## Fractions

### 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

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...
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.
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) = ?
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?
Factorials
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

#### Miscellaneous

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