Questions tagged [fractions]

Questions on fractions, i.e. expressions (not values) of the form $\frac ab$, including arithmetic with fractions. Not to be confused with the tag (rational-numbers): fractions denote rational numbers, but the same rational number may be written in different ways as a fraction.

A fraction is simply an expression $\frac{a}{b}$, where $a$ and $b$ are typically integers (where $b\neq 0$). This tag may be used, when $a$ and $b$ are more general expressions or algebraic objects; however, consider adding a more specific tag also:

Fractions are distinct from rational numbers because they are a representation: $\frac 34$ and $\frac{30}{40}$ are different fractions that happen to represent the same rational number.

For arithmetic with fractions, this tag is appropriate along with .

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How to compare these two large fractions without any calculation

I want to compare two following fractions $$ a = \frac{1}{411}\cdot\frac{1}{412}\cdot\frac{1}{413}; b = \frac{1}{63990006} $$ I want to compare these without any calculations, but simply manipulating, something like this: $$ a=\frac{791}{993};…
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Why is a machine producing 7 nails in 9 seconds faster than one producing 11 nails in 14 seconds?

Machine A Produces 11 nails in 14 seconds. Machine B produces 7 nails in 9 seconds. Which is faster? I thought that $\frac{11}{14}$ is larger than $\frac{7}{9}$ and thus A is the answer. However, the answer says that $\frac{9}{7}$ is bigger than…
User3910
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Fractions Help.

My mother and I have been wrecking our brains trying to figure this out. Q: when is the next time these three fraction won't be whole simultaneously. Our manual working: $2/2, 3/3, 4/5$ - start $1/2, 1/3, 5/5$ $2/2, 2/3, 1/5$ $1/2, 3/3, 2/5$ $2/2,…
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How can we quickly find t in $\frac xt = \frac x a + \frac x b$ for larger integer values of a and b?

If $$\frac x t = \frac x a + \frac x b$$ is there a faster way of finding the value of t instead of finding a common denominator?
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How do I simplify the rational expression $\frac1{2b} + \frac b2$

So the expression is: $$\frac1{2b} + \frac b2$$ Apparently the answer is $\frac{1 + b^2}{ 2b}$. I came up with $2b$ by multiplying the first fraction by $2$ and the second fraction by $2b.$ I got: $$\frac{2+ 2b^2}{2(2b)}$$ which simplifies to…
SunGone
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How to resolve this fraction with change of divisors

If i have an equation like this: $$\frac{a}{b\cdot{\frac{c}{d - \frac{f}{g}}}}$$ With what should I multiply to get $g$? Should I multiply it with $d-\frac{f}{g}$ ? Or is this incorrect?
N22
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Simplify fraction in numerator when fraction is part of sum

I have the result of a quotient rule: $$ \frac{\frac{x}{8+x}-ln8}{x^2} $$ Should I just leave it? Would it be appropriate to separate the fraction into a product of fractions with denominator $x^2$, simplify the left side, and give the result as a…
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Simplify the fractions

First, I apologize for the title, but if I write my question then the characters will be too long. Here's my question: Why…
student
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Simplifying exponential fraction

I have this exponential fraction $$\frac{2^{n+1}}{5^{n-1}}$$ I was wondering how we simplify something like this. I know if the top and bottom had the same like $\frac{2^{n+1}}{2^{n+1}}$, you would just subtract the exponent. But in my situation,…
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Why does cross-cancellation of fractions in multiplication work?

I'm reviewing my arithmetic and right now I'm at fractions, I'm just having a little bit of problem "visualizing" why cancellation works in multiplying fractions, I know how it works, it's just the why. Take for…
Johnsy
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How does $(0.06 + 0.06 + 0.06)$ + $(0.02 = 0.06/3)$ help intuit $0.20/0.06$?

[ Source : ] This passage$^1$ exemplifies Posner's pragmatic, one might at times say cynical, appraisal of human and judicial nature -- a perspective that is enlightening even if not always persuasive. But this particular passage also contains,…
user53259
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Raising fractions to powers using only whole numbers

I need to be able to raise fractions to a power, eg: $f^2=a$ $1.5^2=2.25$ $f^3=b$ ($1.5^3=3.375$) But the challenge I'm facing is that the coding language I'm using only supports whole numbers for f: 1, 2, 3, 4, 5, 6... (i.e. the left side of the…
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Fraction transformation

I am not sure how it is possible to write $\frac{\frac{t}{\sqrt{t^2+1}}+1}{\sqrt{t^2+1}+t}$ as $\frac{1}{\sqrt{t^2+1}}$ Does anyone know how I can transform this? I would appriciate a step-by-step solution.
Maddude
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Why is $\frac{1}{2}\frac{b^{2}-a^{2}}{b-a} = \frac{a+b}{2}$?

Why is $$\frac{1}{2}\frac{b^{2}-a^{2}}{b-a} = \frac{a+b}{2}$$?
M.Mac
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Reading Fractions Using Quantities

I am on the Academic Challenge team for my school, and we were given a basic math problem, in which the answer was $\frac{5x-21}{30x^2}$. I stated this exact answer, although I said "the quantity of 5x minus 21", which they deemed incorrect because…