|Yale-New Haven Teachers Institute||Home|
Lauretta J. Fox
These and similar items which appear daily in newspapers indicate the importance of understanding the concept of percentage. In order to live more insightfully, wisely and enjoyably every citizen should be able to perform simple mathematical calculations which enable him to compute discount and sales tax on purchases, gratuities for services performed, interest on savings accounts and loans, deductions from his weekly pay and other consumer related problems which he encounters from day to day.
Although percentage has been taught through the years in a variety of ways, many people do not understand the concept and do not know how to use it. When specific problems are assigned to be solved in a class, often students remark, “I do not know how to begin. What shall I do first? Shall I multiply or shall I divide? Shall I use the formula B x R = P, or P Ö B, or PÖ B= R?”
In this unit of study we will try to improve the students’ understanding of percentage by providing them with a consistent and meaningful method of solution for general problems. After reviewing the fundamental arithmetic skills involved in percentage, the students will apply them to significant problems faced by the consumer in his daily affairs.
There are several basic objectives for this unit of study. Upon completion of it, the student will be able to:
— understand the meaning of percentage. change any percent to its equivalent decimal or fractional form. find a percentage of a given number.
— find what rate of percent one number is of another.
— find a number when a percentage of it is given.
— find the rate of percent greater or smaller one number is than another.
— solve consumer related problems by using percentage.
The material developed here may be used at several levels of instruction for the following purposes: (1) to introduce the concept of percentage and its applications to middle school students who have not been exposed to the topic previously, (2) to improve the skills of high school students who have not mastered the techniques adequately, (3) to serve as remedial work in adult basic education classes.
7% = 7/100, 6 1/3% = 6 1/3 Ö 100 = 19/3 x 1/100 = 19/300,
82% = 82/100 = 41/50, 4.8% = 4.8/100 = 48/1000 = 6/125.
The fraction 13/100 means 13 divided by 100. Every time a number is divided by 100 the decimal point moves two places to the left. To change a percent to a decimal, divide the given number by 100, that is, move the decimal point two places to the left and remove the percent sign.
|57$ = .57||125% = 1.25||.39% = .0039||87.5% =. 875|
|1/4 = 1/4 x 100 = 25%||.25 = 25||100= 25%|
Any of these percentage problems may be solved in the following manner. Read the problem carefully to determine what is being asked. Represent the number that you wish to find by the letter N, and change the percent to a fraction. Let the word “of” denote multiplication. Translate the problem into a mathematical sentence, and solve the resulting equation to find the desired number N.
|Example 1:||Find 6% of 80.|
|Solution:||Let N represent the required number.|
|Change 6% to 6/100.|
|Let the word “of” denote multiplication.|
|N =6/100 x 80/1|
|6% of 80 = 4.80|
|Example 2:||16 is what percent of 48?|
|Solution:||Let N represent the required percent.|
|Change N% to N/100.|
|Let the word “of” denote multiplication,|
|16/1= N/100 x 48/1|
|48N = 1600|
|N = 33 1/3|
|16 is 33 1/3% of 48.|
|Example 3:||12 is 75% of what number?|
|Solution:||Let N represent the required number.|
|Change 75% to 75/100.|
|Let the word “of” denote multiplication.|
|12/1= 75/100 x N/1|
|12 is 75% of 16.|
|Example 4:||What percent greater than 75 is 100?|
|Solution:||100 75= 25|
|1/3 x 100/1 =100/3 = 33 1/3%|
|100 is 33 1/3% greater than 75.|
|1. 18% of 750 is ___.||2. 86 is ___% of 129.|
|3. 9 1/4% of 112 is ___.||4. 45 is 90% of ___.|
|5. 7 is ___% of 56.||6. 3 is 7.5% of ___.|
|7. 3.6% of 248 is ___.||8. 15 is ___% of 60.|
- 9. Last month gasoline sold for $1.10 per gallon. This month it is selling for $1.32 per gallon. Find the percent of increase in the price per gallon.
- 10. There are 180 days in the school year. Mary was present 95% of the total days. How many days was she present?
- 11. On an arithmetic test John received a grade of 80%. He solved 24 problems correctly. How many problems were there on the test?
- 12. Mr. Brown receives a weekly salary of $200. He spends $60 per week for food. What percent of his weekly salary does he use for food?
- 13. Last year City High School had an enrollment of 1800 students. This year the enrollment is 1600 students. What is the percent of decrease in the student population?
- 14. A number changed from 24 to 36. What was the percent of increase in the number? Later the number changed from 36 to 24. What was the percent of decrease in the number?
To compute the amount of sales tax to be added to the purchase price of any item, multiply the rate of percent by the price of the item. The result usually is rounded to the nearest cent. To round any amount of money to the nearest cent, look at the digit to the right of the hundredths place. If this digit is less than five, drop it and keep the cents obtained in the product. If the digit to the right of the hundredths place is five or greater, drop it and add one to the cents obtained in the product.
|Example 1:||Round the following to the nearest cent:|
|a) $5.2763||b) $.6419|
|Solution:||a) $5.2763 = $5.28||b) $.6419= $.64|
|Example 2:||Find 7.5% of $495.|
|Solution:||7.5/ 100 x 495/1 = 3712.5/100 = $37.125 = $37.13|
Fifteen percent of any amount may be calculated mentally by using the following three steps: (1) Find ten percent of the amount by moving the decimal point one place to the left. (2) Find five percent of the amount by taking one half of the answer in step one. (3) Add the answers in steps one and two.
|Example 1:||Calculate 15% of $3.89 mentally.|
|Solution:||10% of $3.89 = $.389|
|1/2 of .389 = .1945|
|.389 + .1945 =.39 + .19 = .58|
|15% of $3.89 is $.58.|
|Example 2:||Mary bought a purse for $10.95 and a pair of shoes for $18.95. What total amount did she pay if a 7.5% sales tax was added to the price of the items?|
|Solution:||$l0.95 + $l8.95 = $29.90||Cost of the items|
|7.5/100 x 29.90/1 = 224.25/100 = $2.24 Sales Tax|
|$29.90 + $2.24 = $32.14 Total Paid|
|Example 3:||Jim had luncheon downtown last week. He ordered a hamburger at $.95, potatoes at $.65, and a coke at $.50. What total amount did he pay if a 7.5% sales tax and a 15% tip were added to the price of the food?:|
|Solution:||.95 + .65 + .50 = $2.10||Cost of the luncheon|
|7.5/100 x 2.10 =$.16||Sales Tax|
|10% of $2.10 = $.21|
|5% of $2.10 = $.105 = $.11|
|$. 21 + .11 = $.32||Tip|
|$2.10 + $.16 + $.32 = $2.58||Total Paid|
- 1. Using a 7.5% sales tax, find the amount of tax and the total paid for the following items:
|Item||Price||Sales Tax||Total Paid|
|greeting card .||40|
- 2. During November Mrs. English made the following purchases at a department store.
- Find the total amount of the bill including a sales tax of 7.5% on each item.
- 3. Mrs. Green bought the following items at the Orange Market: 6 lb. roast beef $2.39 lb., 2 lb. coffee $3.29 lb., 3 bottles of soda .39 each (taxable item plus 5¢ deposit on each bottle), 8 bars soap 4 bars for .69 (taxable item), 1 bottle shampoo .99 (taxable item), 10 lb. potatoes $1.19. Find the total amount of the purchase including a 7.5% sales tax on taxable items and deposits on bottles.
- 4. Fifteen office workers are planning to go out to dinner. Using the menu below select a dinner for each person. Find the total cost of the bill including a 7.5% sales tax and a 15% tip.
|Alaskan King Crab Legs||$4.55|
|Iced Cherrystones on half shell||$3.50|
|Baked Stuffed Clams||$3.50|
|Whole Lobster one and onehalf lbs.||$17.34|
|Baked Stuffed Shrimp||$9.95|
|Baked Stuffed Sole||$9.95|
|Alaskan King Crab Legs Fra Diavolo||$10.95|
|Scallops with Lobster Sauce||$9.95|
|Broiled Swordfish Steak||$9.95|
|Reef & Beef||$11.95|
|New York Cut Sirloin Steak||$10.75|
|Filet Mignon with Stuffed Tomato||$12.95|
|Prime Rib Au Jus||$9.95|
|Roast Leg of Spring Lamb||$9.95|
The amount that an article is reduced in price is called a discount. The rate of discount is the rate of percent that is taken off the original price of the article. The original price of an article is known as the list price or marked price, while the amount for which the article sells after the discount has been subtracted is the net price or sale price.
To find the net price of an item which is being sold at a discount, first multiply the rate of discount by the marked price to obtain the discount, then subtract the discount from the marked price to obtain the net price. To find the rate of discount, divide the discount by the list price and multiply the result by one hundred.
|Example 1:||Janet bought a coat which usually sells for $98.00 at 25% off. What did she pay for the coat?|
|Solution:||25/100 x $98.00 = 1/4 x 98.00 = $24.50||discount|
|$98.00 $24.50 =$73.50||net price|
|Janet paid $73.50 for the coat.|
|Example 2:||A blouse marked $10.00 was on sale for $6.00.|
|Find the rate of discount.|
|Solution:||$10.00 $6.00 = $4.00|
|4/10 x 100/1 = 400/10 = 40%|
|The rate of discount is 40%.|
|Example:||The list price of a lawn mower is $189.95. What is the net price if discounts of 20% and 10% are allowed?|
|Solution:||20/100 x $189.95/1 = 3799/100 = $37.99|
|$189.95 $37.99 = $151.96|
|10/100 x $151.96/1 = 1519.6/100 = 15.196 = $15.20|
|$151.96 $15.20 = $136.76|
|The net price of the lawn mower is $136.76.|
- 1. During a post holiday sale, a department store advertised 40% off all merchandise. What was the sale price of a sofa that originally cost $395?
- 2. A factory outlet offers 20% to 50% reductions on sweaters. If a sweater marked $18 sells for $11.70, what is the rate of discount?
- 3. During the month of February, a retailer is selling electric typewriters at a 10% discount. On Washington’s birthday he is offering an additional 22% off the reduced price. What is the sale price of a typewriter with both discounts if the price was $299.99 before the February sale?
- 4. John paid $6.31 for a book marked 25% off the regular price. What was the regular price of the book?
- 5. On Tuesday the five and ten cent store offers a ten percent discount to senior citizens. With the discount what was the price of the following items? 2 spools of thread .33 each, 4 washcloths .59 each, 1 lb. candy $1.29 lb., 2 sheets $2.99 each.
- 6. One store was offering a 15% discount on all dresses. Another store reduced the prices of all dresses by five dollars. Which was the better buy on a dress marked $25? How much better was it?
Sometimes a salesman receives a graduated commission. This means that the rate of commission increases as the amount of sales increases. For example, the rate of commission may be 8% on all sales up to $10,000 and 10% on all sales over $10,000. In many cases a salesman receives a fixed salary plus a commission.
|Example 1:||A salesman who works on a commission basis receives 12% of his sales. How much was his commission on a sale amounting to $265?|
|Solution:||Find 12% of $265.|
|Let N= the missing number.|
|Change 12% to 12/100.|
|Let “of” denote multiplication.|
|12/100 x 265/1= N|
|N = $31.80|
|The salesman earned $31.80 commission.|
|Example 2:||A salesclerk receives a weekly salary of $120 plus a commission of 5% on all sales above $500 per week. During three weeks her total sales were $1540, $1235, and $1040. What were her total earnings for the three weeks?|
|Solution:||$120 x 3 = $360||salary for three weeks|
|$1540 $500 = $1040|
|$1235 $500 = $ 735|
|$1040 $500 = $ 540|
|$1040 + $735 + $ 540 = $2315|
|5/100 x 2315/1 = 11575/100 = $115.75||commission|
|$360.00 + $115.75 = $475.75||total earnings|
|Her total earnings were $475.75.|
|Solution:||$1300 $800 = $500||sales above $800|
|8/100 x 800/1 = 6400/100 = $64||commission|
|14/100 x 500/1 = 7000/100 = $70||commission|
|$64.00 + $70.00 = $134.00||total commission|
|His total commission was $134.|
2. Complete the following table.
- 1. A real estate agent receives a commission of 6% of the selling price of a house. If he sells a house for $135,000 what is his commission? What are the net proceeds?
|Selling||Rate of||Amount of||Net|
- 3. A salesman receives a weekly salary of $160 plus a commission of 4% on all sales above $1800. If his sales for four successive weeks were $2180, $1950, $2472, and $2200, find his total earnings from salary and commission for the four weeks.
- 4. A department store clerk receives 12 1/2% commission on all merchandise sold. Last week she received $52 commission. What were her sales for the week?
- 5. Mary earned a 4% commission on the first $650 of weekly sales, and a 6% commission on all sales over $650. Last week her total sales were $1165. How much commission did she receive?
The amount of money borrowed or invested is the principal. The rate of interest is the percent of the principal charged for the use of the money. It is expressed as an annual rate, unless stated otherwise. The period of time for which the money is borrowed or invested may be expressed in years, months, or days, however, the unit of time must correspond to the rate. If the rate is given as an annual rate, the time must be expressed in terms of a year, that is, days or months must be converted to a fraction or multiple of a year. To calculate simple interest, multiply the principal by the rate of interest, then multiply that product by the time. The formula for finding simple interest states: Interest = Principal x Rate x Time.
|Example 1:||Find the simple interest on $540 at 6% per year for 2 years.|
|Solution:||Principal = $540 Rate = 6/100 Time 2 years|
|Interest = $540/1 x 6/100 x 2/1 = $6480/100|
|Interest = $64.80|
|Example 2:||Find the simple interest on $980 at 12% per year for 10 months.|
|Solution:||Principal = $980 Rate = 12/100 Time = 10/12 yr.|
________=980 /1 x 1/10 x 1/1= 980/10 = $98.00
|Example 3:||Steve borrowed $1200 at 14% per year simple interest for 8 months. What was the total amount due when he repaid the loan?|
|Solution:||Principal = $1200 Rate of Interest = 14/100|
|Time = 8/12 year|
|Interest = Principal x Rate x Time|
|Interest = $1200/1 x 14/100 x 8/12 = $112.00|
|Amount Due = Principal + Interest|
|Amount Due = $1200 + $112 = $1312.|
- 1. Ordinary simple interest in which an “interest year” consists of 360 days or twelve months of thirty days each. The number of days used in computing ordinary simple interest is known as approximate time. This method is used frequently to compute interest on installment loan.
- 2. Banker’s or commercial interest in which an “interest year” consists of 360 days, but the exact number of days in the interest period are used. This method results in a greater return to the lender.
- 3. Accurate interest in which an “interest year” consists of 365 days, and the exact number of days in the interest period are used. This method is used by the federal government and some banks.
|Example 1:||Find the approximate time and the exact time from June 12, 1980 to October 24, 1980. Exclude the first day, but include the last day.|
|Month||of Days||of Days|
|June 12 30||18||18|
|Total||132 days||134 days|
|Example 2:||Find the ordinary, banker’s and accurate interest on $1560 at 15% per year for a loan dated March 21 and due June 6 of the same year.|
|Principal = $1560 Rate = 15% Time 75 days|
|Interest = 1560/1 x 15/100 x 75/360|
|= 39/1 x 5/4 = 195/4 = $48.75|
|Ordinary Interest = $48.75|
|Principal = $1560 Rate = 15% Time = 77 days|
|Interest = 1560/1 x 15/100 x 77/360|
|= 13/1 x 77/20= 1001/20 = $50.05|
|Banker’s Interest =$50.05|
|Principal = $1560 Rate = 15% Time = 77 days|
|Interest = 1560/1 x 15/100 x 77/365|
|= 78/1 x 3/1 x 77/365|
|= 18018/365 =$49.364 = $49.36|
|Accurate Interest = $49.36|
|Example 3:||The banker’s simple interest paid on a sum of money invested at 8% per year for 90 days was $18. How much money was invested?|
|Solution:||Principal = N Rate 8% Time =90/360 year Principal x Rate x Time = Interest|
|N/1 x 8/100 x 90/360 = $18|
|The amount invested was $900.|
- 1. Find the approximate time and exact time for the following: a) March 17, 1980 to July 8, 1980 b) September 3, 1979 to February 26, 1980.
- 2. Jack borrowed $5,000 to buy a car. He repaid the loan at the end of four years. The ordinary simple interest on $5,000 for 4 years was $300. What was the annual rate of interest?
- 4. Complete the table.
|$1 800||8%||70 days||(ordinary)|
|12%||10 months||$ 40|
|$ 675||10%||80 days||(banker’s)|
- 5. Bob invested $2,700 at 11% per year simple interest for 15 months. How much interest did he receive on the investment?
- 6. A man borrowed $500 on June 1 at 18% per year exact simple interest. He repaid the loan in full on the following August 30, What was the total amount repaid?
To calculate compound interest, divide the annual rate of interest by the number of interest periods in the year. Multiply the result by the principal to obtain the interest for the period. Add this amount to the principal to find the new principal at the beginning of the next
interest period. Repeat this procedure for the desired number of interest periods. The difference between the final amount and the original principal is the compound interest accumulated.
|Example:||John invested $500 at 6% compounded monthly. How much compound interest did he earn during the tour months?|
|Solution:||Monthly rate of interest is 6% 12 .5%|
|Balance at||Balance at|
|2||$502.50||.5/100 x 502.50/1||= 2.51||$505.01|
|3||$505.01||.5/100 x 505.01/1||= 2.53||$507.54|
|4||$507.54||.5/100 x 507.54/1||= 2.54||$510.08|
Solve the following set of problems.
- 1. How many interest periods are there in six years if interest is compounded and paid annually? Quarterly? Semiannually? If the interest rate is 10% per year, what is the interest rate per period of time for each of these interest periods?
- 2. If a person deposits $500 in a savings account, how much will he have at the end of two years if the interest is compounded semiannually at 5.5% per year? ‘What amount of compound interest will he earn?
- 3. Mary invested $2,500 at 8% interest per year compounded quarterly. How much did she have at end of 1 1/2 years? How much compound interest did she receive?
- 4. How much more compound interest will $850 earn at 6.5% compounded quarterly as compared to semiannually at the rate of 14% per year?
Using the table, compound interest may be found in the following way:
- 1. Divide the annual rate of interest by the number of interest periods in one year. The result is the rate of interest for each period.
- 2. Multiply the number of interest periods in one year by the number of years given in the problem. The result is the total number of interest periods to be used.
- 3. Read down under the column headed “interest periods” until you reach the total number of interest periods to be used. Follow this line across until you reach the column with the rate of interest found in step 1 for each period. The amount shown in this position in the table is the amount to which $1.00 has grown in the given number of years at the given percent.
- 4. Multiply this amount by the principal to obtain the total amount to which the principal has increased.
- 5. Compound Interest Total Amount Original Principal.
|Example:||How much will $925 amount to in ten years at 6% per year compounded semiannually? What amount of compound interest will be accumulated?|
|Solution:||1. The rate of interest for one interest period is 6 Ö 2 =3%.|
|2. The number of interest periods is 10 x 2 = 20.|
|3. In the compound interest table below, read down under the column headed “interest periods” to 20. Read across the line on which 20 appears to the column headed 3%. The amount shown is 1.8061. This means that $1.00 will increase to $1.8061 in ten years when compounded semiannually at 6%.|
|4. To find the amount to which $925 will increase, multiply $1.8061 by 925. $1.8061 x 925= $1670.6425 = $1670.64.|
|5. The compound interest earned is $1,670.64 $925.00 = $745.64.|
(figure available in print form)
Betz, William, et al. Everyday General Mathematics Book One. Revised Edition. Boston: Ginn and Company, 1960.
A mathematics textbook designed for noncollege bound students. While offering a substantial amount of mathematical training, it is concerned directly with immediate application in everyday life. Special attention is given to historical and cultural background.
———. Everyday General Mathematics Book Two. Boston: Ginn and Company, 1960.
Book two carries forward the type of work begun in Book one.
Bolster, Carey, and Woodburn, H. Douglas. Mathematics in Life. Glenview, Illinois: Scott Foresman and Company, 1977.
A fine textbook for general mathematics classes on the high school level. It stresses basic skills and gives pretests and posttests to measure the progress made by students. Following each skill are consumer applications of the skill. Recreational puzzles are provided to capture the interest of students.
———. Consumer and Career Mathematics. Glenview, Illinois: Scott Foresman and Company, 1978.
A textbook for students enrolled in business arithmetic courses on the high school level. It features problems faced by the consumer in everyday life.
Guthrie, Mearl H., and Selden, William. Today’s Business Mathematics. New York: Pitman Publishing Company, 1967.
A textworkbook developed for high school students in business education. It presents an explanation of the skills, techniques and procedures necessary for employees of any store or office.
Grossnickle, Foster E. Fundamental Mathematics. New York: Holt, Rinehart and Winston, Incorporated, 1971.
An excellent book on elementary arithmetic. Essential portions of modern mathematics as well as informal geometry and beginning algebra have been included. The book provides a good background for further study of modern mathematics.
Hart, Walter W., et al. Mathematics in Daily Use. Boston: D. C. Heath and Company, 1966.
Contains subject matter, instruction and practice in mathematics that the majority of people use in their daily affairs. Maximum attention is given to arithmetical processes and to problems solved by them.
Parsky, Larry M. Target Mathematics for the Worker. Johnstown, Yennsylvania: Mafex Associates, Incorporated, 1969.
One in a series of workbooks designed for practice in mathematics used in daily life. Problems faced by the worker are emphasized.
Piper, Edwin B., and Gruber, Joseph. Applied Business Mathematics. Eighth Edition. Cincinnati: SouthWestern Publishing Company, 1965.
This book was designed to have value for consumer use as well as for business use. It develops the fundamental principles and operations through a study of problems faced by every individual daily. Optional topics have been included to challenge the faster or more capable students.
Rosenberg, R. Robert. Business Mathematics Principles and Practice. Fourth Edition. Chicago McGrawHill Book Company, Incorporated, 1953.
A comprehensive text on commercial arithmetic. Practice is provided in all important topics of fundamental arithmetic with emphasis on the arithmetic of business. Its major purpose is to equip the student with a working knowledge of the basic principles of business mathematics that will play a significant part in his adult activities.
Shaw, Bryce H., et al. Mathematics Plus. Boston: HoughtonMifflin Company, 1979.
A textbook for secondary level students who will terminate their formal education at the end of high school. It contains many exercises in basic skill as well as consumer applications, business applications and technical applications.
Stein, Edwin I. Fundamentals of Mathematics. Modern Edition. Boston: Allyn and Bacon, Incorporated, 1968.
A comprehensive basal textbook in contemporary general mathematics for the Junior and senior high schools. It contains all the basic topics of mathematics and includes computational practice and related enrichment materials. It is ideal for use in consumer mathematics, commercial arithmetic, and shop mathematics classes in the high school.
Thompson, Linda L. Consumer Mathematics Encino,California: Glencoe Publishing Company, Incorporated, 1978.
An excellent book to help the student recognize what is involved in being a good consumer and to develop the mathematics skills needed to solve consumer problems.
An arithmetic book suitable for students at the seventh grade level. The author stresses seeing, thinking and doing. In addition to fundamental arithmetic skills the book contains applications of mathematics in the home and in business.
Kravitz, Wallace W., and Brant, Vincent. Consumer Related Mathematics. New York: Holt, Rinehart and Winston, 1976.
A nicely illustrated book for high school studenta enrolled in consumer mathematics. It endeavors to develop, apply and maintain mathematical skills. Included are descriptions of occupations using these skills and a glossary of terms used throughout the book.
Linder, Bertram L. Economics for Young Adults. New York: W. H. Sadlier, Incorporated, 1973.
A nicely illustrated, practical guide to understanding the realities of economics as the student will actually experience them in life.
Marks, John L., et al. Mathematics We Need. Boston: Ginn and Company, 1965.
An arithmetic book for students at the junior high school level which includes basic mathematical skills as well as an introduction to topics of modern mathematics and geometry. Many extra examples and problems are included for practice.
Skeen, Kenneth C. and Whitmore, Edward H. Modern Mathematics Book One. Syracuse, New York: The L. W. Singer Company, 1965.
A careful transition from arithmetic to algebra is provided. It features a review and maintenance program. Careful attention has been given to readability. An introduction to geometry is included.
Stokes, C. Newton, et al. Arithmetic in My World. Boston: Allyn and Bacon, Incorporated, 1958.
A series of arithmetic books designed for the junior high school student. To establish desirable growth patterns, problems have been organized into units of work composed of activities which are in reality areas of development.
Thordarson, T. W., and Anderson, R Perry. Basic Mathematics for High Schools. Chicago: Allyn and Bacon, Incorporated, 1965.
A textbook to help high school students acquire basic mathematical skills and knowledge which will enable them to make accurate comparisons and computations. Explanations are clear and accompanied by examples drawn from experiences common to most high school students.
Contents of 1980 Volume VII | Directory of Volumes | Index | Yale-New Haven Teachers Institute