Master Fraction Math Problems with Expert Help

Prove this inequality $\sum_{c y c} \frac{a}{a b + 1} \geq \frac{3}{2}$
For $a > 0$ $b > 0$ and $c > 0$ such tat $a + b + c = 3$ prove that:
$\frac{a}{a b + 1} + \frac{b}{b c + 1} + \frac{c}{c a + 1} \geq \frac{3}{2}$
By AM-GM
$\frac{a}{a b + 1} = a - \frac{a^{2} b}{a b + 1} \geq a - \frac{a^{2} b}{2 \sqrt{a b}} = a - \frac{a \sqrt{a b}}{2}$
$\Rightarrow L . H . S \geq a + b + c - \frac{a \sqrt{a b} + b \sqrt{b c} + c \sqrt{c a}}{2}$
$\geq 3 - \frac{a^{2} + b^{2} + c^{2} + a b + b c + c a}{4}$
Help me to continue

Elliana Molina 2022-11-06

Reducing combinations in fractions
$\frac{b * (\binom{b - 1}{k - 1}) (\binom{r}{n - k})}{n * (\binom{b}{k})} * \frac{(\binom{r + b}{n})}{(\binom{r}{n - k}) (\binom{b}{k})}$
How would I go about reducing this into $\frac{k}{n}$ ?
My attempt was $\frac{b!}{n * (k - 1)! (b - k - 2)!} * \frac{k!}{b!} = \frac{k!}{n * (k - 1)! (b - k - 2)!} = \frac{k}{n * (b - k - 2)!}$
Don't know where the errors are / where to go from here

Laila Murphy 2022-11-06

Prove the next cyclic inequality
Let $a, b, c > 0$ such that
$a + b + c = 1$
prove that
$c y c l i c \sum \frac{a b}{\sqrt{c + a b}} \leq \frac{1}{2}$
By symetrie, i proved it by assuming $a = b = c$ but i cannot justify this hypothesis.

Frances Pham 2022-11-05

Why $\frac{\frac{1}{2} \times \frac{k}{k + 7}}{\frac{1}{2} \times \frac{k}{k + 7} + \frac{1}{2} \times \frac{2}{5}} = \frac{k}{k + \frac{2}{5} (k + 7)}$ ?
My attempt:
$\frac{\frac{1}{2} \times \frac{k}{k + 7}}{\frac{1}{2} \times \frac{k}{k + 7} + \frac{1}{2} \times \frac{2}{5}} = \frac{\frac{1}{2} \times \frac{k}{k + 7}}{\frac{1}{2} (\frac{k}{k + 7} + \frac{2}{5})} = \frac{\frac{k}{k + 7}}{\frac{k}{k + 7} + \frac{2}{5}}$

figoveck38 2022-11-05

How would I simplify this fraction easily?
Original question :
$\frac{\tan (x)}{\log_{e} x}$
I used the quotient rule and got the following fraction.
Fraction :
$\frac{\frac{x \cdot \sec (x)^{2} \cdot (\log_{e} x) - \tan (x)}{x}}{\log_{e} x}$
apologies in advance for the fractions, i tried to put in $\log_{e} x$ underneath the denominator of the other fraction but for some reason it wasn't working.

Fahdvfm 2022-11-05

3.31 Game of dreidel. A dreidel is a four-sided spinning top with the Hebrew letters nun,1d shin, one on each side. Each side is equally likely to come up in a single spin ofthe dreidel. Suppose you spin a dreiler three times. Calculate the probability of getting
(a) at least one nun?
(b) exactly 2 nuns?
(c) exactly 1 hei?
(d) at most 2 gimels?

Karley Castillo 2022-11-05

Simplifying a recursive fraction involving Lambert function
Is there a simplification for the following recursive fraction :
$\frac{\frac{\frac{n}{W (n)}}{W (\frac{n}{W (n)})}}{W (\frac{\frac{n}{W (n)}}{W (\frac{n}{W (n)})})}$
The above formula uses a recursion 3 times. I'm looking for a simplification when we have such a finite recursion, for instance when this one appears $i$ times. I would like to remove the recursion, i.e. obtain a single fraction.
Thank you.

Alvin Parks 2022-11-02

Attempt to view irrational number as a fraction
I am wondering if an irrational number can be represented as a fraction in this way:
For example (to represent $π$ ):
$π = 3.14159265359... = \frac{314159265359...}{100000000000...}$
In the fraction $\frac{314159265359...}{100000000000...}$ , the numerator is an integer whose digits have the same order as digits of $π$ , and the denominator is simply $10^{(# o f d i g i t s o f n u m e r a t o r - 1)}$ . Isn't an irrational number represented as a fraction in this way? Probably I misunderstand the concept of the irrational number. Thanks in advance.

Audrey Arnold 2022-11-02

Why is $\frac{1}{2} \frac{b^{2} - a^{2}}{b - a} = \frac{a + b}{2}$ ?

Seettiffrourfk6 2022-10-29

How do I compute the limit of a radical outside of a fraction?
How do I compute this?
$lim_{x \to \infty} (\sqrt[3]{n^{2} + 5} - \sqrt[3]{n^{2} + 3})$
I tried multiplying it by the conjugate exactly as if it were in a fraction, but it didn’t quite work out, because you end up with a way bigger expression.
Arriving at the solution likely uses the sandwich theorem, but with two separate expressions. I find it to be quite the challenge.

Martin Hart 2022-10-29

Reducing Radian Angle
How would I reduce a large radian angle into a smaller one?
For example, if you're asked to find a related angle for $\cos \frac{3177 π}{12} .$ I usually try to break it down to $\frac{π}{2},$ $\frac{π}{3},$ $\frac{π}{4},$ or $\frac{π}{6}$ to visualize it in a unit circle, but I wouldn't know where the terminal arm would lie in such a big angle.
I know that a period for one revolution in a unit circle is equal to $2 π .$ So I could divide the big angle $\frac{3177 π}{12}$ by $2 π .$
So that's $\frac{3177 π}{12} \cdot \frac{1}{2 π} = 132.375.$ So I found out the number of rotations. But I still don't know how to interpret that in a unit circle. If I were to draw the angle $\frac{3177 π}{12},$ which quadrant would the terminal arm lie?

Deon Moran 2022-10-28

Positive Integer values of a fraction
I am trying to find all strictely positive integers $x, y$ so that
$z = \frac{4 x + 1}{4 y - 1}$
is a strictely positive integer.
Some cases are possible when choosing special values of $x$ and $y$ . However, I am not able to find the general solution.

Jack Ingram 2022-10-27

Simplification of an algebra
How can this fraction be simplified?
$\frac{\frac{1}{c} + \frac{1}{d}}{c + d}$
.My friend argues with my answer despite working out the answer,.Any other methods applicable

Kendrick Finley 2022-10-27

Solve:
$(1 / 2) b + 4 = (1 / 8) b + 88$

tikaj1x 2022-10-26

$\frac{a}{b} + \frac{c}{d} = \frac{a + c}{b + d}$
Please help me to find $a, b, c, d$ such that $\frac{a}{b} + \frac{c}{d} = \frac{a + c}{b + d}$ ? I take $a, b$ for example $2, 3$ to find $c, d$ but I get stuck on this. I know that is diophantine equation , but I am unable to find suitable $a, b, c, d$
$a, b, c, d \in Z, a, c \neq 0$
Thanks in advance.

Francis Oliver 2022-10-26

Write the partial fraction decomposition of the rational expression.
$\frac{x + 4}{x^{2} - x - 30}$

Mariyah Bell 2022-10-25

How can this expression be calculated? $\frac{1 + \frac{3 \dots}{4 \dots}}{2 + \frac{5 \dots}{6 \dots}}$
I can't see any obvious way this could be calculated. It seems to converge to a value of approximately 0.6278...
$\frac{1 + \frac{3}{4}}{2 + \frac{5}{6}} \approx 0.6176$
$\frac{1 + \frac{3 + \frac{7}{8}}{4 + \frac{9}{10}}}{2 + \frac{5 + \frac{11}{12}}{6 + \frac{13}{14}}} \approx 0.6175$
Going all the way up to 62 gives a result of 0.627841944566, so it seems to converge.
Is it possible to find a value for this? Will it have a closed form solution?

Dylan Nixon 2022-10-25

Represent improper fraction as a sum of unique unit fractions
Is it possible to represent an improper fraction as a finite sum of unique unit fractions (Egyptian fractions)?

Raiden Barr 2022-10-25

Can we find integers x and y such that f,g,h are strictely positive integers
Let $a > 2$ and $b > 2$ two strictely positive integers. Let us consider the following quantities:
$f = \frac{x y + a y + a^{2}}{b y}$
$g = a (y + a) \frac{x y + a y + a^{2}}{b y^{2} x}$
$h = \frac{y + a}{b}$
My question is:
Can we find integers x and y (not necessarly positive) such that f,g,h are strictely positive integers. Or at lest how one can proves that they are exist.

Sariah Mcguire 2022-10-24

Simplifying Fractions Containing Variables (Basic)
I appreciate this is very simple, but I'm experiencing a very basic problem with fractions containing variables and I'd just like to check I'm along the right lines. In the following instance:
$\frac{2 (x + 7) (3 x + 1)}{2} = \frac{2}{2} \cdot \frac{2 (x + 7) (3 x + 1)}{1} = (x + 7) (3 x + 1)$
Does the $\frac{2 (x + 7) (3 x + 1)}{1}$ have a denominator of 1 because we have factored out the 2 in the prior step?

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