Middle School Level Questions: Ask Dr. Math

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Below are some questions and answers from our archive of questions sent via e-mail by K12 students and teachers to Ask Dr. Math. You may also browse the entire Middle School Level Archive by category.

Rational Numbers

About Rational Numbers
Can you explain 'rational numbers' to me? How do you express them? What kind of operations can you perform on them?
Different Types of Numbers
I don't understand how to tell the difference between the different kinds of numbers. i.e. reals, integers...
Questions about .999...
Is 0.999... a rational number? Can .999... ever be equal to one?

Fractions and Algebra

Factoring Fractions
A student asks how to divide (12ax + 16x) by (4x).
Factor and Simplify
How would you simplify the following problem: (1/x^2-1/y^2)/ (1/x^2+2/xy+1/y^2) ?
Fraction Equation Problem
Let 1/a = a + 1/b = b + 1/c = c + 1/d = 2. What is the value of abcd?
Fraction Problems
I am having trouble with some fractions and fractional equations.
Solving an Equation with Fractions
A student asks how to work the problem: (x + 5)/4 = (x-4)/(2/5).
Solving for x
I need help on this problem: x/x+4 = x+1/x+8.
Solving Another Equation with Two Variables
6y = 3D = 54. I am supposed to solve the equation, but I don't even understand it!
Solving Equations
A student has a number of 'word problems' that require solving equations.
Solving an Equation with Fractions
v = 6 - 5/4 v. I don't know if this question is asking 6 minus 1 1/4 or 6 * 1 1/4.
Algebra using Fractions
Please show me how to solve this equation: (5/8)X = 9.
Algebra Test Questions
1. Find 3 consecutive integers such that the sum of the first and third is 84. 2. The length of a rectangle is 5 less than twice the width, the perimeter of the rectangle is 80; find the dimensions of the rectangle. 3. Solve these equations: c + 2/5 = 3/7 and 1/6 = a + 3/a - 4.
Adding Fractions in Algebra
A student needs help with adding fractions in equations.
Fractional Coefficients
x-4/8 - 5-2x/3 = 7/12 ?
Simplifying Fractional Equations
am supposed to perform the indicated operations and express results in lowest terms: 1: ((x^2 - 2xy + y^2) / 6xy) * ((3x^2y^2) / (x^2 - y^2)) = ?
Simplifying: Put it Together
I am supposed to perform the indicated operations and express the results in lowest terms. Here's the problem: (5/x-1) - (4x/x-1) - (2-x/x-1) = ?
Order of Algebraic Operations
How do I solve: (9*4-7*5+6*7-10) / ( 6/6+7*3-4*3-9) and 2+2*3-(2/2)*(4-3)+1 ?
Another Order of Operations Problem
(1+1*8-9*4) / (8*9+7*2) = ?
Sierpinski Gasket
What is the area of a Sierpinski gasket?

Fractions in Word Problems

Dimes and Quarters Puzzle
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?
Combinations of Coins
If I had 6 coins in my pocket, how many combinations could I have?
Trains and Fractions
A non-stop train leaves city A for city B at 60 m.p.h.... If to the numerator and denominator of the fraction 1/3 you add its denominator... A train moving at 45 m.p.h. meets and is passed by a train moving at 36 m.p.h...
Speed, Distance, and Time
If you drive one mile in 45 seconds, how fast are you driving?
Rates and Fractions
Find each rate as a fraction in simplest form: 1) 24 laps in 12 minutes, 2) 180 miles with 15 gallons...
Using Improper Fractions in a Word Problem
Mr. Smith owns 1/4 of the stock of a business. Mrs. Smith owns 5/6 as much as Mr. Smith. How much do they not own of the stock? Word Problem with Fractions
Jack can finish a job in 5 days, and Richard can finish the same job in 7 days. A job is to be done by Jack and Richard together. How long will it take them to finish it?
Proportions of Men to Women
In a certain village 2/3 of the men are married to 3/5 of the women. What proportion of the population as a whole is married?
Solve for x in a Percentage Equation
Express as a function of x. 1) The volume of water that must be added to 5L of antifreeze to make a solution that is x% antifreeze...

Fractions and Division

Why Divide Fractions?
Where, in this wide world of ours, would one divide fractions? For example: 3 2/5 / 7/8. Can you give me a real life example?
How to Divide Fractions?
When dividing fractions, for example: 3/11 divided by 5, I know that you reciprocate the '5' and multiply, but I don't understand why except that it works! Can you explain it?
A Fraction Divided By a Fraction
What is 18/6 divided by 2/3?
Inverting and Multiplying When Dividing Fractions
I have always been taught that to divide fractions I must invert and multiply. Why does this algorithm work? Is there any other way to divide fractions?
Dividing Fractions
Tell me a shortcut for dividing fractions.

Fractions and Multiplication

Multiplying Fractions
x over 2y multiplied by 4y over 2 = ?

Fractions and Zero

Can You Have Zero in the Denominator?
My (seventh grade) math teacher and I are in the midst of an argument over whether any number can be placed over zero in a fraction. I know that when you take a fraction like 10/5, it equals 2, and 5 * 2=10, and that works out for all fractions where the numerator is larger than the denominator, but it doesn't work out for n/0. I know that you'll probably say that it isn't a legitimate fraction, but.. I still say that it is. :)
Zero as an Exponent
2^3 is 2 times itself 3 times or 2x2x2. But 2^0 isn't 2 times itself 0 times, is it? Can you explain 2^0 in those terms?
Division by Zero
Why can't you divide a number by 0?
One day when I was playing on a calculator, I found that 1 and 0 both have a factorial of 1. Can you explain this?

Fractions and Decimals

Converting Decimals to Fractions
A certain fraction is converted to a decimal and the decimal representation is 0.6428571428571428571... What is the fraction?
Writing Repeating Decimals as Fractions
When you are expressing a repeating decimal as a fraction, why do you use 9, 99, 999, or 9999 etc. in the denominator? For example, 0.5555...(5 repeating) equals 5/9.
Prime Factors in Unit Fractions
Why does the denominator of a unit fraction have to have prime factors of 2 and/or 5 in order to terminate?
Reduced fraction
Write as a reduced fraction: -------/.428571


History of Fractions
How did the Babylonians, Ancient Chinese, Egyptians, Greeks, or Hindus use fractions?
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Mara Landers

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