Recent questions in Fractions

Fractions
Answered

kennadiceKesezt
2022-09-26

Let $f(x,y)=xy+\frac{50}{x}+\frac{20}{y}$, Find the global minimum / maximum of the function for $x>0,y>0$

Clearly the function has no global maximum since $f$ is not bounded. I have found that the point $(5,2)$ is a local minimum of $f$. It seems pretty obvious that this point is a global minimum, but I'm struggling with a formal proof.

Fractions
Answered

Joyce Sharp
2022-09-26

Using all numbers 0 to 9 only once, form two fractions whose sum is 1.

I have tried every possible combination but with no luck. I believe the fractions must be xx/xx + xxx/xxx but I am not sure. Any ideas are most welcome. I even tried getting all different ways to select 3 out of 10 digits, then omitting all the primes and trying to make fractions with simple values 1/3, 1/4, 2/5 etc using only different digits but again with no luck! By the way, two years ago I was given a similar one, with three fractions and without the digit 0, for which I found a solution 7/68+9/12+5/34 but now I am stuck!!

Fractions
Answered

Kaila Branch
2022-09-25

I read in a textbook that the explanation to this rule lies in the fact that division is the reverse operation to multiplication. Unfortunately, the author did not elaborate on this point. Based on this, can someone help me to understand why

$\frac{a}{b}$ / $\frac{c}{d}$ = $\frac{a}{b}$ * $\frac{d}{c}$?

Fractions
Answered

Sara Fleming
2022-09-25

While calculating some integrals I happened to face the following estimate:

${\int}_{m}^{m+1}{\int}_{n}^{n+1}\frac{dydx}{x+y}\ge \frac{1}{m+n+1}.$

After some tedious calculations, I figured that this estimate follows from the inequality

$(x+1)\mathrm{log}(x+1)-2x\mathrm{log}x+(x-1)\mathrm{log}(x-1)\ge \frac{1}{x}\phantom{\rule{1em}{0ex}}\text{for}x1.$

(If we interpret $0\mathrm{log}0$ as $\underset{\u03f5\to 0}{lim}\u03f5\mathrm{log}\u03f5=0$, then the inequality also holds for $x=1$.)

But how do we prove this inequality?

What I have tried: Let $f(x)=x\mathrm{log}x-(x-1)\mathrm{log}(x-1)$ for $x>1$. Then the RHS equals $f(x+1)-f(x)$ so by the Mean Value Theorem there exist some $\xi $ between $x$ and $x+1$ such that

$f(x+1)-f(x)={f}^{\prime}(\xi )=\mathrm{log}(1+\frac{1}{\xi -1}),$

and it suffices to show that $\mathrm{log}(1+\frac{1}{x})\ge \frac{1}{x}$ ... which is unfortunately not valid !

I think some clever use of the MVT can solve this problem, but I don't see how I should proceed. Please enlighten me.

Fractions
Answered

Ignacio Casey
2022-09-25

Fractions
Answered

furajat4h
2022-09-25

I'm trying to figure out the equality

$$\frac{1}{y(1-y)}=\frac{1}{y-1}-\frac{1}{y}$$

I have tried but keep ending up with RHS $\frac{1}{y(y-1)}$.

Any help would be appreciated.

Fractions
Answered

unjulpild9b
2022-09-25

This was given to me as a practice problem as preparation for the final exam. The only thing is I have no idea how to do the question, as we did not go over generating fractions in class. I googled what a generating fraction was, and I found some information about generating functions, but nothing too helpful.

The hint says to find the sum of the series. But I'm not sure how to find the series in the first place.

Any help would be much appreciated, thank you.

Fractions
Answered

Orion Cervantes
2022-09-25

Let $m=(abc{)}^{\frac{1}{3}}$, where $a,b,c\in {\mathbb{R}}^{\mathbb{+}}$. Then prove that

$\frac{b}{ab+b+1}+\frac{c}{bc+c+1}+\frac{a}{ac+a+1}\ge \frac{3m}{{m}^{2}+m+1}$

In this inequality I first applied Titu's lemma ; then Rhs will come 9/(some terms) ; now to maximise the rhs I tried to minimise the denominator by applying AM-GM inequality.But then the reverse inequality is coming Please help.

Fractions
Answered

Hagman7v
2022-09-24

I am having trouble remembering/finding simple source to review a few fundamentals. I know I need to factor and try getting x alone. Though not able to recreate answer. Seems it would have multiple roots. Would be great to see the proper steps to solve.

$$\frac{1.6}{3.01}=\frac{x}{1000+x}$$

Fractions
Answered

videosfapaturqz
2022-09-24

I have trouble solving following problem:

Given the sequence:

$${a}_{n}={\textstyle (}1+\frac{1}{n}{{\textstyle )}}^{n}$$

We have to show that

$$\frac{{a}_{n+1}}{{a}_{n}}={\textstyle (}1-\frac{1}{(n+1{)}^{2}}{{\textstyle )}}^{n+1}\frac{n+1}{n},n\in \mathbb{N}$$

I would really appreciate any help and I hope everything is correctly written.

Greetings

Fractions
Answered

ghulamu51
2022-09-24

I have done problems where I need to get the parameter so the inequality is right for real values of x. But what do I have to do to make it right for every value of x? What condition? I am not asking for anyone to throw me the answer of the whole thing. Just to clarify what the condition needs to be so I can finish on my own.

Fractions
Answered

Lustyku8
2022-09-24

If $A,B,C\in (0,\frac{\pi}{2})$. Then prove that

$\frac{\mathrm{sin}(A+B+C)}{\mathrm{sin}(A)+\mathrm{sin}(B)+\mathrm{sin}(C)}<1$

Fractions
Answered

besnuffelfo
2022-09-24

I happen to be stuck trying to simplify this:

$\left[\frac{(3x+2)(x+1{)}^{\frac{3}{2}}-(\frac{3}{2}{x}^{2}+2x)(\frac{3}{2})(x+1{)}^{\frac{1}{2}}}{(x+1{)}^{3}}\right]$

Fractions
Answered

clasicaacyx
2022-09-24

Taken from local contest. I have no clue on how to approach this problem.

Fractions
Answered

David Ali
2022-09-24

There are terminologies for natural numbers, whole numbers and so on. (If the meaning of the terms can be found, it becomes easier to understand. For natural numbers, the term "natural" refers to the naturally occurring set of numbers in nature like $2,3,4$ and not $-2$, $-3$, and $-4$).

But I didn't find any information about why the numerator is called "numerator" and denominator is called "denominator".

Is it just a simple terminology given by mathematicians (like "addition" in addition) or is there any special "meaning" behind these terms (like "natural" in natural numbers)?

Thanks for your time. If any doubt please comment.

Fractions
Answered

Kaila Branch
2022-09-23

If $\sqrt{xy}+\sqrt[3]{xyz}<a(x+4y+4z)$ and $x,y,z>0$ then $a$ is

with the help of am gm inequality

$\sqrt{xy}\le \left(\frac{x+y}{2}\right)$ and $\sqrt[3]{xyz}\le \left(\frac{x+y+z}{3}\right)$

so $\sqrt{xy}+\sqrt[3]{xyz}\le \frac{x+y}{2}+\frac{x+y+z}{3}=\frac{5x+5y+2z}{6}$

want be able to go further, could some help me

Fractions
Answered

Lyla Carson
2022-09-23

Make a formula to find the perimeter of the rectangle, the perimeter is 24 units.

The longer side is $\frac{5}{x+1}$ and the shorter side is $\frac{2}{x}$

I know that the answer is $\frac{1}{4}$ but no idea how.

Any ideas how to solve it?

Fractions
Answered

kjukks1234531
2022-09-23

How does one add these two terms with exponents in the numerator like ${h}^{2}+\frac{{h}^{2}}{4}$?

According to my lesson on Khan Academy, one should get ${h}^{2}(1+\frac{1}{4})$

However, intuitively, it would seem that one would get $\frac{4{h}^{2}}{4}+\frac{{h}^{2}}{4}$ having first taken a common denominator and then $5\frac{{h}^{2}}{4}$

After having searched for clarification, none of the search results really helped me to derive the answer. Hopefully this will not add, as such, a redundant post. Please clarify.

Fractions
Answered

Hana Buck
2022-09-23

$$2y\le \frac{{y}^{2}}{{x}^{2}}+{x}^{2}$$

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