Chapter 8 Sections 8.1 and 8.2

84% 5% 37.5% 172%

.84 .05 .375 1.72

Finding a Certain Percentage of a Whole

According to minoritynurse.com, just 5.8% of all registered nurses in the United States are men. The average hospital in 2012 had 335 registered nurses. How many would likely be male? First, write 5.8% in decimal form, as 0.058. The question is "what is 5.8% of 335?" which we translate into symbols: 5.8% of 355 is .058 x 335 = 19.43 You can't have 0.43 nurses, so we interpret the answer as 19 male nurses at an average hospital.

Computing Monthly Payments Example

Admiral Chauffeur Services borrowed $600 at 9% simple interest for years to repair a limousine. Find the interest, future value, and the monthly payment. Step 1 Find the Interest I= Prt = (600)(0.09)(1 1/2) = $81 Step 2 Find the Future Value of the Loan A = P+I A= 600 + 81 = 681 Step 3 Divide the future value of the loan by the number of months. Since years = 18 months, divide $681 by 18 to get $37.83. The monthly payment is $37.83. The Lookout Restaurant took out a loan for $5,000. The simple interest rate was 6.5%, and the term of the loan was 3 years. Find the interest, future value, and monthly payment. Step 1 I = Prt = (5000)(0.065)(3) = $975 Step 2 A= P + I = 5000 + 975 5975 Step 3 5975/36 = 165.97

42%

42/100=21/40

.74 .05 1.325 5.463

74% 5% 132.5% 546.3%

Recognizing Misuse of Percents in Advertising

A department store advertised that certain merchandise was reduced 25%. Also, an additional 10% discount card would be given to the first 200 people who entered the store on a specific day. The advertisement then stated that this amounted to a 35% reduction in the price of an item. Is the advertiser being honest? Let's say that an item was originally priced at $50.00. (Note: any price can be used.) First find the discount amount. Discount = Rate x Selling Price = 25% x 50.00 =.25 x 50.00 =12.50 Then find the reduced price. Reduced Rate = Original Price - Discount = 50.00 - 12.50 = 37.50 Next find 10% of the reduced price. Discount = rate x reduced price = 10% x 37.50 = .10 x 37.50 = 3.75 Find the second reduced price. Reduced Price = 37.50 - 3.75 = 33.75 Now find the percent of the total reduction. Reduction/ Original Price 12.50 + 3.75/ 50.00 16.25/50.00 = 0.0325 = 32.5% The total percent of the reduction was 32.5%, and not 35% as advertised.

Finding a Percent Change (Decrease)

A large latte at the Caffeine Connection sells at a regular price of $3.50. Today it is on sale for $3.00. Find the percent decrease in the price. Step 1 Find the amount of decrease. $3.50 − $3.00 = $0.50. Step 2 Make a fraction Amount of Decrease / Original Price .50/350 Step 3 .1428= 14.3 (rounded) The decrease in price is 14.3%.

Finding a Whole Amount Based on a Percentage

A medium-sized company reported that it had to cut its work force back to 70% of what it was last year. If it has 63 workers now, how many did it have a year ago? Convert 70% to a decimal: 70% = 0.70. Now write as a question and translate to symbols: 70% of what number is 63 .70 * x = 63 This gives us the equation 0.70x = 63, which we solve for x. ..70x=63 Divide both sides by .70 x=63/.70 = 90 The company had 90 workers a year ago.

Converting Percents to Fractions

A percent can be converted to a fraction by dropping the percent sign and using the percent number as the numerator of a fraction whose denominator is 100.

Calculating Cost of Sale from Commission

A real estate agent receives a 7% commission on all home sales. How expensive was a home if she received a commission of $5,775.00? In this case, the problem can be written as $5,775.00 is 7% of what number? 5775 is 7% of x 5775 = .07x Divide .07 by both sides 5775/0.07= 82,500 The home was purchased for $82,500.00.

Finding the True Rate of a Discounted Loan

A student obtained a 2-year $4,000 loan for college tuition. The rate was 9% simple interest and the loan was a discounted loan . Find the discount. Find the amount of money the student received. Find the true interest rate. a) The discount is the total interest for the loan. P = 4000 r = .09 t = 2 years I= Prt = (4000)(.09)(2) =$720 The discount is $720. b) The student received $4,000 − $720 = $3,280. c) The true interest rate is calculated by finding the rate on a $3,280 loan with $720 interest. I = Prt 720 = (3,280)(r)(2) Multiply 720 = 6,560r Divide both sides by 6,560 r = $720 -------- 6,560 = 0.1098 (rounded) The true interest rate is approximately 10.98%.

Finding a Percentage from a Discount

Cat finds the perfect dress at Macy's, with a selling price of $79. She has two coupons to choose from: one offers $15 off any purchase of $50 or more, the other offers 20% off any item. Which is the better choice? The question really asks if the $15 discount is more or less than 20%, so we need to answer this question: 15 is what percent of 79? 15= x * 79 This is the equation 79x = 15, which we can solve for x. 79x=15 Divide both sides by 79 x=15/79+ x= .19 The decimal 0.19 corresponds to 19%, so the 20% off coupon will be a little better.

Using the Banker's Rule

Find the simple interest on a $1,800 loan at 6% for 120 days. Use the Banker's rule. P = 1,800 r = .06 t = 120/360 I = Prt = (1,800)(.06)(120/360) =$36 The interest using the Banker's rule is $36.

Computing Simple Interest

Find the simple interest on a loan of $3,600 for 3 years at a rate of 8% per year. Change the rate to a decimal and substitute into the formula I = Prt: I= Prt =(3,600)(.08)(3) = 864 The interest on the loan is $864. Find the simple interest on a $12,000 loan for 5 years at 7%. I=Prt = (12,000)(0.07)(5) =4200 The interest on the loan is $4,200

Finding Future Value

Find the simple interest on a loan of $3,600 for 3 years at a rate of 8% per year. Find future Value = Substitute into the formula A=P+I A=P+I =3600+864 =4,464 The total amount of money to be paid back is $4,464. ALTERNATE SOLUTION A= P(1+ 0.08 x 3) =3600 (1+ 0.08 x 3) = $4,464

Finding a Percent Change (Increase)

In 2012 the population of the town of Oak Creek was 23,258. In 2013, the population increased to 23,632. Find the percent increase in the population. Step 1 Find the amount of increase 23,632-23,258=374 Step 2 Make a fraction Amount of increase / Original Price 374/23,258 Step 3 Change Fraction to a Percent 374/23,258 = .0160 The increase in Population is 1.6%

Converting Percents to Decimals

In order to change a percent to a decimal, drop the % sign and move the decimal point two places to the left.

Formulas for Computing Simple Interest and Future Value

Interest = principal × rate × time: I=Prt Future value = principal + interest: A=P+I or A P(1+rt)

Computing the Term of a Loan

Judi and Laura borrowed $4,500 at to put in a hot tub. They had to pay $2,756.25 interest. Find the term of the loan and the monthly payment. P = 4,500 r = 8 3/4 % = .0875 I = 2,756.25 2,756.25 = (4500)(.0875)(t) Divide both sides by (4500)(.0875) 2,756.25 = (4500)(.0875)(t) ---------- ---------------- (4500)(0.0875) (4500)(0.0875) 7 = t The term of the loan was 7 years, which is 84 months The total amount paid is the principal plus the interest: 4,500 + 2,756.25 = 7, 256.25 Divide by 84 months to find the monthly payment 7, 256.25 / 84 = 86.38

Computing Principal

Phillips Health and Beauty Spa is replacing one of its workstations. The interest on a loan secured by the spa was $93.50. The money was borrowed at 5.5% simple interest for 2 years. Find the principal. I = 93.50 r= 5.5% or .055 t= 2 93.50 = P (.055)(2) Divide both sides by (0.055)(2) 93.50 -------- (0.055)(2) P = $850 The amount of the loan was $850.

Computing Interest Rate

R & S Furnace Company invested $15,250 for 10 years and received $9,150 in simple interest. What was the rate that the investment paid? P = 15,250 t = 10 I = 9,150 9, 150 = 15,250 x r x 10 Divide both sides by (15,250)(10) 9,150 ------ (15,250)(10) r = .06 or 6% The interest paid on the investment was 6%.

The Banker's Rule

Simple interest for short term loans is sometimes computed in days. For example, the term of a loan may be 90 days. In this case, the time would be 90/365 since there are 365 days in a year. However, many lending institutions use what is called the Banker's rule. The Banker's rule treats every month like it has 30 days, so it uses 360 days in a year. They claim that the computations are easier to do. When a lending institution uses 360 days instead of 365, how does that affect the amount of interest? For example, on a $5,000 loan at 8% for 90 days, the interest would be I = Prt = (5,000)(0.08)(90/365) = 98.63 Using the Banker's rule, the interest is I = Prt (5,000)(0.08)(90/360) 100.00 We can see why this is called the Banker's rule and not the customer's rule!

Discounted Loans

Sometimes the interest on a loan is paid upfront by deducting the amount of the interest from the amount the bank gives you. This type of loan is called a discounted loan. The interest that is deducted from the amount you receive is called the discount

Procedure for Finding Percent Increase or Decrease

Step 1 Find the amount of the increase or the decrease. Step 2 : Amount of Increase Divided by original amount or Amount of Decrease Divided by Original Amount Step 3 Change the fraction to a percent.

Computing Monthly Payments

Step 1 Find the interest. Step 2 Find the future value of the loan Step 3 Divide the future value of the loan by the number of months

An outdated Rule

The Banker's rule is very old, and there was a time when it made sense. Originally, interest on savings and loans had to be calculated by hand, and standardizing every month to 30 days did in fact make the calculations a lot simpler. But those types of calculations haven't been done by hand for over 50 years, and in the age of computers, it's just plain silly to worry about how difficult it is to compute the interest. That's exactly why computers are called computers—they're really good at computing things! So why do some lenders still use the Banker's rule? Because they can, probably, and it makes them more money.

Calculating Sales Tax

The sales tax in Allegheny County, Pennsylvania, is 7%. What is the tax on a calculator that costs $89.95? What is the total amount paid? Find 7% of $89.95: Write 7% as 0.07 in decimal form, then multiply. .07 x 89.95= 6.30 (rounded) The sales tax is $6.30. The total amount paid is 6.30 + 89.95 = 96.25

Converting a Decimal to a Percent

To change a decimal to a percent, move the decimal point two places to the right and add a percent sign.

Changing a Fraction to a Percent

To change a fraction to a percent, first change the fraction to a decimal, then change the decimal to a percent.

Computing Simple Interest for a Term in Months

To meet payroll during a down period, United Ceramics Inc. needed to borrow $2,000 at 4% simple interest for 3 months. Find the interest. Change 3 months to years by dividing by 12, and change the rate to a decimal. Substitute in the formula I = Prt. I = (2,000)(0.04)(3/12) = 20 The interest is $20. Marta needs some quick cash for books at the beginning of spring semester, so she borrows $600 at 11% simple interest for 2 months. How much interest will she pay? I= (600)(.11)(2/12) =$11 The interest is $11

P (Principal)

is the amount of money borrowed or placed into a savings account.

Future value (A)

is the amount of the loan or investment plus the interest paid or earned.

I (Interest)

is the fee charged for the use of money.

Time (t) or term

is the length of time that the money is being borrowed or invested. When the rate is given as a percent per year, time has to be written in years.

Rate (r)

is the percent of the principal paid for having money loaned, or earned for investing money. Unless indicated otherwise,