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Middle School Archive

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.

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Questions about Zero - Undefined or Indeterminate? [8/16/1995]

Is 0,0,0,... a geometric sequence? What is the value of 0/0? What is the value of 0^0 (^ means exponent)? Can it be considered to be the limit of a^0 as a approaches 0?

Questions Answered Correctly [06/11/2003]

There are 30 multiple-choice questions in a contest. Five marks are awarded for each correct answer, and three marks are deducted for each incorrect answer. No marks are awarded for questions left unanswered. If a student scores 78 marks, what is the greatest possible number of questions answered correctly?

Raising Sides of an Equation to the 0 Power [12/04/1997]

If you raise both sides of an equation to the 0 power, will both sides then be equal to one?

Rates of Paddling [04/25/2000]

I can travel 12 miles downstream in six hours less than it takes me to travel the same distance upstream... How fast is this river's current?

Rationalizing the Denominator [05/27/1997]

Why do we have to rationalize the denominator?

Rational Number and its Reciprocal [03/14/2002]

A rational number greater than one and its reciprocal have a sum of 2 1/6. What is this number? Express your answer as an improper fraction in lowest terms.

Rational Root Theorem [08/27/1998]

List all possible rational zeros of the each function, then determine the rational zeros: f(x) = x^3 - 4x^2 + x + 2.

Ratio of Anti-Freeze to Water [02/05/2003]

A barrel is filled with 45 gallons of anti-freeze. 9 gallons are removed and the barrel is refilled with water. 9 gallons of the mixture are removed and again, the barrel is refilled with water. What is the final ratio of anti-freeze to water in the barrel?

Ratio of Ratios [06/05/2002]

In a community, 3/5 of the men are married to 2/3 of the women. What fraction of the residents are married?

The Ratio of Water to Wine [10/22/1995]

A cask is filled with 45 gallons of wine. Nine gallons are removed and the cask is refilled with water. The nine gallons of the mixture are removed and the cask is again refilled with water. What is the ratio of water to wine in the final mixture?

Ratios and Mixtures [05/10/2001]

Sue needs to mix up a potion for her client... in what ratio must Sue mix the honey and turpentine?

Rearranging Variables [10/01/1998]

Why when you use a variable in a problem does the answer sometimes still have a variable, so you can't get an actual number answer?

Rectangles: Area and Perimeter the Same [4/15/1996]

There are two different rectangles whose sides are integers and whose area and perimeter are the same. Find the dimensions.

Relationship Between Commutative and Associative Property? [06/30/2002]

Are both the commutative and associative properties necessary to prove a + b + c = c + b + a?

Relations versus Functions [10/27/1998]

What is the difference between a relation and a function? What about the domain and range of a function?

Restrictions on Variables in Division [8/4/1996]

What are the restrictions on the variables in (y^3+8)/(y^2+3y+2)?

Reversing the Inequality [09/21/1998]

Why does the inequality sign reverse when you multiply by a negative number?

Reversing the Inequality Symbol [03/09/1999]

Why do we reverse the inequality symbol when dividing/multiplying by a negative number?

A Right Triangle of Points [01/14/1999]

Determine the values of x that would make the points (x,0), (-2,1), and (3,4) the vertices of a right triangle.

Rule of Three [06/15/2002]

Please explain to me briefly (and with an example) the concept of the rule of three.

Rules of Exponent Manipulation [1/22/1996]

f sub r equals the reciprocal of ...

Running Rates, Line Segments, and Basic Algebra [6/26/1996]

(1) Steven ran a 12-mile race at an average speed of 8 miles per hour. If Adam ran the same race... (2) Given a line made of three segments, which statements are true...? (3) If 3 more than x is 2 more than y, what is x in terms of y?

The Sailor's Reward Problem and Prime Numbers [12/11/2004]

Does the answer to the classic Sailor's Reward problem have to be a prime number?

Same Day Off? [12/08/2002]

One person has every ninth day off and started on Sunday, and another person has every sixth day off and started on Monday. When will they both be off on the same day?

Savings on Gasoline [05/22/2001]

Your old car averaged 22 mpg and you new car should average 37 mpg. Suppose you drive 12,500 miles per year and the cost of gas averages $1.85 per gallon. How much will you save on gasoline each year?

Sets and Subsets [1/23/1995]

My teacher said that integers are a subset of reals, and whole numbers are a subset of integers, and counting numbers are a subset of whole numbers, and so on and so forth. What does that mean?

Several Ways to Solve a Rate-Time-Distance Problem [02/14/2006]

A bus leaves point A travelling at a speed of 10 mph. 22 minutes later a car leaves point A travelling in the same direction as the bus at 40 mph. At what distance will the car catch the bus, and how much time will have elapsed since the car left point A?

Shadow Game [01/28/2001]

Tom is 6 feet tall and Carol is 5 feet tall. If Carol stands at the "shadow top" of Tom's head, their two combined shadows total 15 feet. How long is each shadow?

Sharing the Cost of the Weekend Trip [07/24/2003]

A group of people planned to rent a large beach house for a weekend trip. They were to share the $800 cost equally. However, two people were unable to go and this increased the cost for each person by $20. How many persons were in the original group?

Shortcut for Comparing Fractions [01/14/2003]

My teacher showed us how to cross multiply to compare fractions.

Sierpinski Gasket [8/29/1995]

What is the area of a Sierpinski gasket?

Sign Switching [01/28/2002]

Can you explain sign switching in multiplying and dividing inequalities?

Simple Algebraic Equations [5/26/1996]

How do you logically solve simple algebraic equations?

Simple Equation: 3y + 2 over 4 equals 7 [10/17/2001]

I think you have to subtract 2 on both sides, but I'm lost.

Simplification Problems [8/20/1996]

Simplify 2x(4-2x+y) - 5x(y^2+x-y)

Simplify an Expression [6/15/1996]

Simplify (9a+2b)-(4a-3b)

Simplifying an Algebraic Fraction [05/09/2000]

How can I simplify 1 + 1/(x + 1/x)? I can't figure out the common denominator.

Simplifying an Expression [04/29/1999]

Can you help me figure out (x+3)+(4x-7)+(x-20) ?

Simplifying an Expression [05/04/1999]

How do you simplify 9x - 3x + 7 - 4x + y - 3 ?

Simplifying Expressions [8/31/1996]

How do I simplify (y + 3) - 2(y + 7) ?

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