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From the Dr. Math FAQs

Age Word Problems
A problem with one variable: Al's father is 45. He is 15 years older than twice Al's age. How old is Al? But what if we can't express all the information in the problem using only one variable? Sometimes we must use two variables to express the given information, which means that we need two equations to solve for these variables.

Classic Problems
Classic problems from the Ask Dr. Math FAQ (Frequently Asked Questions) files, including: age word problems, birthday probability, boy or girl?, camel and bananas, coin problems, doubling pennies, grazing animals, liars and truthtellers, the missing dollar, Monty Hall (three doors), three houses three utilities, the Tower of Hanoi, two trains, two trains and a fly, and working together.

Missing Dollar
Three men go to stay at a motel, and the man at the desk charges them $30.00 for a room. They split the cost ten dollars each... where is the missing $1.00?

Negative Times a Negative
Why does a negative times a negative equal a positive? Does this ever happen in the real world?

From the Dr. Math Library

0 Divided by 0 - 11/27/1997
What is the answer to 0 divided by 0? I think it is undefined...

1000 $1 Bills in 10 Envelopes - 12/11/2002
Put 1000 $1 bills into 10 envelopes in such a way that someone can ask you for any amount of money from $1 to $1000 and you can give it to him through a combination of the envelopes.

1+1 Theorem - 8/5/1996
What is the 1+1 theorem?

15 Percent Annual Interest - 8/24/1996
How much money will I will have in 29 years if I save $120 per month at 15 percent annual interest?

25 Coins in a Dollar - 05/07/2001
Carlos has 25 coins whose total is $1.00. What are the coins?

50 Coins = One Dollar - 02/24/1997
How can you make up one dollar using 50 coins?

55 Cents, Seven Coins - 05/15/2001
How can you get 55 cents with seven coins (quarters, dimes, nickels or pennies only) eight ways?

8 dimes, 5 quarters, 4 half dollars - 1/13/1995
Jacob has 8 dimes, Laura has 5 quarters, and Mellisa has 4 half dollars. How much money is this all together?

Accumulating Interest - 06/26/2003
Michael deposits $355 each month into an account that pays 3% interest, accrued monthly. How much money will he have after 3 months? 6 months? 12 months?

Adding and Multiplying to Get 7.11 - 01/13/1999
What four monetary values, when added or multiplied, equal $7.11? (Find A, B, C, D, such that A+B+C+D = A*B*C*D = 7.11.)

Adding and Subtracting Fractions - 10/25/1999
Can you help me learn to add and subtract fractions?

Adding Decimals - 05/26/1998
I can't figure this out: 894.56 + 4563.5 = ?

Adding Four 3-Digit Numbers - 08/13/1998
How do I add: 457 + 458 + 954 + 958?

Adding Negative Numbers - 03/12/1997
What are examples of adding negative numbers in the real world?

Adding Numbers in the 100s - 7/31/1996
How do you add 500 + 995?

Adding or Dividing Letters - 10/26/2002
Solve for x: ax + bc = cx - df. I don't understand how you can divide and add and subtract letters.

Adding, Subtracting, Multiplying, Dividing Fractions - 6/7/1996
I am ten years old and I don't understand lower-term fractions.

Adding Two-Digit Numbers Using Money - 09/13/2001
I don't understand how to add 2-digit numbers like 24 + 56 + ?

Age and Money - 12/13/1994
Frank is eight years older than his sister. In three years he will be twice as old as she is. How old are they now? (and other similar problems)

Amount of Interest Earned in a Year - 11/28/1995
If Li Fong had earned one fourth of a percent more in annual interest on an investment, the interest for one year would have been $45 greater. How much did she invest at the beginning of the year?

Annuities - 12/31/2001
If I have $100,000 at 5% interest and withdraw from it for 20 years, leaving a zero balance, how much monthly income will I have?

An Annuity - 12/13/1995
Compound interest with monthly installments.

Applications of Logarithms - 04/15/1998
How can I calculate the logarithm function on a number without a calculator? What are some of the uses of the logarithm function?

At the Box Office - 03/08/1999
A "random walk" probability problem.

Average Speed and How It Relates to Average in General - 10/07/2008
The average speed of an object is found by dividing distance by time. A basic numeric average is found by adding up the numbers in the set and dividing by how many there are. How does this basic average relate to average speed, or are they unrelated?

Baccarat; a Casino Game - 5/22/1996
Information on the probabilities of baccarat.

Bar Graphs and Histograms - 04/30/2008
What is the difference between a bar graph and a histogram? What are the similarities? Which graph explains more details?

Base Number Equivalence Tables - 10/30/2001
My son needs to show tables for base 3 and base 7.

A Better Mathematics Curriculum -
What should we be teaching students today? Dr. Tom provides his take on what would make a better math curriculum.

Betting Strategy - 05/02/2003
Suppose 2 teams play a series of up to 7 games in which the first team to win 4 games wins the series, and then no more games are played...

Birdseed Formula - 6/5/1995
Someone has 3 dollars and wants to buy two kinds of bird seed. One kind is $1.49/lb. and the other is $2.88/lb...What is the formula for figuring out how much bird seed she can buy? (Includes tax.)

Bits (Money) - 02/26/2001
In the cheer: Two bits, four bits, six bits, a dollar; All for (school), Stand up and holler! what are bits?

Borrowing in Subtraction - 12/01/1998
When doing subtraction on problems like 100-99, why do the zeros become nines? How does that borrowing work?

Box A or Box B? - 11/21/2000
A professor shows you two boxes. Box A has $10K; Box B has $1M or $0. You can take Box B only or both boxes... Which do you choose?

Break-Even Algebra - 10/24/2014
A teen struggles to interpret a word problem into an algebraic inequality. By relating inequalities to equalities and applying some dimensional analysis, Doctor Ian steps through the question.

Buying and Selling Horses - 10/22/1995
A man buys a horse for $60, sells the horse for $70, rebuys the horse for $80, and sells the horse for $90. Did the man break even, lose money, or gain money? If he lost or gained money, how much?

Calculating Interest on Auto Loans - 7/10/1996
How do the the lien holders for auto loans calculate interest on the principle?

Can He Buy Enough Hose? - 5/12/1995
George needs 33 yards of hose to water his lawn. He has $25. Can he get 50 feet of hose for $12.99?

Carrying and Borrowing - 01/14/1998
My son has a problem understanding addition and subtraction when he has to do regrouping, what I call carrying and borrowing...

The Case of the 90 Apples - 11/19/2002
A Cadi gives 90 apples to three girls and says: you must sell these apples in the market. Fatima, the eldest, takes 50, Cunda 30, and Siha, the youngest, the remainder (10). If Fatima sells 7 apples for 1 dinar the others will sell their apples at the same price. The three girls must sell all their apples and must get the same amount of money.

Catalan Numbers and Probability - 05/29/2003
Twenty persons want to buy a $10 ticket each. Ten of them have a $10 note and others have a $20 note. The person at the ticket counter has no money to start with. What is the probability that the person at the ticket counter will not have a change problem?

Change for a Dollar - 01/22/2001
Could you please tell me the two hundred plus ways in which coins are used to make change for one dollar?

Checking Account Balance - 05/08/1997
Each of your parents writes you a check for a random amount less than $400. What is the probability that more than $80 will remain in their joint account after the checks have been written?

Chefs and Algebra - 02/05/2002
I hope to become a chef someday, and I would like to know how algebra (specifically polynomials, the quadratic formula, factoring, etc.) could possibly relate to my future career and my life beyond high school.

Choosing between Median and Mode as the Best Representation - 03/02/2009
A set of data contains the numbers 1, 3, 9, 10, 13, 15, 25, 39, 58, 63. Which measure of central tendency best represents the data?

Civil Engineering and Math - 02/25/1997
What forms of math or physics do civil engineers commonly use?

Closed Envelopes Paradox - 09/13/2001
You must choose between two closed envelopes, one with a certain amount of money and the other with twice as much...

Coin Combinations Quickened by Process of Elimination - 11/04/2013
A student seeks a method for efficiently attacking a puzzle that involves partitioning sums of coins. Doctor Greenie suggests he reverse his thinking and work backwards.

Coin Recognition - 11/20/2002
My son is working on money in class and I am looking for something that shows what each coin looks like.

Coin Specifications - 09/24/2001
Approximately how many nickels does it take to make a pound? How many dimes? Pennies?

Combinations of Coins - 1/28/1996
If I had 6 coins in my pocket, how many combinations could I have?

Comparing Individual Discounts to a Collective Discount - 07/19/2005
I am purchasing two books that each have a 10% discount. If the discount is taken on each book first or taken on the total price of the two books, will the result be same or different?

Compound Inequality - 9/15/1996
5x-1 > 4 and 6x+3 < 4x-1

Compounding Interest and e - 11/11/1998
How is the number e derived from compounding interest? Does it come from the series expansion of e?

Compound Interest - 07/30/1997
I want to buy a car that costs $20,058.00 and would like to know how they calculate the interest on my loan.

Compound Interest - Rule of 70 - 11/29/1994
There is a famous "Rule of 70" for compound interest. What is it and why does it work?

Compound interest with Repayments - 07/24/1997
I need a compound interest formula with repayments, such as in a bank situation.

Concepts of Adding in Base 2 - 08/18/1998
I don't understand the whole concept of base 2.

Congruent Fractions - 04/03/1997
What are congruent fractions?

Converting a Mixed Decimal to a Fraction - 11/09/2001
Write 4.16 2/3 as a mixed number. We know the answer is 4 1/6; we just can't figure out why.

Cost of a Ruler - 08/01/1997
Alice and Betty want to buy the same kind of ruler but Alice is 22 cents short and Betty is 3 cents short...

Cost of Cleaning a Building - 07/03/2003
If I clean a 3200 square foot building five nights per week for a sum of $575.00 per month, what is the cost per square foot?

Counterfeit Money - 05/28/1999
How can I find the amount of money a shoestore owner lost when he was passed a counterfeit $20 bill?

Counting Money - 06/28/1997
How can I learn to count money better?

Counting Sheep the Traditional Way - 10/13/2003
Do you know anything interesting about mathematics in England, say from 400 to 1200 AD?

Craps and Betting Systems - 03/17/2003
In craps, can I bet on the 7 and win (more than I lose) if I assume a seven normally comes up within 32 rolls?

Creating a Mathematical Model of a Complicated Situation - 10/09/2005
I am riding a bike and have several possible routes, each of which contains various traffic lights, amounts of traffic, the possibility of being stopped by a policeman for running a light, and other factors which influence the time it will take to ride each route. How can I model the routes to predict which will be the best choice?

Creating a Payoff Matrix - 06/04/2003
Shoe Town and Fancy Foot are both vying for more share of the market.

Crypto-Number Puzzle - 01/21/1997
A Pascal program to find numerical values for the letters in: ONE + TWO + TWO + THREE + THREE = ELEVEN.

Currency Exchange Rates - 01/02/1999
Do you know how much half a crown, a crown, and a shilling are worth in American money?

Darts and Wilensky's Paradox - 03/17/1999
What is the probability of scoring 25 points with 2 darts if the dartboard is made up of 4 concentric circles?

Data Compression - 08/10/2003
Is there an algorithm that can reduce any binary number to a much smaller binary number, then later be reversed to regain the original number? It has to work for any binary number.

Declining Balance Interest - 03/22/1999
Can you explain declining balance interest to a high school business class?

Defining Counts - 07/05/2016
An adult struggles to disambiguate a unit label from the variable that represents the amount of that unit. Doctor Peterson clears up the confusion before reconciling the different uses with explicit labeling and dimensional analysis.

Derivative of e^x - 10/28/1999
How can I prove that the derivative of e^x is e^x?

Dice Game and Expected Value - 04/18/2005
You pay $6 to play a game where you roll a die with a payoff as follows: $8 for a 6, $7 for a 5, and $4 for any other result. What are your expected winnings? Is the game fair?

Diesel or Gasoline Engine? - 10/16/1996
You can buy a car with a diesel engine for $427.65 more than the same car with a gasoline engine...

The Difference Between Log and Natural Log - 2/8/1995
What is the difference between log and natural log?

Dimes and Quarters Puzzle - 11/5/1994
Jessica has 16 dimes and quarters. Whitney has twice as many dimes and 1/3 as many quarters as Jessica has. They both have the same amount of money. What coins does each have?

Direct and Inverse Variation - 04/25/2001
An example of each type of problem.

Discrete Functions, Broken Down - 09/14/2016
Given written descriptions of everyday situations, a teen struggles to tell the difference between discrete functions and continuous ones. Doctor Peterson guides her to solve a simpler problem.

Discrete or Continuous Problem - 2/13/1995
Mae spends $65 a week on food. After 3 weeks she has $415 left in her food budget. What my math class wants to know is: Is this continuous or discrete?

Discrete versus Continuous - 08/24/1998
What is discrete math? How do you use it?

Dividing Investments for Maximum Return - 09/26/1999
How should one invest $60,000 in mutual funds with a return of 10% and municipal bonds with a return of 12% in order to maximize income?

Dividing the Cost of a Hotel Room - 11/20/1995
A student tells a story about the division of the cost of the hotel room, but the charge doesn't seem to add up. Where did the extra dollar go? (See 'Tricky Word Problem', below.)

Dividing the Prize Money - 05/24/2000
Mrs. Shoe gave her prize winnings to her children in order... and all the prize money was divided equally amongst her children. How many children were there?

Dividing with the Same Units, Such As Money - 06/03/2008
What is $30.00 divided by $5.00? Is it 6 or $6.00? Can I divide two things with the same units, like dollars?

Dollars and the Metric Prefixes - 01/11/2002
I can't do problems that involve km, dm, mm, dam, m, etc.

Dollarwise Division - 02/20/2013
What could it mean to divide dollars into dollars? Doctor Peterson provides examples of contexts where it makes sense to see units in a divisor, then introduces dimensional analysis.

Doubling Pennies for a Year - 06/03/2001
How many pennies will we have at the end of 365 days if we begin on January 1 and double our money each day?

Do We Need a Zero before the Decimal? - 01/26/2005
Do I need to use a zero to hold the dollar place if the amount is less than a dollar? I see money amounts written both ways, as $.45 or $0.45. Are both correct? If one is better, why is it better?

Egyptian Numerals - 10/14/1998
Can you give me some information or references on Egyptian numerals?

Employee Pay Schedule - 07/12/2003
For a particular job, Moe should get 40% of the gross pay, Larry should get 30%, and Curly should get 30%, assuming they all work the same number of hours. But how should the money be divided if they work different numbers of hours?

Estimation in 3rd Grade - 12/16/1996
How do you learn to estimate? What is its importance?

Euler's Number - 01/28/1998
Why is Euler's number referred to as the natural base?

Exceeding the Monthly Payment - 05/28/2010
When a borrower pays in excess of an installment amount, how do you calculate the resulting pricipal and interest? Doctor Wilko explains that lenders service interest first, then apply additional funds to lower any outstanding balance, illustrating with an example.

Expected Number of Plays to End the Game - 06/19/2009
Bob and Kyle each have $18. They flip a fair coin repeatedly. If the coin comes up tails, Bob pays Kyle $1. If the coin comes up heads, Kyle pays Bob $1. The game is finished when either one of them has no money left. What is the expected number of coin flips this game will last?

Explaining Subtraction of Integers as Adding the Opposite - 02/16/2007
I'm trying to help my 7th grade class understand why subtracting an integer is the same as adding its opposite, particularly in cases like (-5) - (-3). Can you give real life examples or ideas on how to explain this to them?

Exp, Log, and Ln Functions Explained - 7/31/1996
What is the exp function? When is it needed? Also, how do I calculate Log and Ln functions with basic arithmetic and logic?

Exponents in the Real World - 11/23/1997
How come we have to know how to add, subtract, multiply, and divide exponents? When do you add them?

Extra Loan Payments, Amortization Tables - 03/31/1998
If one extra payment is made each year, how much money will be saved over the life of a 30-year mortgage and how much sooner will it be paid off?

Fair Value of a Coin Toss Game - 01/21/2001
Suppose you play a game where you toss a coin until tails turn up...

Feeding Chickens - Arithmetical Progression - 7/6/1996
A farmer has 3000 hens. Each week he sells 20... what is the total cost of feeding the hens...?

Fencing Two Back Yards - 01/27/2001
Sarah and Jane want to put a fence around their two rectangular back yards, with a single fence separating the yards... If they share the cost of the fence equally, how much will each of them pay?

Finding the Digits of SEND + MORE = MONEY - 05/19/1998
Representing and solving SEND + MORE = MONEY as a system of equations and inequalities.

First Calculation of E - 12/18/1997
How did Euler first calculate the value of e?

Formula for 8 Percent Interest - 5/31/1996
I put $15 a month into a saving account and at the end of each year I get 8 percent interest...

Fourth Dimension - 05/13/1997
Can you help me understand the fourth dimension?

Front End Estimation - 08/08/2005
My math class is working on front end estimation, and I just don't get it. Can you explain it to me?

A Gazillion - 06/25/1997
What power of 10 is a gazillion?

Girls and Coins - 01/29/2001
Five girls stopped for a soda on the way home from school. The girls were amazed to find that each of them had exactly six coins...

Golf Balls - 04/13/1997
Angie bought some golf balls for $14. If each ball had cost $0.25 less...

Guessing Your Way to Algebra: On Rates, Remainders, Relationships, and Restraint - 09/08/2013
John spent 2/5 of his money on exercise books, and 1/3 of the remainder on three ballpoint pens, which cost eight times as much per. Doctor Ian steps through this word problem by using numeric guesses as placeholders before declaring variables — and making a key insight.

Half Life of Money - 04/13/2001
What is the standard accepted formula for determining the amount of time it takes for money to lose half of its value due to inflation? I've found two, they are: t = t = log2/r and t = log2/log(1+r).

Hard Powers - 12/11/1997
We have a challenge in our math class: nine to the ninth to the ninth power.

The History of Numbers and Numerals - 12/08/1998
Can you give me some information on the history of numbers? How were they developed and where are they used?

Horse Racing Odds - 02/21/2002
In horse racing the odds often add up to more than one. Why?

How can .999999.... equal 1? - 03/21/2001
If .99999999.... goes on forever, wouldn't it be just a little below one?

How Does Math Relate to Economics? - 02/20/2007
Economics has to do with the consumption and production of goods, so how else besides simply using numbers does math relate to it?

How Do Lawyers Use Math? - 01/26/2001
I have to write a two-page paper on how lawyers use math.

How Do You Know That Events Are Equally Likely? - 07/03/2008
How do you determine whether the outcomes of an event are equally likely? The outcomes of a fair coin toss are considered equally likely so each has a probability of 1/2. But how do you first determine that the probability of getting heads or tails really is equally likely?

How High? How Fast? - 07/17/2002
If you throw a ball straight up at 100 mph, how high would it go? And how fast would it be moving when it came back down?

How Long Will the Money Last? - 03/31/2007
If someone has $1,000,000, interest is 5%, inflation is 2%, interest income is taxable at 33.3%, and the person needs $75,000 per year on which to live, how long will it take to deplete the entire principal?

How Many Coins Did Each Originally Have? - 10/22/2001
Natalie has some nickels, Dirk has some dimes, and Quincy has some quarters. Dirk has five more dimes than Quincy has quarters... How many coins did each originally have?

How Many Different Combinations of Shares Could She Buy? - 03/05/1998
Mrs. Ridero has $240 that she wants to invest in three companies traded on the stock market ...

How Many Men, Women, and Children at the Circus? - 10/18/2000
A circus performance is witnessed by 120 people who have paid $120. The men paid $5, the women $2, and the children $.10 each. How many of each went to the circus? A solution without algebra.

How Many Mice, Cats, and Dogs? - 01/13/2003
You must spend $100 to buy 100 pets, choosing at least one of each pet. The pets and their prices are: mice @ $0.25 each, cats @ $1.00 each, and dogs @ $15.00 each. How many mice, cats, and dogs must you buy?

How Much Money? - 08/16/2001
A man has 3 daughters, 4 sons, and a sum of money. If he divides the money equally among his daughters $2 will be left over...

How Much Money Entering the First Store? - 09/26/2002
Timothy spent all of his money at five stores. At each store, he spent $1 more than half of the amount he had when entering the store. How much money did he have when he entered the first store?

How Much Money is Left? - 5/1/1995
A lady has a certain number of dollar bills in her purse. She has no other money. She spends half the money on a hat and gives $1 to a beggar outside the store. She spends half the remaining dollars for lunch and tips the waiter $2. She then spends half the remaining dollars for a book, and just before she goes home she spends $3 on a hot fudge sundae. She now has $1 left. How many dollars did she begin with?

How Much Popcorn did Paul Sell? - 01/28/2002
Paul made $44.14 selling 27 items (beer and popcorn). If he made $1.22 selling popcorn and $2.62 selling beer, how many boxes of popcorn were sold?

How Numbers Will Change in the Future - 10/29/2003
I'm asking my students to develop new number systems. I was hoping to show them what other people are speculating about future number systems, but I was unable to find anything.

How Old at $4,000? - 10/15/2002
Aunt Isabella gives each niece and nephew $10 on his or her first birthday, and on each birthday thereafter the children get $20.00 more than on the birthday before. How old will a child be when he or she receives a total of $4,000.00?

The Hungarian Job Assignment - 03/10/2011
A company owner writes in for help cost-efficiently assigning tasks to different employees when each one commands her own fee for every job. Invoking a little graph theory, Doctor Jacques introduces the Hungarian algorithm and walks through an application to an example assignment.

If It Doesn't Follow a Form, Don't Draw a Conclusion - 09/22/2011
A student of syllogisms and detachments struggles with statements that do not map to those forms. By rhetorically extending her examples, Doctor Peterson reveals that sometimes, the only thing you can conclude is "No conclusion possible."

The Importance of Number Sense and Estimating Answers - 04/28/2004
I don't understand how to solve problems using number sense. For example, if an employee at the deli counter slices a 2 foot long salami, about how many slices will he get? How can I figure that out? Why is it important?

Importance of Zero - 12/19/1995
Has anyone developed an inquiry-based activity for the classroom in which students in grade 5 can explore the importance of zero?

Impossible Problems - 08/07/2001
Are there prizes for finding solutions to impossible problems?

Income from Winning the Lottery - 7/30/1996
How much money would I need to win in the lottery to have an income of $75,000.00 year until I die? That should be in about 30 more years.

Installments without Formulas - 11/15/2016
Wouldn't it be simpler to calculate monthly payment amounts by just adding total interest to loan principal and dividing by the term? An adult questions the necessity for a formula — one with a negative exponent, to boot. After pointing out how the interest depends on the remaining balance, which itself changes over time, Doctor Peterson derives it all by carefully following the money.

Integers in Daily Life - 05/31/2002
How are integers (especially negative integers) used in daily life?

Integers in Everyday Life - 01/16/2002
How do we use integers in everyday life?

Interest Continuously Compounded or Compounded Daily - 10/11/2002
I lent a friend $40 on the condition that he pay 3% interest per day until he decided to pay me back. What formula should I use to calculate the amount he owes me at any day x?

Interest: Definitions - 9/25/1996
What is compound interest? What is continuous compounding?

Introducing Place Value to Children - 10/01/2007
Do you have any ideas on how to introduce children to the concept of place value? I'm working with my son and finding it hard to explain.

Investing at Two Different Interest Rates - 11/24/2003
If Pedro invested $5000 for one year, part at 12% annual interest and the balance at 10% annual interest. His total interest for the year was $568. How much money did he invest at each rate?

Investments - 5/30/1996
Nancy has inherited $80,000... her annual return on three investments is $6800. How much is invested in each?

Irrational Pi - 7/15/1996
Why is pi considered an irrational number when it's defined as the ratio of the circumference of a circle to its diameter?

I Solved It. Now What? - 10/16/2003
I can solve an equation like 12 = 5x + x + 6 to get x = 1, but I don't know what it means, or what I'm supposed to do next.

K as an Abbreviation for Thousands - 05/29/2002
I was taught that 30K meant thirty thousand. Is that correct?

Kindergarten Math Problems - 11/1/1994
I am a kindergarten teacher who has some very interested and interesting math students. Are you willing to talk with us about math? Maybe you have a problem for us to work on so we can get acquainted?

Kumon Math - 3/24/1996
Do you know how I can find information on the Kumon method of teaching mathematics? It was developed by Mr. Toru Kumon of Japan in 1958...

Law of Random Numbers - 12/22/1997
I've recently been debating over the lottery...

Learning Decimals - 10/19/1998
Can you explain decimals to me?

Learning Math Formulas - 02/22/2001
Why is it important to develop math formulas, and why is it important to memorize them?

Let's Make a Deal Probability - 06/01/1999
In "Let's Make a Deal," does Monty's showing you a door, or his knowing which door the prize is behind, affect the probability?

Linear Expressions - 08/17/1999
What are linear expressions, and how will I use them in my life?

Linear Programming - 01/30/1997
Maximize the profit of a nutrition center given information about the energy trail mixes they sell.

Linear Programming - 04/10/2003
A detailed solution, including graphs, of a linear programming problem involving buying file cabinets to maximize storage capacity.

Links from 1 to 63 - 02/19/2002
You are given 1 chain of 63 links to use instead of money. Your task is to cut the chain in 3 places so that you'll be able to hand a person any number of links he asks for, from 1-63.

Logarithms of the Zeros of a Quadratic - 03/24/2002
If p and q are the zeros of 2x^2-mx+1=0, what is the value of log(base 2)p + log(base 2)q?

Lowest Integer That Can't Be Made - 04/05/2004
You have an unlimited number of 'a' cent stamps and 'b' cent stamps, where both a and b are relatively prime positive integers. I am interested in finding a postage value above which any postage amount can be made. I need to find a formula involving a and b, that will give the cutoff point for any such pair of numbers a and b.

Making Change - 04/15/1999
What is the greatest amount of money John can have in his pocket?

Mark-ups, Merchant Fees, and Multiplication ... or Division? - 01/06/2011
Marking up a price by p percent through division by (1 - p) yields a different result than does multiplication by (1 + p). Doctor Wallace explains methods for marking up and for offsetting credit card merchant fees, and along the way distinguishes between markup and margin.

Math and Fashion - 01/27/2003
How is math used for fashion?

Math Anxiety - 04/04/1997
Do you have any information about math anxiety?

Mathematics in the Applied Sciences - 02/24/1997
How do civil engineers and others use math in their work?

Math in Card Games - 10/06/2000
How is math involved in creating and playing card games?

Math in Everyday Life - 09/22/2000
I need to write a story about all the ways in which math relates to everyday life over the course of one day. Could you help me with some examples?

Math in Everyday Life - 10/17/2002
How do you use math in your everyday life?

Math in the Real World - 2/19/1996
I would like to know some examples of mathematical applications in the real world.

Math Logic - 6/5/1996
Sally, Ron, Jim, and Meghan are President, VP, Treasurer, and Captain of the cheerleading squad, but not necessarily in that order. Who is what?

Maximizing Revenue by Analyzing a Parabola - 03/13/2007
When priced at $30 each, a toy has annual sales of 4000 units. The manufacturer estimated that each $1 increase in cost will decrease sales by 100 units. Find the unit price that will maximize total revenue.

Maximizing Window Area - 02/24/1997
Maximize the area of a Norman window (rectangular with a semicircle on top) while minimizing the length of the perimeter.

Metric Conversion - 10/23/1996
How many meters is 67 centimeters equal to?

Metric vs. Standard System - 08/19/2001
Compare and contrast the advantages and disadvantages of the metric system and the standard system using weights and measures.

The Missing Dollar Gets Dressed Up - 12/27/2014
Two adults dispute how to solve a seemingly innocuous word problem. Doctor Ian reveals the red herring in this arithmetic puzzle related to the classic accounting riddle of a motel bellboy and a missing dollar.

The Missing Million - 06/12/2001
If I have six million dollars and my friend has three million dollars, I am wealthier than my friend by three million dollars...

Money Puzzle - 01/05/1998
A man goes to the bank and asks for x dollars and y cents.

Monte Carlo Approximation of Pi - 11/24/2001
Using the circle-in-square model, find an algorithm to determine the value of pi.

The Motel Bellboy and the Missing Dollar - 01/28/1998
The bellboy decides to give each of the men only one dollar back and keep the other two dollars for himself...

Mr. Thomas' Salary - 9/6/1996
Mr. Thomas spent 1/12 of his salary to buy a watch and 3/11 of the remaining to pay for a new television set... what was his salary?

Multiplying by 0 - 03/09/2001
Why, when multiplying any number by 0, do you get 0?

Multiplying Negative by Negative - 11/6/1994
Neg x Neg = Pos just doesn't make sense. Does it ever happen in real life?

Multiplying Two Negative Numbers - 01/16/1997
Can you give me an example why when you multiply two negative numbers you get a positive number?

Natural Logarithms - 11/15/1999
Why are natural logarithms called "natural"?

The Nature of e - 02/23/1997
Why is e considered natural?

Negative 5 and Minus 5 - 10/05/2002
Can you tell me the difference between negative five and minus five?

Negative Numbers in Real Life - 06/25/2008
I'm finding it very difficult to imagine negative numbers outside of math. In the real world, I can't have -$100 in my bank account. I can't give away 6 apples when I only have 5 and be left with -1 apple. How can the math be accurate but impossible?

Negative Numbers in the Real World - 7/2/1996
I am finding it difficult to give concrete examples of negative numbers.

Negative Ratios and Dividing by Zero - 6/8/1995
How can 1 : -1 = -1 : 1 ? Is x divided by 0 = infinity? Are there different infinities?

Negatives - 1/10/1995
Can anyone explain why two negatives make a positive in sign multiplication?

Negative Two Kids - 09/27/2002
Let's say 3 x 2 = 6. Three is how many kids there are in a 'group.' Two is how many 'groups' there are, and six is the answer. Now let's say 3 x (-2) = -6. How can you have negative two 'groups' of three kids?

Nonsense Solutions to Word Problems - 04/28/2003
Bill, Simon, and John are brothers. Bill is as many years younger than one brother as he is older than the other. Simon is 7 years younger than twice the age of John. John is 5 years older than half the age of one of his brothers. I keep getting -3 for Simon's age. What am I doing wrong?

Number of Quarters - 9/11/1996
Tony has eleven more nickels than quarters. How many quarters does he have if the total value of his coins is $2.65?

On Spins and Surprises - 10/15/2011
Nine consecutive spins of a roulette wheel surprise a gambler unfamiliar with how to determine the likelihood of independent events. Doctor Vogler obliges with the requested combinations and probabilities, but only after questioning our arbitrary preference for some events over others that may have the same -- or even longer -- odds.

Open Box Problem - 06/22/2003
Find the formula for the greatest volume box you can make from a sheet of cardboard with different-sized corners cut out of it.

Optimizing Advertising - 10/15/2003
Doubling the amount spent on advertising increases total income by 20 percent. What is the optimum level of advertising?

Ordering Fractions by Changing Them to Decimals - 12/11/2008
How can I put the fractions 3/10, 1/2, 7/8, 4/5, 1/6, 3/4, 9/10, 1/3, 3/5, 3/8, 1/4, and 1/10 in order from least to greatest?

Order of Operations and Adding Negative Numbers - 08/08/1997
What does (-9 + -10) mean?

Organizing a Word Problem - 03/21/2000
Backstreet Boys front-row tickets cost $35, center stage $25, balcony $15, t-shirts $10, and posters $8. John buys 2 front-row tickets, 2 t- shirts, and 2 posters. How much does he spend?

Paradox Involving Swapping Dollars - 08/01/2005
A discussion of a seeming paradox involving two people comparing amounts of money in their wallets with the greater amount being given to the holder of the lesser amount. At first glance it seems that the game favors both participants, but a closer look reveals that is not really the case.

Paradox of the Unexpected Exam - 03/26/1998
A teacher announces that a test will be given next week on one of the five weekdays. Why won't the test ever be given?

Parentheses from the Inside Out - 03/03/2003
I don't understand order of operations and evaluating expressions.

Paying Off a Loan - 11/14/1997
You want to borrow $25,000 at 6.9% to buy a new car...

Percent Greater Than vs. Increased - 11/22/2002
Is there a difference between the following statements? My profits are 200% greater than they were last year; My profits from last year have increased 200%.

Powerball Lottery: Odds of Winning a Prize - 09/02/2001
How are the Powerball lottery odds established?

Powerball Odds When Buying More Tickets - 12/31/2001
If the odds are 80,000,000 to 1 of buying a winning ticket, do they go to 40,000,000 to 1 if you buy 2 tickets?

Principal or Interest? - 08/08/2002
Given a new car loan for $20,000 at 9% simple interest, how do I calculate how much of the monthly payment is going to principal vs. interest?

Prize Money - 09/04/1997
If first prize wins $1,000 out of $6,000 and twentieth prize wins $100, how much money do second through nineteenth place win? Is this a geometric sequence?

Problem-Solving: Systems of Equations - 4/27/1996
Figuring the cost of printing business cards.

Profit on Investment - 4/16/1996
Ann, Bill, Cynthia, and Don invest $2000, $3000, $8000, and $7000 in a joint business venture...

Project Ideas - 06/30/2002
Can you suggest any science projects involving math and the stock market?

Pumpkin Price - 06/28/2001
Peter Peter has 40 pumpkins to sell. He will sell them all if he sets his price at $1.00 each, but will sell one fewer pumpkin for each 5-cent increase in price...

Putting Fractions in Order - 01/24/2001
I am trying to order fractions from least to greatest, but am having no luck.

Quadratic Equation - 7/10/1996
Why is a quadratic equation "quadratic"?

Questions about Fourier Series - 11/18/2000
Must the integrals defining the coefficients of a Fourier series exist? If so, what type of integral (Riemann, Lebesgue, etc.) is considered, and how is convergence of the series determined?

Range, Mean, Median, and Mode - 11/17/1998
Why do we study range, mean, median, and mode? How will these concepts help me in real life?

Rate vs. Ratio - 10/23/2001
What is a rate? What is a ratio? Are all rates ratios, and are all ratios rates?

Rate vs. Yield - 01/31/2002
What is the interest amount for the period from Sept. 05, 2001 to Dec. 04, 2001, at 5% interest, compounded daily, on principal of $356,964.15?

Reba's Physics Problems - 12/09/2001
Reba's dad said he'd pay her a quarter for each correct answer and fine her a dime for each incorrect answer. She received $3.80 after doing 25 problems. How many problems did she answer correctly?

Retirement Savings - 05/07/2001
How can I calculate how much money I'll need to save each month from age 25 to age 65 at 5% compound interest so that I can withdraw $1,000 each month from age 65 to 80 without running out?

Return on Investments - 10/14/1998
What is the formula for calculating the return on investments (ROI) in a percentage?

Roulette Probability - 12/07/1997
What is the probability of winning a certain amount of money using only the even money bets and the two to one bets?

Roulette Questions - 01/26/1998
What is a safer bet: to bet on a number, or to bet on color? Which bet maximizes your wins: betting on numbers, or betting on colors?

Rule 78 Amortization with Balloon Payment - 09/29/1999
An amortization schedule, using the RULE-78 method, for a loan with a balloon payment.

Rule of 72 - 01/26/1999
Why does the rule of 72 work? For what interest values does it work best?

Ruth Has $2300 More Than Ken - 04/15/1999
...If Ken gives Ruth $2000, Ruth will have eight times as much money as Ken. How much money does each of them have?

Salary at 5 Percent Yearly Increase - 9/11/1996
Martin starts work at $18,000/year and receives an annual increase of 5 percent. What is his salary after 30 years?

Savings, Superannuation, and Investments - 04/18/1999
You invest $400 per month at 8 percent per annum calculated monthly. Calculate the amount accumulated...

Selling the Bike - 3/1/1995
Billy bought a bike for $40 and sold it for $50. He plans to buy it back for $60 and sell it again for $70. Will he make money, lose money, or come out even?

Send More Money - 05/12/1997
Find the digit that each letter represents in the equation SEND + MORE = MONEY.

Setting Interest Rate to Maximize Profit - 7/25/1996
A loan company is limited to charging maximum 18 percent interest on a loan. The amount of money available for loans is proportional to the interest rate the company will pay its investors...

Seventy-two and 115: What Do Logs Have to Do with Doubling and Tripling Your Money? - 02/19/2010
Doctor Carter uses logarithms and the Taylor series to show where the 72 in the "law of 72" comes from -- and shows how the same interest rate calculations yield 115/I as an approximation for the number of years it takes an investment that bears I interest rate to triple in value.

Sharing Money - 01/25/1997
Three people have various numbers of coins. How can they share the coins so that each has the same amount of money?

Shopping - 8/26/1996
Which items should a shopper buy if she wants to spend as much of the $62.00 she has as possible?

Simultaneous Equations - 07/01/2002
A man has a total of 20 dimes and quarters. If the dimes were quarters and the quarters were dimes he would have 90 cents more than he has now. How many of each coin does he have?

Slot Machines - 5/29/1996
What are the probabilities and mathematical expectations of slot machines?

Software Sales - 02/01/1997
How do you transfer word problems into mathematical equations?

Solving a Fraction Problem Using Common Sense - 03/21/2004
When the gun sounded, only two-fifths of the racers began to run. If 460 racers began to run, how many racers were there in all? Can you show me how to figure out what a fraction is without knowing the total number?

Solving a Typical Rate-Time-Distance Problem - 09/02/2005
A look at how to use algebra to solve problems involving rates, times, and distances. The problem in this example centers around walking and running at two different rates for a total of 3 hours.

Solving Equations to Find Profit - 8/10/1996
You buy A and B for 40,000 Won; sell A for a 0.3 percent and B at a 0.2 percent discount. You profit 5000 Won - what was the original price of A?

Solving for Multiple Unknowns - 03/22/2003
Three chickens and one duck cost as much as two geese. One chicken, two ducks and three geese cost $25. What is the cost of each bird?

Solving Problems by Making Organized Lists - 04/09/2008
To find 11 coins that total $1.37, Dr. Ian makes organized lists which reduce the problem to smaller and smaller problems until it can be solved. This general strategy is useful in many math problems.

Solving SEND + MORE = MONEY - 04/18/2002
I have tried logical reasoning and can't get it.

Sonya Kovalevskaya - 11/19/1995
Could you please advise me on where I can find information on a mathematician by the name of Sonya Kovalevskaya?

Square Dance Partners, Trip Budget - 6/30/1996
1) Sally invited 17 guests to her party...Everyone had to have a partner so that the sum of each couple's number was a perfect square... 2) Mary budgeted $31 a day for her trip...

Squaring Two-Digit Numbers Ending in 5 - 09/10/2001
Take the first digit, multiply it by the next consecutive number, and place it in front of 25. Can you prove this shortcut?

Stock Market Scam - 02/08/2001
Every Monday a firm sends you a letter correctly predicting the behavior of a given stock over the coming week. In the 8th letter the firm offers to sell you more predictions for a fee. You refuse. Why?

Strategy and Expected Value of Dice Game - 09/03/2005
After rolling a die, you can collect the amount that shows on the face of the die, or you can reject that payoff and roll again. You collect only the amount of the most recent roll, and you may choose to roll up to three times. What is the expected payoff of this game?

Subtracting Mixed Numbers with Borrowing - 03/31/2009
A question on how to subtract 6 1/8 - 3 1/3 leads to examples of borrowing and regrouping in various contexts, including time, money, liquid volume, and of course mixed numbers. An alternative approach of producing an equivalent but easier problem is also presented.

Subtracting Negative Integers - 09/12/2001
I have questions like: -3 - -13. My teacher said to do it in a money way....

Subtracting Positive and Negative Numbers - 5/21/1996
How would you figure out the answer to -13 - 6 = ? and how can you subtract a negative number from a positive number or a positive number from a negative number?

Subtracting Taxes from Earnings - 11/21/1995
Customers at Flo's Meat Market must pay 6% state sales tax. One day Flo takes in $2650, including tax. How much of the $2650 does Flo have left after sending the tax money to the state?

Subtraction Using Nine's and Ten's Complements - 05/27/2000
How does subtraction using the "method of complements" work? Why does it give the correct answer all of the time?

Switching Dollars and Cents - 10/07/1997
How do I find an equation?

Take Aways - 03/28/2002
56 - 18 = ? I don't know how to do it.

Tea and Cakes - 11/13/2002
A cafe sold tea at 30 cents a cup and cakes at 50 cents each. Everyone in a group had the same number of cups of tea and the same number of cakes. The bill came to $13.30. How many cups of tea did each person have?

Teaching Large Number Concepts - 11/05/1997
I have to do a project on how to teach the concept of millions, billions, and trillions to elementary school children.

Teaching Rounding Rules - 10/17/2002
I was taught that when one rounds numbers like 3.5 or 4.5 to the nearest even integer, so that 3.5 should be rounded up, and 4.5 should be rounded down. Is there any good reason why grade-schoolers are taught to always round up?

Time Value of Money - 03/20/2002
Determine the annual percentage rate of yield or return on an investment.

Tricky Word Problem - The Missing Dollar - 2/26/1996
A student asks for help with what seems to be an impossible word problem (more on 'Dividing the Cost of a Hotel Room', above).

Trigonometry in a Nutshell - 04/11/2001
What is trigonometry? Can you give me some problems and examples? What does trigonometry have to do with circles? with waves?

T-Shirts and Shorts - 8/24/1996
In the evening, a sporting goods store sells twice as many as many T- shirts as shorts...

The Two Envelope Paradox in Probability and Expected Return - 09/20/2005
A classic problem in which two envelopes contain money, one double the other. After choosing one at random, should you trade it for the other? It appears there should be no advantage, but the expected values suggest otherwise. Or do they?

The Two Envelopes - 09/08/1997
Given two sealed envelopes, one containing twice as much money as the other...

Types of Correlation - 06/25/2002
Can you explain positive, negative, strong, and weak correlation?

Understanding Graphs - 10/30/2002
How do we use algebra or graphs in real life?

Understanding Negative Numbers - 5/18/1996
I don't understand negatives - can you help me?

Using Addition and Subtraction to Check Answers - 02/25/2003
How can you use addition to check subtraction? How can you use subtraction to check addition?

Using Graph Theory to Count Routes - 07/15/1998
How do you use a diagram to count the number of different direct routes that connect five cities?

Using Variables to Solve Word Problems - 05/10/1998
Susan puts only dimes and quarters in her coin bank ...

The Value of Two-Column Proofs - 12/19/2000
What is the point of learning two-column proofs?

What Are Variables For? - 02/09/2001
Why is it important to be able to figure out the values of variables?

What does 9,600,000,000,000,000,000,000 mean? - 11/20/1996
How do you understand such a big number?

What Does It Mean to Distribute - 08/25/2007
I understand the basics of the distributive property, but I'm confused about what the 'over' means, like in the distributive property of multiplication over addition or over subtraction.

What is a Derivative? - 06/07/1998
What is the concise definition of a derivative? How do we use derivatives?

What is Algebra? - 08/24/1997
Could you explain algebra from the ground up?

What is Mathematical Modeling? - 11/08/2002
I am looking for documentation that would describe the difference between calculations and mathematical modeling.

What is multiplication? - 11/09/1997
What does multiplication have to do with addition, geometry, and real life?

What Sign Do You Use? - 10/21/2001
If two negatives equal a positive, then how do you do it, and what sign do you take?

What We're Doing When We Do Algebra - 03/25/2002
How do you graph x + y = 5 on a graph sheet?

Which is More? - 05/21/2002
Given a choice between nickels stacked to your height, or quarters placed end to end to your height, which should you choose?

Why Do We Have to Study Math in School? - 02/25/2005
A freshman in high school questions the point of learning lots of math that he will never use in real life, and Dr. Ian replies with some interesting thoughts about the benefits of studying math.

Why Estimate? - 08/16/2002
Estimate to the nearest thousand a solution to Dave's subtraction problem. The numbers are: 6,107 - 2,980 = 4,127.

Why is Math Interesting? - 01/16/2001
Why does math have to be so boring and hard?

Why Is Mental Math Important? - 03/25/2004
How can I convince a 14 year old girl who is in 8th grade the importance of mental math? I think that skills like mentally adding and subtracting 2 digit numbers and being able to estimate multiplying 2 digit large numbers are critical. My daughter's teacher says that such skills aren't needed because of calculators and computers.

Why Learn to Factor? - 07/08/2002
Why do I need to learn to factor expressions if I'm going to be a fashion designer?

The Why of Algebra - 11/13/1998
What is algebra for? Why do you need the letters and variables?

Why Order of Operations? - 5/17/1995
7th-graders wonder why we use the order of operations at all. Why not just set up problems to read from left-to-right? Can you suggest some "real-life" examples that show the necessity of using the order of operations?

Why Study Prime and Composite Numbers? - 01/25/2001
My husband and I wonder why this is taught in school at all. Who uses this in the real world? Why does someone need to know whether a number is a prime number or not?

Why the Decimal Point Goes Where It Does - 09/21/2003
My daughter is having trouble understanding why the first place before the decimal point is 'ones', and the first one after is 'tenths'.

Winning the Lottery - 01/30/1997
Can you find a formula that predicts the outcome of the lottery?

With How Much Money Did Each Start the Game? - 04/24/2002
Ferne, Donna, and Susan have just finished playing three games... After three rounds, each has $24.00. How much money did each have at the start of the game?

Working Out Betting Odds - 06/27/2001
I am doing a horse racing night in a local pub, and as part of the committee I have been asked to try to find out how to work out the betting odds for each horse...

Working with Two Equations using Graphs - 01/16/2002
You estimate that you can make muffins for $.30 each. Advertising will cost $18.00. You sell the muffins for $.75 each. Write and graph equations to represent income and expenses. Find the break-even point.

The World Series - 10/22/2000
What is the probability of the Yankees and the Mets playing each other in the World Series?

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