0
votes
3answers
46 views

Show that $\, 0 \leq \left \lfloor{\frac{2a}{b}}\right \rfloor - 2 \left \lfloor{\frac{a}{b}}\right \rfloor \leq 1 $

How can I prove that, for $a,b \in \mathbb{Z}$ we have $$ 0 \leq \left \lfloor{\frac{2a}{b}}\right \rfloor - 2 \left \lfloor{\frac{a}{b}}\right \rfloor \leq 1 \, ? $$ Here, $\left \lfloor\,\right ...
0
votes
3answers
63 views

What is the correct way to divide a fraction?

This is a very basic question, but i'm struggling with it. Can someone explain the rules when dividing a fraction like this: $$\frac{\cos(\pi x)\sin(\pi x)}{\Large{\frac{\cos(\pi x)}{\sin(\pi x)}}}$$ ...
0
votes
1answer
40 views

Calculate $X$ of a math problem

I am trying to learn some more math and I got stuck on this: $$\frac{0.2}{X} = 140$$ How do I calculate $X$? EDIT Sorry I meant to calculate $$\frac{28}{X} = 140$$ So that $X = 0.2$, but how do I ...
3
votes
2answers
50 views

Simplifying a square root fraction

Simplify the following $$\frac{\sqrt{3}}{\sqrt{2}(\sqrt{6} - \sqrt{3})}$$ Apparently the answer is $\frac{1}{2} (2 + \sqrt{2})$ but can't for the life of me see how to get it. Any help is massively ...
1
vote
1answer
35 views

Simplifying an expression written as the sum of three fractions

Specifically, I don't know what to do first given the following expression: $$ \frac{4x - 2}{6} - \frac{2 - x}{4} + \frac{x + 3}{3} $$ So I think of it as $\frac 16(4x-2) - \frac 14(2-x) + \frac ...
3
votes
2answers
55 views

Show that $\frac 1{1+x+y^{-1}}+\frac1{1+y+z^{-1}}+\frac1{1+z+x^{-1}}=1$ if $xyz=1$

If $x.y.z=1$ show that $\dfrac 1{1+x+y^{-1}}+\dfrac1{1+y+z^{-1}}+\dfrac1{1+z+x^{-1}}=1$ My attempt - L.H.S$=\dfrac 1{1+x+y^{-1}}+\dfrac1{1+y+z^{-1}}+\dfrac1{1+z+x^{-1}}$ $=\dfrac y{y+xy+1}+\dfrac ...
0
votes
1answer
56 views

$\frac{x}{y} \ge \frac{a_1}{b_1} \ge \frac{a_2}{b_2}$ and $b_1 \le b_2 \implies \frac{x+a_1}{y + b_1} \ge \frac{x+a_2}{y + b_2}$?

Given $\frac{x}{y} \ge \frac{a_1}{b_1} \ge \frac{a_2}{b_2}$, where $x,y,a_i,b_i$ are positive numbers. I would like to prove the following: Claim: If $b_1 \le b_2$, then $\frac{x+a_1}{y + b_1} ...
0
votes
1answer
19 views

Solving $\frac12 (3y+2)-\frac58=\frac34y$ for $y$ using LCD method

I am solving $$\frac12 (3y+2)-\frac58=\frac34y$$ for $y$ using LCD method. Can't figure out what I did wrong! The answer in the back of the book is $-1/2$. PS: In the first line that is a $1/2$ in ...
2
votes
3answers
221 views

Simplifying nested/complex fractions with variables

I have the equation $$x = \frac{y+y}{\frac{y}{70} + \frac{y}{90}} $$ and I need to solve for x. My calculator has already shown me that it's not necessary to know y to solve this equation, but I ...
4
votes
4answers
517 views

What is the non-trivial, general solution of these equal ratios? [closed]

Provide non-trivial solution of the following: $$\frac{a}{b+c}=\frac{b}{c+a}=\frac{c}{a+b}$$ $a=?, b=?, c=?$ The solution should be general.
0
votes
1answer
35 views

Comparing Fractional Numbers

Does a formula exist for comparing two fractional numbers, without resolving to using anything other than integers and fractions? (Thus not real numbers). In other words: given $\dfrac{a}{b}$ and ...
0
votes
2answers
79 views

How does $\sqrt {\frac{{4 + \sqrt {15} }}{8}} = \frac{{\sqrt {8 + 2\sqrt {15} } }}{4}$

I have the follow answering to a question from my textbook: $\sqrt {\frac{{4 + \sqrt {15} }}{8}}$ However my textbook simplifies it to: $\frac{{\sqrt {8 + 2\sqrt {15} } }}{4}$ I've checked and my ...
4
votes
7answers
788 views

When the numerator of a fraction is increased by 4, the fraction increases by 2/3…

When the numerator of a fraction is increased by $4$, the fraction increases by $2/3$. What is the denominator of the fraction? I tried, Let the numerator of the fraction be $x$ and the denominator ...
3
votes
3answers
114 views

How does $-\sqrt {\frac{{2 - \sqrt 2 }}{{2 + \sqrt 2 }}} $ simplify to $1 - \sqrt 2 $?

I've the answer for a question in my textbook to be: $-\sqrt {\frac{{2 - \sqrt 2 }}{{2 + \sqrt 2 }}} $ which i've then simplifed to: $-\sqrt {3 - 2\sqrt 2 } $ However my textbook states $-\sqrt ...
0
votes
1answer
55 views

Using all types of fractions

Is their a website that teaches you everything you need to know about fractions, just fractions. I ask this because I do calculus...and I suck at fractions. I hate them so much. I have no idea how to ...
1
vote
3answers
31 views

Fraction exponents in division

if I have $\frac{a^{6/5}}{b^{1/5}}$, I know you subtract exponents when dividing so $6/5 - 1/5$ is $5/5$, so since that's just one, is this equal to $a/b$?
3
votes
2answers
63 views

Absolute value on the top of a fraction

What is the answer to a question similar to this one, where the absolute value bars are only around the numerator of the fraction? $$\frac{|2+4(2)|}{5-10}$$ Would the fraction be equal to ...
0
votes
1answer
40 views

cancell common factors

p³ - PQ² --------- Divided by (P+Q)² Apparently the answer is P(P-Q) --------- Divided by P+Q But how? - What I was thinking P-P (P-Q), (P+Q) ------------------ Divided by (P+Q) which is P-P ...
0
votes
3answers
66 views

Splitting the numerator

Can someone explain how we can get the second fraction by splitting the numerator? $$\frac{x^3}{x^2+x+1}=x-1+\frac{1}{x^2+x+1}$$ I can get the LHS from the RHS but not the other way around. What ...
1
vote
3answers
56 views

Basic algebra, isolating the variable

So I have the equation $$\tan30=\frac{4.9t-\frac{10}{t}}{\frac{8.77}{t}}$$ And I want to find t, but my algebra has failed me. This is my working so far. ...
0
votes
1answer
57 views

Rearranging algebraic formula when subject is on both sides

I have run into some difficulty with a question on making a variable the subject of an equation where the variable is on both sides. I am really struggling to find a method for making "a" the ...
0
votes
5answers
89 views

Guys, hard limit, please help.

Here is the limit I'm struggling with: $$\lim_{x\to0}\cfrac{x\tan x-x\sin x}{x\sin^2x/\cos x}.$$ Worked so hard to find it, but couldn't.
3
votes
5answers
115 views

How do I prove $\frac{ \sqrt{x+h}-\sqrt{x} }{ h}=\frac{1}{\sqrt{x+h}+\sqrt{x}}$?

$$\frac{ \sqrt{x+h}-\sqrt{x} }{ h}=\frac{1}{\sqrt{x+h}+\sqrt{x}}$$ I know I just asked a question and I did figure out how that one worked but I'm not sure how I would go about this one.
0
votes
1answer
42 views

What's the difference between “continued fractions” and “compound fractions”?

What should we call a fraction which includes another fraction in its numerator or denominator, like $${ab\over {c \over d}}$$?
4
votes
4answers
116 views

why do equations work and how do they relate to each other?

Ok, so I understand that an equation is something like 15 = 15 , and that the only criteria as far as I can tell for it being an equation is that both sides are equal to each other. I have a few ...
1
vote
2answers
43 views

How to solve $\dfrac{7x}{8}+4-\dfrac{2x}{3}=4x-3$?

$$\frac{7x}{8}+4-\frac{2x}{3}=4x-3$$ I do not understand how to simplify this. Could anyone here help me, please? Thanks.
1
vote
0answers
21 views

MultiEquations (with fractions)

Can you please help me solve these equations i don't understand how to solve them with fractions. 1=n-2/15 151/20 =2a+1 3/4 -3/5 -2 1/5k = - 26/25
2
votes
1answer
71 views

$\sqrt[\large m]{(x+y)}\over \sqrt[\large k]{(x+y)}$ $=\sqrt[\large m-k]{(x+y)} $?

Is it always true that: $\sqrt[\large m]{(x+y)}\over \sqrt[\large k]{(x+y)}$ $=\sqrt[\large m-k]{(x+y)} $ where $m,k \in \mathbb N$ ? I tried it with a few numbers and it seems to work every time.
-3
votes
1answer
170 views

How to multiply, divide, add and subtract fractions

I've spent hours on this and I keep getting mixed answers. I need to know the rules for multipling, dividing, adding, subtracting equations involving fractions. I google search but the information is ...
7
votes
2answers
97 views

Why does partial fraction decomposition always work?

Say you have a function $p(x)/q(x)$ for some polynomials $p(x)$ and $q(x)$ and $p$ has a lower degree than $q$. Say $q$ has degree three and $p$ has degree two. If you partially decompose it, you'll ...
2
votes
2answers
61 views

Proper decimal fraction for $\frac{4n+1}{n(2n-1)}$

Assume I have a function $f(n) = \frac{4n+1}{n(2n-1)}$ with $n \in \mathbb{N} \setminus \left\{ 0 \right\}$. The objective is to find all $n$ for which $f(n)$ has a proper decimal fraction. I know ...
7
votes
2answers
218 views

Rationalizing the denominator 3

It is a very difficult question. How can we Rationalizing the denominator? $$\frac{2^{1/2}}{5+3*(4^{1/3})-7*(2^{1/3})}$$
5
votes
8answers
348 views

If $\,\,x+\dfrac{1}{x}=5,\,\,$ find $\,\,x^5+\dfrac{1}{x^5}$.

If $x>0$ and $\,x+\dfrac{1}{x}=5,\,$ find $\,x^5+\dfrac{1}{x^5}$. Is there any other way find it? $$ \left(x^2+\frac{1}{x^2}\right)\left(x^3+\frac{1}{x^3}\right)=23\cdot 110. $$ Thanks
1
vote
3answers
52 views

Simplifying long fractions

How would I go about simplifying long fractions, such as the likes of this: $((8+\frac{3}{4}) + (3\frac{2}{3}))$ / $((4+\frac{2}{5}) - (1\frac{7}{8}))$ The correct answer is ($4 + \frac{278}{303}$) ...
0
votes
2answers
57 views

solve ratio word problem without algebra

Four gallons of yellow paint plus two gallons of red paint make orange paint. I assume this makes six gallons. So the ratio is 4:2, or 2:1. Question: how many gallons of yellow paint, and how many ...
0
votes
3answers
50 views

How do I compute the individual terms of a polynomial to the power of -1?

If my polynomial $p$ is: $x+1$, obviously $p^{-1} = \frac{1}{x+1}$. Is it possible for me to split $\frac{1}{x+1}$ into a sum of two terms? In other words, is there an algorithm to write $p^{-1}$ as ...
1
vote
2answers
58 views

Formula help with this equation

I don't know what the answer to this formula is, can someone please help me. I've tried lots of things but getting no where. If $x=\dfrac56+\dfrac{15}{18}-\dfrac{10}{12}$, then $(x-1)3=$ ?
0
votes
1answer
52 views

Linear Equation Problem

Evaluating the expression below: $\displaystyle \frac{2(6-x)}{3} = \frac{9(x+5)}{6} + \frac{1}{3}$ The answer is $-\frac{23}{13}$ but to obtain this answer what specific method do you use?
1
vote
3answers
42 views

mixed numbers subtraction vertically

In the following subtraction we are subtracting $2$ mixed numbers vertically. I know how it works except the last step. $$ 7 \frac{1}{3} - 4 \frac{1}{2} = 3 + \frac{-1}{6} = 2 + \frac{5}{6} = 2 ...
0
votes
1answer
73 views

Graphs of functions with fractional powers: $x^{p/q}$

How does changing the value of $\dfrac{p}{q}$ affect the drawing of the graph (domain/range/shape, etc.) How do you calculate asymptotes? Below is a question dealing with this type of function. ...
2
votes
3answers
159 views

Rules for cancelling fractions with exponents

I have an expression that I need to simplify, I know the answer (wolframalpha) but I'm not sure of the rule that gets me there. $\dfrac{(\alpha) X_1^{\alpha -1} X_2^{1-\alpha}}{(1-\alpha)X_1^\alpha ...
14
votes
4answers
313 views

Show the identity $\frac{a-b}{a+b}+\frac{b-c}{b+c}+\frac{c-a}{c+a}=-\frac{a-b}{a+b}\cdot\frac{b-c}{b+c}\cdot\frac{c-a}{c+a}$

I was solving an exercise, so I realized that the one easiest way to do it is using a "weird", but nice identity below. I've tried to found out it on internet but I've founded nothingness, and I ...
1
vote
3answers
97 views

Mixed number fractions vs regular fractions? $3\frac{1}{6}-1\frac{11}{12}$

I just passed Calculus 2 in college with an A and I'm rather embarrassed that I'm asking this question. My wife is taking an intermediate Algebra course in college and they gave her the below ...
0
votes
4answers
105 views

Fraction Problem

The product of two fractions is 1/9. The larger fraction divided by the smaller fraction is 4. What is the sum of the two fractions? $\frac{a}{b}\frac{c}{d}=\frac{1}{9}$ I will assume that ...
0
votes
4answers
101 views

Why does $ \frac {\frac {1}{\sqrt{x}}}{x} = \frac {\sqrt{x}}{x^2} $?

A homework question recently asked for me to simplify: $\frac{1}{\sqrt{7}} \div {7}$ It's easy to see that this becomes $\frac{1}{7\sqrt{7}}$ But according to wolfram alpha this is also equal to ...
0
votes
1answer
64 views

simplify cube root of a fraction

I asking this question here to just check the work and that I have simplified it correctly. Given the formula $\Large \sqrt[3]{\frac{12m^4n^8}{5p^4}}$ We need to rationalize the denominator by ...
5
votes
4answers
3k views

What's the algebraic property where you can flip the fractions in an equation?

Earlier in algebra, we spent over 20 minutes trying to figure out $$ \frac{1}{R_1} + \frac{1}{R_2} = \frac{1}{R_e} \,\,\,\, \text{solve for }R_2 $$ when the teacher said "What you start out with is ...
0
votes
2answers
121 views

Algebraic Manipulation question - trying to get alternate form

I'm currently working on algebraic manipulation, changing algebraic fractions into a chosen alternate form but I've hit a brick wall. I'm trying to get: $$\frac{2(3^x - 2^x)}{3^{x+1} - 2^{x+1}}$$ ...
1
vote
3answers
64 views

Rationalizing a denominator.

The question instructs to rationalize the denominator in the following fraction: My solution is as follows: The book's solution is which is exactly the numerator in my solution. Can someone ...
0
votes
1answer
59 views

Unknown terms of the proportion

please help me solving this problem. The question is, find the unknown terms of the proportion $$\frac 23 = \frac x{12} = \frac y{15}.$$