P +Prt A = 900 + (900)(.3)(2) A = 900 + 540 A = 1440, So the total amount Ray would pay for his television is $1440.
Credit Cards and Simple Interest Lessons Credit Cards are used throughout the world to purchase goods and services instead of cash money. This material is to discuss some of the concepts of the credit card. Annual Percentage Rate (APR): The annual interest rate associated with the credit card. Credit Limit: The maximum amount allowed charged on a credit card. Transaction Fee: Fees charged to the customer for late payments, exceeding the credit limit, or other non-interest fees. Periodic Rate: The Annual Percentage Rate divided by the period used to calculate the balance (usually 365). Purchases: Charges on a credit card from purchasing goods and services. Cash Advances: Charges on a credit card from cash received. Balance Transfers: Charges on a credit card from transfers from other accounts. Finance Charge: The total amount to be paid in transaction fees and interest. Grace Period: The period of time in which if the new balance of a credit card is paid off, there is no finance charge. How is the interest on a credit card calculated? Most widely used credit cards (Visa, MasterCard, etc.) use the Average Daily Balance approach to finding the amount of interest on purchases during a period of time. In this section we will explain how to calculate the Average Daily Balance.
Example 9: Let's say we have this credit card statement:
Tingling National Bank Credit Card Covering the Period April 15 through May 14 APR = 22.9% Beginning Balance: $240.00
DATE
TRANSACTION
AMOUNT
4/18
Peasant Pastures Gas
$10.00
4/23
Blue Mollusk Seafood Restaurant
$35.00
5/01
Payment Received
$100.00
5/10
New Army Clothing Store
$65.00
What is the amount of interest for this period? The first thing we must find is the Average Daily Balance. This credit card statement is for 30 days. From April 15 through April 17, the balance is $240.00 (3 days) From April 18 through April 22, the balance is $240.00 + $10.00 = $250.00 (5 days) From April 23 through April 30, the balance is $250.00 + $35.00 = $285.00 (8 days) From May 1 through May 9, the balance is $285.00 - $100.00 = $185.00 (9 days) From May 10 through May 14, the balance is $185.00 + $65.00 = $250.00 (5 days) To find the Average Daily Balance, we must average the above: ($240 3 + $250
5 + $285
8 + $185
9 + $250
5) ÷ 30
= (720 + 1250 + 2280 + 1665 + 1250) ÷ 30 = (7165) ÷ 30 = $238.83 The amount of interest is obtained using the Simple Interest Equation: I = Prt. The principal is the average daily balance. The rate is the annual percentage rate. The time is in years, thus the time is the number of days divided by 365. I = Prt I = 238.83× (0.229) × (30/365)
I = 4.50 Therefore the interest for this account for this period is $4.50.
Example 10: Let's look at another statement: Weripoff National Bank Credit Card Covering the Period July 23 through August 24 APR = 19.8% Beginning Balance: $174.76 DATE
TRANSACTION
AMOUNT
7/25
McReynold's Restaurant
$11.45
7/28
Washington Merlins Basketball Tickets $212.00
7/31
Payment Received
$200.00
8/12
High's Black Swamp Theatres
$31.75
8/20
WEW Cooked Wrestling Tickets
$150.00
What is the amount of interest for this period? From July 23 through July 24, the balance is $174.76 (2 days) From July 25 through July 27, the balance is $174.76 + $11.45 = $186.21 (3 days) From July 28 through July 30, the balance is $186.21 + $212.00 = $398.21 (3 days) From July 31 through August 11, the balance is $398.21 - $200.00 = $198.21 (12 days) From August 12 through August 19, the balance is $198.21 + $31.75 = $229.96 (8 days) From August 20 through August 24, the balance is $229.96 + $150.00 = $379.96 (5 days) The Average Daily Balance is: ($174.76×2 + $186.21×3 + $398.21×3 + $198.21×12 + $229.96×8 + $379.96×5) ÷ 33 = $249.11
The amount of interest is: I = Prt I = 249.11*(0.198)*(33/365) I = 4.46 Therefore the interest for this account for this period is $4.46. These two examples show the amount of interest for one month, I would say, reasonable amounts. Of course this is only one month's worth of interest and many of us have higher balances than the ones above. Remember, the Average Daily Balance approach is used to determine the interest on purchases. Interest obtained from the use of cash advances and balance transfers may be obtained another way, usually with an entirely different Annual Percentage Rate. In fact, most cash advances use the Compound Interest Formula compounded daily.
6.4 Financial Literacy Exercises: Read each problem carefully. Write your answer to the point and in complete sentences. 1. A bank charges 6% simple interest. How much interest must you pay on a loan of $2000 for 3 years? 2. A bank charges 5% simple interest. How much interest must you pay on a loan of $700 for 2 years? 3. A bank charges 8.75% simple interest. How much interest must you pay on a loan of $4000 for one year? 4. A bank charges 7.50% simple interest. How much interest must you pay on a loan of $8000 for one year? 5. What is the interest on a $650 simple interest loan with a 10% rate for 18 months? 6. What is the total amount made on a $1000 simple interest investment if the rate is 8% and the term is 3 years? 7. How much would Jerry totally pay if he borrows $2000 at 15% simple interest for 5 years? 8. A computer costs $1100. If a company is charging 30% simple interest for 3 years, how much is totally paid for the computer?
9. Find the average daily balance and the amount of interest from purchases for the credit card account below: Tingling National Bank Credit Card Covering the Period January 11 through February 10 APR = 15.5% Beginning Balance: $200.00 DATE
TRANSACTION
AMOUNT
1/17
Payment
$80.00
1/20
Purchase
$35.00
2/01
Purchase
$100.00
2/04
Purchase
$45.00
10. Find the average daily balance and the amount of interest from purchases for the credit card account below: Tingling National Bank Credit Card Covering the Period November 17 through December 17 APR = 16.9% Beginning Balance: $1231.24 DATE
TRANSACTION
AMOUNT
11/25
Purchase
$65.21
11/30
Purchase
$11.48
12/6
Purchase
$163.95
12/13
Payment
$300.00
11. Find the average daily balance and the amount of interest from purchases for the credit card account below: Tingling National Bank Credit Card Covering the Period October 10 through November 9 APR = 9.9% Beginning Balance: $569.18 DATE
TRANSACTION
AMOUNT
10/19
Purchase
$23.99
10/22
Purchase
$100.00
10/31
Purchase
$65.50
11/2
Payment
$20.00
11/6
Purchase
$78.06
12. Find the average daily balance and the amount of interest from purchases for the credit card account below: Tingling National Bank Credit Card Covering the Period February 11 through March 7 APR = 12.0% Beginning Balance: $0.00 DATE
TRANSACTION
AMOUNT
2/11
Purchase
$100.00
2/14
Purchase
$100.00
3/02
Credit
$25.00
13. Find the average daily balance and the amount of interest from purchases for the credit card account below: Tingling National Bank Credit Card Covering the Period January 1 through February 1 APR = 9.9% Beginning Balance: $1,987.65 DATE
TRANSACTION
AMOUNT
01/01
Beginning Balance
$ 1,987.65
01/02
Purchase
$256.78
01/04
Payment
$497.15
01/05
Purchase
$78.99
01/07
Purchase
$22.21
01/20
Payment
$800.00
6.4 Applications of Percents Exercise Answers: 1. The interest is $360. 2. The interest is $70. 3. The interest is $350. 4. The interest is $600. 5. The interest is $97.50 6. The interest is $240.00. So the total amount is $1240.00 7. Total amount paid is $3500 8. The total amount paid for the computer was $2090 9. The average daily balance is $202.74; The interest for the period is $2.67 10. The average daily balance is $1301.36; The interest for the period is $18.68 11. The average daily balance is $673.54; The interest for the period is $5.66 12. The average daily balance is $183.00; The interest for the period is $1.50 13. The average daily balance is $1930.73; The interest for the period is $16.76
6.5 Circle Graphs Statistical information is often presented in a table form or in the form of a graph such as, line graph, bar graph, or circle graph. These graphs are a way of presenting or visualizing statistical data for easy comparisons. For circle graphs, also known as pie charts, the information is presented in the form of a circle divided into sectors, each sector representing a part of a whole as percent. Example 1: A class of 25 students took a mathematics test. The results are presented by the circle graph below.
Test Results
A, 20% B, 32%
B F C
C, 40%
F, 8%
A
Use the circle graph above to answer the following questions. a. How many students obtained a C grade? Answer: The largest sector of the circle is 40% of the whole circle. Therefore, we calculate 40% of the 25 students who took the test. 40% of 25 = .4×25 = 10 students. b. How many students received an A or a B on the test? Answer: 20% of the students obtained an A, and 32% obtained a B. 20% + 32% = 52% 52% of 25 = .52×25 = 13 students. c. What percent of the students failed the test? How many were they? Answer: The smallest sector of the graph indicates that 8% of the students failed test. 8% of 25 = .08×25 = 2. Two students failed the test. d. How many more students obtained a C grade than a B grade? Answer: 40% of 25 = .4×25 = 10 C students 32% of 25 = .32× 25 = 8 B students 10 – 8 = 2 more students obtained C than B.
e. What percent of the entire class passed the test? Answer: Either, Subtract the 8% who failed from the 100% 100% - 8% = 92% Or, 20% + 32% + 40% = 92% passed
6.5 Circle Graph Exercises 1 – 4 Ms. Blossom is a school teacher. She earns $3,500 per month for her hard work as a teacher. Her monthly expenses are represented by the circle graph below.
Budget Savings, 10% Utilities, 12%
Food, 25%
Food Rent Transportation
Transportation, 8%
Utilities Savings Rent, 45%
Use Ms. Blossom’s circle graph to answer the following questions about her budget. 1. How much does Ms. Blossom spend on rent? 2. How much does she save every month? 3. What percent of her monthly income does Ms. Blossom spend on utilities and transportation? What is the amount? 4. What is Ms. Blossom’s total expenditure other than rent?
5-9. A company named Compassion employs people of all ages, from twenty-year-olds to seventy-year-olds. The circle graph below shows the number of employees in each age group. Use Compassion’s circle graph to answer questions 5- 9.
Ages 70 yr olds, 6 20 yr olds, 12
20 yr olds 30 yr olds
60 yr olds, 15
50 yr olds, 10
40 yr olds 30 yr olds, 24
50 yr olds 60 yr olds 70 yr olds
40 yr olds, 8
5.
Find the total number of employees of this company.
6. What age group constitutes the largest group of employees? 7. What percent of employees are in their sixties and seventies? How many are they? 8. How many employees are under fifty years of age? 9. What age group makes up 16% of the total number of employees/
10-16. Assuming there is a very small country with population of 24 million. This country is divided into six ethnic groups, Akan, European, Ewe, Ga, Gurma, and Moshi. This information is presented in the circle graph below. Use the graph to answer questions 10 –16
Ethnic Groups
Moshi, 16%
Moshi Ga
Akan, 44%
Ga, 8% Ewe, 13%
Ewe European Gurma Akan
European, 16% Gurma, 3%
10. Which is the predominant ethnic group in this population? 11. What percent is represented by the largest ethnic group? 12. What is the population of the largest group of people? 13. Which ethnic groups have the same percentage of people? 14. How many people are in each group of people with the same percent? 15. The Ewe group is larger than the Ga group by what percent? 16. How many more people belong to the Ewe group than the Ga group?
17-20. A survey asked 160 mathematics students what they thought were the most pertinent contributory factors to their success in mathematics education. Among the factors considered were, the role of the instructor, taking responsibility for their homework assignments, study skills and time management, being tutored on one-on-one basis, and study groups. The results of the survey are presented in the circle graph below Use the graph to answer questions 17 – 20.
Math Students Tutoring, 5%
Study Groups, Study Skills, 10% 15%
Study Skills Instructor
Doing Homework, 45%
Instructor, 25%
Doing Homework Tutoring Study Groups
17. What was the most predominant thought of the students surveyed on how to be successful in learning mathematics? 18. How many students thought instructors play a major role in getting them to be successful in mathematics? 19. What percent of the students believed that study skills and time management are very important in their mathematics education? 20. How many students believed in study skills and time management?
6.5 Circle Graph Exercise Answers 1. $1,575 2. $350 3. 20%, $700 4. $1,925 5. 75 6. 30 year-olds 7. 28%, 21 8. 44 9. 20 year-olds 10. Akan 11. 44% 12. 10,560,000 13. European and Moshi 14. 3,840,000 15. 5% 16. 1,200,000 17. Taking responsibility in doing homework assignments 18. 40 19. 15% 20. 24
