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Word Problems

Browse Middle School Algebra
Stars indicate particularly interesting answers or good places to begin browsing.

Selected answers to common questions:
    Direct and indirect variation.
    Inequalities and negative numbers.
    Positive/negative integer rules.
    Solving simple linear equations.

Subtracting One Equation from Another [05/30/2001]
Why can you subtract one equation from another and get something meaningful that you can use to solve the equations?

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?

Summing Three Consecutive Numbers [12/03/1998]
How many natural numbers from 10 to 40 can also be written as a sum of three consecutive natural numbers?

Sum of a Number and Its Reciprocal [01/28/2002]
The sum of a number and its reciprocal is 5 1/5. What is the number?

Sum of the Angles in a Star [09/21/1999]
How can I find the sum of the measures of the five acute angles that make up a star?

Sum of the Digits of Multiples of 9 [04/08/2001]
Why do the digits of nine times a single-digit number add up to 9?

The Sum of Two Numbers is 20... [11/03/1997]
Twice one number is 4 more than four times the other. Find the numbers.

Swimming Laps [04/15/2002]
John decides to swim a certain number of laps of the pool in five days. On the first day he covers one fifth of the total. The next day he swims one third of the remaining laps...

Swimming Upstream and Down [09/19/1997]
A man swims 200 meters upstream in 3 minutes (180 seconds) and downstream 150 meters in .7 minutes. How fast is the man swimming and how fast is the water?

Symbol for Integers [5/19/1996]
Why is the symbol for integers 'z'?

System of Equations and the Substitution Method [06/07/1998]
Can you explain the "substitution" method of solving systems of equations, and provide an example?

Teaching Inequalities [04/01/2009]
We're looking for the best way to help students understand the idea of greater than or equal to, including some real-life examples of how it might be used. Do you have any thoughts?

Testing for Triangular Numbers [06/09/1999]
How do I figure out whether 12456, 1225, or 13456 are triangular or square numbers?

Thinking about the Mediant of Two Fractions [12/10/2009]
One of my students noticed that when given 1/a and 1/b, the fraction 2/(a+b) always seems to be in between 1/a and 1/b. Is this a new fraction rule, or has this been discovered and proved before?

Thirds of Seven [07/24/1997]
Seven is equal to how many thirds of seven?

Thoughtful Trial and Error [05/21/2002]
Arrange the numbers 1 through 10 in a triangle so that all the rows of three numbers have the same sum, and all the rows of four numbers have the same sum.

A Three-Legged Stool [06/26/2001]
Why is a three-legged stool steady, while a four-legged stool can be wobbly?

Three Less Than a Number [08/19/2001]
What is "three less than a number"? I believe it is x-3 but I am being challenged that it is 3-x.

Three Pieces of Wire [01/07/1998]
A florist cuts a wire into 3 pieces. The 1st piece is 20% of the 2nd; the 3rd piece is 110% of the 2nd. The wire is 23 cm long; how long is each piece?

Three Weights [12/07/1997]
A boy selling fruits has only three weights, but with them he can weigh any whole number of pounds from 1 pound to 13 pounds inclusive. What three weights does he have?

Times Greater Than, Times As Much As [05/02/1999]
A number is five times greater than x. Will this number be 6x or 5x?

Top 10 Algebra Mistakes [05/23/1998]
What do you think are the top 10 mistakes that are made in algebra?

Total Test Questions [09/21/1997]
On her biology test, Cathy answered 5/6 of the questions correctly...

Train and Tunnel [7/18/1996]
A train passes completely through a tunnel in 5 minutes. A second train, twice as long, passes through the tunnel in 6 minutes...

Train B Overtakes Train A [01/17/2002]
Train A leaves the station traveling at 30 miles per hour. Two hours later train B leaves the same station travelling in the same direction at 40 miles per hour. How long does it take for train B to catch up to train A?

A Train Breaks Down [06/30/2001]
A train breaks down an hour after starting its journey. After repairs, it can only continue at half its original speed and arrives at its destination two hours late...

Train Overtaking Another Train [10/19/1996]
Two trains leave a station, one an hour after the other...

Translating English to Algebra [06/27/2002]
Write an algebraic expression for 'Seven more than twice a number'.

Translating Word Sentences to Math Sentences [03/10/2004]
"Six more than three times x is four times two less than x" was translated in our textbook to 6 + 3x = 4(x - 2). Some of my students felt it should be 6 + 3x = x - (4*2). Is there a true rule or law of mathematics or some definite tradition as to the interpretation of sentences?

Traveling Trains and Nonlinear Systems [07/16/1998]
A train leaves town A and travels 540 miles at a uniform speed, followed an hour later by another train.... What were the speeds of the trains?

Triangle and Trapezium Ratio [10/26/1998]
ABCD is a square with sides of length 4 cm. Find M on BC so that the ratio of triangle ABM and trapezium ADCM is equal to 1/3.

Turning an Equation Upside-Down [07/09/2001]
After this step: 2/h = (a+b)/ab, a book shows this: h/2 = (ab)/(a+b). Can you simply invert the fractions and maintain the equivalency?

Turning Word Problems into Equations [04/15/2002]
I don't like turning word problems into equations because it is too confusing. Is there a technique that might help me out?

Turning Word Problems to Equations [03/08/2001]
How do you turn English sentences into math equations?

Two Cars on Crete [08/15/1997]
Find the speed of the car from Phaistos.

Two Numbers with Equal Sum, Product, and Quotient? [05/07/2007]
Find two numbers such that when you find the sum, product, and quotient of the two numbers they are all equal.

Two Step Equations [02/24/1998]
How to solve two-step equations.

Two Steps Forward, One Step Back [10/14/2003]
A salmon swims 30 feet upstream in 5 seconds, then rests for 2 seconds. If she moves 10 feet downstream during each rest, how long does it take her to swim 1010 feet?

Two Trains with a Twist [9/11/1996]
The local train heading for Westford at 60km/h and the express train heading for Eastford at 80 km/h pass each other at Centerville while heading in opposite directions...

Two Ways to Solve an Equation [1/23/1996]
x + 3 = 7

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