Master Fraction Math Problems with Expert Help

What is 2.8 as a fraction?

Kareem Mejia 2022-11-16

A geometric inequality in a triangle
If $x, y, z > 0,$ could it be:
$\frac{z (2 z + x)}{z + x} b^{2} + \frac{y (2 y + x)}{y + x} c^{2} > \frac{y z}{y + z} a^{2}$
for a triangle $A B C,$ with $a = B C,$ $b = A C,$ $c = A B ?$
If $b^{2} + c^{2} = a^{2}$ , then the inequality holds, since
$\frac{z (2 z + x)}{z + x} - \frac{y z}{y + z} = \frac{z^{2} (x + y + 2 z)}{(z + x) (y + z)}, \frac{y (2 y + x)}{y + x} - \frac{y z}{y + z} = \frac{y^{2} (x + 2 y + z)}{(z + x) (y + z)}$
If $b^{2} + c^{2} > a^{2}$ , then the inequality holds, since
$\frac{z (2 z + x)}{z + x} b^{2} + \frac{y (2 y + x)}{y + x} c^{2} > \frac{y z}{y + z} b^{2} + \frac{y z}{y + z} c^{2} > \frac{y z}{y + z} a^{2}$
I can’t handle the case $b^{2} + c^{2} < a^{2}$ and I need some help

Nico Patterson 2022-11-15

partial fraction decomposition of $\frac{(2 l + 1) (l + 1)}{K + (l + 1) (l + 2) (l - 1)}$
I am struggling with the partial fraction decomposition of
$\frac{(2 l + 1) (l + 1)}{K + (l + 1) (l + 2) (l - 1)},$
where $K$ is a positive real number.
I would like to decompose it in a sum of simpler polynomial fractions.
I tried the Mathematica's function "Apart" but it is returning the fraction so I am thinking there is no simple partial fraction decomposition.
Could you help me?

Josie Kennedy 2022-11-14

Prove that $\frac{1}{15} < \frac{1}{2} * \frac{3}{4} * \dots * \frac{99}{100} < \frac{1}{10}$
My attempt:If we name the value $A$ we have:
$A^{2} < (\frac{1}{2} * \frac{3}{4} * \dots * \frac{99}{100}) (\frac{2}{3} * \frac{4}{5} * \dots \frac{100}{101}) = \frac{1}{101} \Rightarrow A < \frac{1}{10}$
But I don't know, how to prove the other side?

Sophie Marks 2022-11-13

Which fraction is equivalent to $- \frac{4}{5}$ ? (A) $- \frac{- 4}{- 5}$
(B) $- \frac{4}{- 5}$
(C) $- \frac{- 4}{5}$

Joglxym 2022-11-13

What formula or rule has been used here?
I was in between proving a trigonometric identity but couldn't succeed. I went through the solution and saw this in between
$\begin{aligned} \frac{\cos A \cos B}{\sin A \sin B} & = \frac{3}{1} \\ \frac{\cos A \cos B + \sin A \sin B}{\cos A \cos B - \sin A \sin B} & = \frac{3 + 1}{3 - 1} \end{aligned}$
What happened there in the second step?

Kaylynn Cook 2022-11-12

Fractional/rational form of $0.999...$
Is it possible to express $0.999...$ , a repeating number, as a fraction? Or as a ratio of two numbers?
Basically all (my) attempts at the problem cancels all the terms and returns $1$ . Is it even possible? Or has it been proven to be an exercise of futility?

Davirnoilc 2022-11-11

Problem with inequality and number e
Prove that, for every positive integer $n$ , the following inequality holds:
where ${x}$ denotes the fractional part function applied to number $x$
I can;t think of any solution.

Kale Sampson 2022-11-11

Critique reasoning
Molly is wrapping parts of sandwiches to sell at her sandwich cart . She cuts each sandwich in fourths and then wraps each fourth separately. She says that she wrapped 16 fourths, so she wrapped 16 hole sandwiches. What was Molly's mistake?

anraszbx 2022-11-11

Find the value of $\frac{\sin 24 x}{\cos 18 x \cos 6 x}$
Given $\tan x = \sqrt{5} - 2$ , find the value of
$\frac{\sin 24 x}{\cos 18 x \cos 6 x} .$
Now I wrote numerator as $\sin (18 + 6) x$ and I used identity $\sin (A + B)$ , after that I reduced problem to $\tan 18 x + \tan 6 x$
How do I proceed? Thanks

kunguwaat81 2022-11-11

Tan of difference of two angles given as sum of sines and cosines
I could have done the question had it been to find the value of $\tan \frac{α + β}{2}$ . But here the tangent of the difference has been asked. How does one go about it?
If $\sin α + \sin β = a$ and $\cos α + \cos β = b$ , then show that
$\tan \frac{α - β}{2} = \pm \sqrt{\frac{4 - a^{2} - b^{2}}{a^{2} + b^{2}}} .$

Frances Pham 2022-11-11

Limit when $x$ converges to $0$
Can someone please help me with this simple limit problem, I do not know what to do here because its undefined $0 / 0$
$lim_{x \to 0} \frac{e^{1 / x}}{\sqrt{x}} .$

Yaretzi Mcconnell 2022-11-10

What makes this answer unsimplified?
Generally, what are the rules for simplifying answers?
I had this question:
Given $f (x) = \frac{1}{x}$ , evaluate $\frac{f (x) - f (a)}{x - a}$ .
Why is the following a bad answer:
$\frac{\frac{1}{x} - \frac{1}{a}}{x - a}$
The better answer is:
$\frac{- 1}{x a}$
Why is this? Is it generally just bad to leave fractions in either the numerator or denominator? So we should try to eliminate the fraction in the numerator by solving the equation first on the top by using a common denominator?

Davirnoilc 2022-11-09

If $\frac{a}{a + 1} + \frac{b}{b + 1} + \frac{c}{c + 1} = 1$ prove $a b c \leq 1 / 8$
Could I get some help with this?
What I have tried-
Making the denominators equal and adding to get another equation of the form $\frac{N}{M} = 1$ then multiplying by $M$ both the sides and simplifying to get $2 a b c + a b + b c + a c = 1$ , from which we have, $a b c < 1 / 2$
But which obviously isn't enough to prove that $a b c \leq 1 / 8$
I also tried using Jensen's inequality which states that $f (E (X)) \geq E (f (X))$ when $f$ is a concave function and $f (E (X)) \leq E (f (X))$ when $f$ is a convex function, and where $E (X)$ denotes the expectation of X ,
but that didn't really help.

kituoti126 2022-11-09

limit of a sequence
I know from my intuition that the sequence
$x_{n} = {(1 - \frac{1}{3})}^{2} {(1 - \frac{1}{6})}^{2} {(1 - \frac{1}{10})}^{2} \dots \dots {(1 - \frac{1}{\frac{n (n + 1)}{2}})}^{2}, n \geq 2$
is convergent. But i don't know how to prove it.I almost try to apply every theorem I know (for eg ratio test ,monotone convergence theorem,...). Help me to prove this.
Proof or idea is needed.Where does the sequence converge to?

clealtAfforcewug 2022-11-08

Divide using long division. State the quotient, q(x), and the remainder, r(x).
$(8 x^{3} - 2 x^{2} - 14 x - 28) \div (2 x - 4 <mo stretchy=$

Aron Heath 2022-11-08

Rational expression in lowest term - question from Euler’s element of algebra
How do I convert this expression to lowest form?
$\frac{5 a^{5} + 10 a^{4} x + 5 a^{3} x^{2}}{a^{3} x + 2 a^{2} x^{2} + 2 a x^{3} + x^{4}}$

Aryanna Fisher 2022-11-07

Algebraic Fractions grade 10
I am a a student and I am having difficulty with answering this question. I keep getting the answer wrong. Please may I have a step by step solution to this question so that I won't have difficulties with answering these type of questions in the future.
Solve
$\frac{3 x}{4} + \frac{2}{3} = x$
$x$ = ____

odcizit49o 2022-11-07

Find $\arctan (\frac{1}{3}) + \arctan (\frac{1}{9}) + \arctan (\frac{7}{19})$
Firstly used this formula
$\begin{aligned} \arctan (α) + \arctan (β) & = \arctan (\frac{1 - x y}{x + y}), x > 0, y > 0 \\ = \arctan (\frac{1 - \frac{1}{3} \frac{1}{9}}{\frac{1}{3} + \frac{1}{9}}) \\ = \arctan (2) \end{aligned}$
So it is $\arctan (2) + \arctan (\frac{7}{19}) .$

Ty Gaines 2022-11-06

Inequality $\frac{1}{n + 1} + \frac{1}{n + 2} + . . . + \frac{1}{3 n + 1} > 1$
This is a 9th grade problem. I was trying to take the greatest numerator, which is the last numerator of the last fraction. But there are only $2 n + 1$ terms. Right? After that I have no idea. Thx!

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