 # Get help with algebra fraction problems

Recent questions in Fractions kennadiceKesezt 2022-09-26

### Proving a local minimum is a global minimum.Let $f\left(x,y\right)=xy+\frac{50}{x}+\frac{20}{y}$, 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 $\left(5,2\right)$ 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. Joyce Sharp 2022-09-26

### Fractions with sum 1Using 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!! Kaila Branch 2022-09-25

### Why do we 'invert and multiply' when dividing fractions?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$\frac{a}{b}$ / $\frac{c}{d}$ = $\frac{a}{b}$ * $\frac{d}{c}$? Sara Fleming 2022-09-25

### Does $\left(x+1\right)\mathrm{log}\left(x+1\right)-2x\mathrm{log}x+\left(x-1\right)\mathrm{log}\left(x-1\right)\ge \frac{1}{x}$ hold for $x\ge 1$?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(If we interpret $0\mathrm{log}0$ as $\underset{ϵ\to 0}{lim}ϵ\mathrm{log}ϵ=0$, then the inequality also holds for $x=1$.)But how do we prove this inequality?What I have tried: Let $f\left(x\right)=x\mathrm{log}x-\left(x-1\right)\mathrm{log}\left(x-1\right)$ for $x>1$. Then the RHS equals $f\left(x+1\right)-f\left(x\right)$ so by the Mean Value Theorem there exist some $\xi$ between $x$ and $x+1$ such that$f\left(x+1\right)-f\left(x\right)={f}^{\prime }\left(\xi \right)=\mathrm{log}\left(1+\frac{1}{\xi -1}\right),$and it suffices to show that $\mathrm{log}\left(1+\frac{1}{x}\right)\ge \frac{1}{x}$ ... 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. Ignacio Casey 2022-09-25

### Is it true that $\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+a}\ge \frac{3}{2}$ if $a+b+c=1$? furajat4h 2022-09-25

### Prove equality in formula related to logistic growthI'm trying to figure out the equality$\frac{1}{y\left(1-y\right)}=\frac{1}{y-1}-\frac{1}{y}$I have tried but keep ending up with RHS $\frac{1}{y\left(y-1\right)}$.Any help would be appreciated. unjulpild9b 2022-09-25

### Find generating fraction of number 0.21562626262This 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. Orion Cervantes 2022-09-25

### Inequality $\frac{b}{ab+b+1}+\frac{c}{bc+c+1}+\frac{a}{ac+a+1}\ge \frac{3m}{{m}^{2}+m+1}$Let $m=\left(abc{\right)}^{\frac{1}{3}}$, 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. Hagman7v 2022-09-24

### Seems Simple, forgetting some fundamentals - $\frac{1.6}{3.01}=\frac{x}{1000+x}$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.$\frac{1.6}{3.01}=\frac{x}{1000+x}$ videosfapaturqz 2022-09-24

### ${a}_{n}=\left(1+\frac{1}{n}{\right)}^{n}$: Prove the following Equation.I have trouble solving following problem:Given the sequence:${a}_{n}=\left(1+\frac{1}{n}{\right)}^{n}$We have to show that$\frac{{a}_{n+1}}{{a}_{n}}=\left(1-\frac{1}{\left(n+1{\right)}^{2}}{\right)}^{n+1}\frac{n+1}{n},n\in \mathbb{N}$I would really appreciate any help and I hope everything is correctly written.Greetings ghulamu51 2022-09-24

### Values for p so inequality is right for every x. $-9<\frac{3{x}^{2}+px-6}{{x}^{2}-x+1}<6$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. Lustyku8 2022-09-24

### How do I prove this trigonometric inequality?If $A,B,C\in \left(0,\frac{\pi }{2}\right)$. Then prove that$\frac{\mathrm{sin}\left(A+B+C\right)}{\mathrm{sin}\left(A\right)+\mathrm{sin}\left(B\right)+\mathrm{sin}\left(C\right)}<1$ besnuffelfo 2022-09-24

### How to simplify this fraction with different powers?I happen to be stuck trying to simplify this:$\left[\frac{\left(3x+2\right)\left(x+1{\right)}^{\frac{3}{2}}-\left(\frac{3}{2}{x}^{2}+2x\right)\left(\frac{3}{2}\right)\left(x+1{\right)}^{\frac{1}{2}}}{\left(x+1{\right)}^{3}}\right]$ clasicaacyx 2022-09-24

### For $a,b,c>0$ prove that $\frac{a}{a+\sqrt{\left(a+b\right)\left(a+c\right)}}+\frac{b}{b+\sqrt{\left(b+a\right)\left(b+c\right)}}+\frac{c}{c+\sqrt{\left(c+b\right)\left(c+a\right)}}\le 1$Taken from local contest. I have no clue on how to approach this problem. David Ali 2022-09-24

### Why are numerator and denominator called so?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. Kaila Branch 2022-09-23

### values of $a$ in inequalityIf $\sqrt{xy}+\sqrt{xyz} and $x,y,z>0$ then $a$ iswith the help of am gm inequality$\sqrt{xy}\le \left(\frac{x+y}{2}\right)$ and $\sqrt{xyz}\le \left(\frac{x+y+z}{3}\right)$so $\sqrt{xy}+\sqrt{xyz}\le \frac{x+y}{2}+\frac{x+y+z}{3}=\frac{5x+5y+2z}{6}$want be able to go further, could some help me Lyla Carson 2022-09-23

### Using Algebraic Fractions To Find PerimeterMake 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? kjukks1234531 2022-09-23

### Add fractions with exponents in the numeratorHow 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 ${h}^{2}\left(1+\frac{1}{4}\right)$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. Hana Buck 2022-09-23
### Prove that for any two real numbers $x,y$ with $x\ne 0$$2y\le \frac{{y}^{2}}{{x}^{2}}+{x}^{2}$ memLosycecyjz 2022-09-23