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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.

Different Approach to a Set of Equations [11/13/2002]
The sum of four numbers is 22. The first number is twice the difference of the second and the fourth. The second number is five times the difference of the third and the fourth. The third number is twice the difference of the first and the fourth. What are the four numbers?

Digit Problems: Find the Number [01/29/1998]
A two-digit number is 6 more than 4 times the sum of its digits. The digits from left to right are consecutive even integers. Find the number.

Digit Reversal Trick Explained [03/23/2001]
Take a 3-digit number and subtract its reverse. Then, take the result and add its reverse. Why is the answer is always 1089, no matter what the initial numbers were?

Dimensions of a Rectangle [8/26/1996]
Find the original dimensions of a rectangle whose area is increased by 81 square meters.

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

Direct and Inverse Variation in One Problem [03/23/2006]
If s varies directly as r and inversely as t, and s = 10 when r = 5 and t = 3, for what value of t will s = 3 when r = 4?

The Discriminant of Quadratic Equations [08/06/1998]
Can you explain more about the discriminant, b^2 - 4ac? Why is it sometimes represented by a triangle (delta)?

Distance Between a Point and a Line [08/23/1998]
Find the distance between the origin and the line 3x + 4y = 5.

Distance, Time and Speed [8/1/1996]
What distance is traveled by driving 45 minutes at an average speed of 76 km/hr.?

Distributive Property [10/29/1997]
I don't understand the Distributive Property at all.

Distributive Property [10/13/2001]
We are learning the distributive property in school. Why do we have to learn this, and how will we use it, and when?

Distributive Property [12/11/1997]
What is the distributive property?

Distributive Property with Negatives [10/13/2004]
When using the distributive property with negative numbers, I don't understand when, why, and how the operator changes (from addition to subtraction, or the other way). Sometimes it changes, sometimes it doesn't. Can you explain when to change it, and why?

Distributive Property: (x+2)(x+4) [09/18/2001]
I'm confused about the distributive property. Example: (x+2)(x+4).

The Divide between Numbers and Variables [04/25/2011]
An adult student wonders why subtracting the coefficient from a variable expression would not eliminate it. Describing algebraic manipulations as processes of doing and undoing, Doctor Peterson clears up the confusion, and offers some further rules of thumb on simplification.

Dividing a Square [07/31/1999]
A large square is divided into several equal-sized smaller squares using x lines. How can I find a formula for the number of smaller squares produced?

Dividing Compound Fractions [11/16/1999]
Can you give me a simple step-by-step method to simplify expressions like [(3/y)+(5/zy^2)] / [(5/2y)-(4/y^2)]?

Division of Number Line [5/14/1996]
Using the interval from zero to one, how do I explain that when you find half the distance and then half of that distance and so on, you'll never settle on one number?

Doubling Opposite Integers [04/19/1998]
On a number line, a certain integer is 24 units away from the double of its opposite. What is the integer?

Empty Sets [08/15/1997]
Can you help me understand empty sets?

Equal Sum, Product, Quotient of Two Numbers [10/13/2002]
I need to know the sum, product, and quotient, all equal, of two different numbers.

Equation for undefined slope [10/24/1999]
How do you write an equation for a vertical line with an undefined slope? For example, a vertical line with an x-intercept of 8.

Equation in One Variable [04/13/1997]
How do I solve an equation in one variable by adding and subtracting terms?

Equations Involving Fractions [07/07/2000]
How do I solve equations with fractions and decimals in them?

Equations That Make No Sense [01/15/2003]
Solve for x: x-8/5 = x+2/6.

Equations with Absolute Value [01/24/1999]
I know that if an equation has an absolute value, you need to have a positive and a negative part when you solve it. How do you get both answers?

Equations with Fractions [04/21/1997]
How do you solve x/2 + x/3 = 1/6?

Equation vs. Expression [03/19/1999]
How is an algebraic equation different from an algebraic expression?

Equation with Absolute Value [3/25/1996]
I have problems solving equations with absolute value and graphing on a number line. Example: |2n-6| > 5

Equivalent Expressions / Equivalent Equations [07/27/1997]
Could you please explain the difference between equivalent expressions and equivalent equations?

Etymologies of Algebra, Geometry, Trigonometry [10/22/2001]
What are the origins and roots of the words geometry, algebra, and trigonometry?

Evaluating an Expression [03/28/2001]
How do you evaluate an expression like 2x + 3x + 4, when x = 6?

Evaluating Expressions [08/12/1997]
Evaluate the variable expression 5x and 13-x when x = 4.

Examples and Explanations of Basic Properties of Equality [07/31/2004]
Can you explain the substitution, symmetric, and transitive properties of equality in simpler terms?

Exchanging Seats in a Boat [06/08/2001]
Ten women are fishing in a long, narrow boat. One seat in the center of the boat is empty. The five women in the front of the boat want to change seats with the five women in the back of the boat...

Expanding Polynomials [06/16/1999]
How can you expand three binomials like (s+4)(2s-1)(s-2), or trinomials like (z2-2z+3)(2z2+z-1)? FOIL doesn't work on these.

Extraneous Solutions [08/08/2003]
What should an engineer interpret from an equation, or the solutions of that equation, if its solutions prove to be extraneous?

Factor and Multiply to Check [5/21/1996]
Is x to the second power plus 6x plus 5 factorable?

Factor for X; Consecutive Odd Integers [01/19/1998]
(x-2)(x+3) = 6 : solve by factoring for x. Label your variables, write an equation, solve, label answers...

Factorials and Order of Operations [09/21/2000]
Where do factorials go in the order of operations?

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