Questions on fractions, which are expressions (not values) of the form $\frac pq$.

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3answers
84 views

Why does $\frac{49}{64}\cos^2 \theta + \cos^2 \theta$ equal $\frac{113}{64}\cos^2 \theta $?

I have an example: $$ \frac{49}{64}\cos^2 \theta + \cos^2 \theta = 1 $$ then what happens next: $$ \frac{113}{64}\cos^2 \theta = 1 $$ Where has the other cosine disappeared to? What operation ...
1
vote
2answers
30 views

decimal to fractions

When being asked how to solve the Arithmetic Means of 8, 7, 7, 5, 3, 2, and 2, I understand that adding these numbers then dividing by 7 (the amount of numbers) gives me the decimal 4.85714... But ...
33
votes
2answers
2k views

If the decimal expansion of $a/b$ contains “$7143$” then $b>1250$

I recently stumbled upon this really interesting problem: If we have a fraction $\frac{a}{b}$ where $a,b \in \mathbb{N}$ and we know that the decimal fraction of $\frac{a}{b}$ has the numerical ...
2
votes
2answers
60 views

Here are two fractions, $\frac{2}{3}$,$\frac{7}{8}$, which of these fractions are closer to $\frac{3}{4}$?

I've been throwing this question around my family. No one has a clue, therefore can someone help? I'm pretty sure this will be easy to do
2
votes
3answers
72 views

Get rid of the square roots of the denominator: $\dfrac{1}{\sqrt{7}-2\sqrt{5}+\sqrt{3}}$

I squared the whole denominator, but that didn't help. Also I searched for a propriety or identity like $A^2-B^2$, but I didn't see one that could fit. Any help is appreciated. Thank you.
0
votes
2answers
48 views

Fractions of an amount [on hold]

I need help with the following problem: Catalin works in an office. One week he divides his time between these tasks: $\frac{1}{4}$ of his time in meetings $\frac{5}{8}$ of his time writing ...
2
votes
1answer
72 views

Are fractions with zero divisors in the denominator never well defined?

Are fractions with zero divisors in the denominator never well defined? I know that for a fraction in modular arithmetic to be well defined, the denominator must not be a zero divisor, e.g: $$ x ...
0
votes
4answers
59 views

Dividing fractions in real life scenario / application

First of all sorry if this question sounds too stupid or offends anyone. One apple divide by two you get half an apple. $\large{\frac{1}{2} = 0.5}$ I couldn't get my head around with dividing ...
2
votes
4answers
359 views

SAT Maths Question About Fractions

Whilst revising, a problem caught my eye and I cannot seem to find an answer. I am usually bad at these types of questions. On a certain Russian-American committee, $\frac23$ of members are men, ...
0
votes
3answers
68 views

$(5x +1) ÷ (3x)$ is not a polynomial?

On the Mathwarehouse page on polynomial equations, it gives this expression as an antiexample, something that is not a polynomial: $(5x +1) ÷ (3x)$ However, it also says on the same page that if it ...
2
votes
1answer
17 views

Fractions and Largest Common Multiple, Algebra, Numerator and Denominator Identical Numbers?

This is the question find $x$ of equation: $$\frac{5x-2}{5} - \frac{2x+3}{2} = 3$$ I tried multiplying this all by 10, the LCM. It ended with: $x -x=49.$ How do you solve this without cancelling ...
0
votes
5answers
44 views

How does this seemingly-trivial simplification work?

In a section on inductive proofs in the book Modelling Computing Systems: Mathematics for Computer Science (Muller, Struth) there is a simplification that is assumed to be trivial, but that I can't ...
1
vote
1answer
43 views

Can we write “fractional root” symbol in math?

Fractional exponents are legit but I have never seen fractional roots, so I just wonder if we can write fractional roots such as this: It sometimes can be convenient to think about too.
3
votes
6answers
396 views

Is $15/52$ equal to $17/59$?

Is $\frac{15}{52} = \frac{17}{59}$? I typed it into the calculator and found: $$\frac{15}{52} = 0.2884615 $$ $$\frac{17}{59} = 0.2881356 $$ So I thought they were different. But then my friend said ...
0
votes
1answer
25 views

Simplification imaginary fractions

In an exercise, a partial fraction expansion has to be done. I have no problem with that, but one of the last steps includes a simplification as follows: \begin{equation*} \left( -\frac 12 - \frac 16 ...
0
votes
3answers
28 views

Ordering an even and odd fraction that are close

We know that $1/4 < 5/11 < 1/2$. I did it this way from small to large: $$\frac{1 \cdot 3}{4 \cdot 3} = \frac{3}{12}$$ $$\frac{5}{11}$$ $$\frac{1 \cdot 6}{2 \cdot 6} = \frac{6}{12}$$ It is ...
-1
votes
3answers
59 views

How to show simple inequality of fractions

If $$\frac {a}{a+b}<\frac{a'}{a'+b'}$$ then how can I show that $$\frac {a}{a+2b}<\frac{a'}{a'+2b'}\ \forall\ a,b,c>0$$ I tried puitting in a constant k so $$\frac ...
0
votes
1answer
13 views

Farey Sequence implemenatation

I'm trying to use the Farey sequence to get the next lowest reduced fraction in a list. For example, for $n = 8$, we have $\dots, \frac13, \frac38, \frac25, \frac37, \frac12, \dots$ So let's take ...
2
votes
2answers
30 views

Sign of fractional exponent [duplicate]

What is the sign of $-1^{\frac{2}{3}}$? I thought it was positive 1 because it involves squaring, but that doesn't seem to be the case. Why?
1
vote
2answers
44 views

Solving equations including floor function.

I got a little trouble solving equations that involve floor function in an efficient way. For example : $$ \left\lfloor\frac{x+3}{2}\right\rfloor = \frac{4x+5}{3} $$ In the one above, I get that ...
4
votes
4answers
89 views

How does $-\frac{1}{x-2} + \frac{1}{x-3}$ become $\frac{1}{2-x} - \frac{1}{3-x}$

I'm following a solution that is using a partial fraction decomposition, and I get stuck at the point where $-\frac{1}{x-2} + \frac{1}{x-3}$ becomes $\frac{1}{2-x} - \frac{1}{3-x}$ The equations are ...
1
vote
1answer
27 views

rational numbers as upper limit of a summation?

a quick question: Is it a legit way to use a fraction as the upper limit of a summation? Given is a frequency $f$ and a sample rate $f_s$. I want to use a sum like this: $\sum_{k=1}^{\frac{f_s}{2f}} ...
0
votes
5answers
50 views

canceling double fractions how?

I had this example: $$ \frac{\frac{11}{5}}{2} = \frac{11}{10} $$ then: $$ \frac{2\frac{1}{5}}{2} = \frac{11}{10} $$ $$ \frac{1}{5} \not= \frac{11}{10} $$ is this right canceling of double ...
1
vote
2answers
30 views

If $\frac{a}{b}=\frac{b}{c}=\frac{c}{d}$, prove that $\frac{a}{d}=\sqrt{\frac{a^5+b^2c^2+a^3c^2}{b^4c+d^4+b^2cd^2}}$

What I've done so far; $$\frac{a}{b}=\frac{b}{c}=\frac{c}{d}=k\\ a=bk, b=ck, c=dk\\ a=ck^2, b=dk^2\\ a=dk^3$$ I tried substituting above values in the right hand side of the equation to get ...
1
vote
1answer
24 views

Order approximation for rational polynomial

I have this fraction: $\frac{(-12a^3)d^3 + (4wa^3 - 16a^2)d^2 + (5wa^2 - 8a)d - a^2w^2 + 2aw - 1}{(- 12wa^4 + 12a^3)d^3 + (4a^4w^2 - 20a^3w + 16a^2)d^2 + (4a^3w^2 - 11a^2w + 7a)d + a^2w^2 - 2aw + 1}$ ...
0
votes
0answers
20 views

Calculating enrichment

My question concerns how enriched something is as im trying to combine several lists of uneven group size and the answer is escaping me. So basically, I have 6 groups and I want to compare them with ...
0
votes
2answers
47 views

$\frac{6}{4 \times 2} + \frac{7}{5 \times 2} + … + \frac{21}{19 \times 2}$

I got this exercise from school and I have no idea what notion to use, it resumes to Harmonic series, I can't find a generic answer. Do you have any idea? $\frac{6}{4 \times 2} + \frac{7}{5 \times 2} ...
2
votes
1answer
49 views

High School Probability and Contradiction

So I recently came across this question (2(a)) that my friend who teaches high school math posed to me. I thought the solution could be found by using the identities $P(B\,|\,A) = \dfrac{P(A\cap ...
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votes
4answers
74 views

How to prove that $\frac{\ln 12}{\ln 18}$ is irrational witout using the change of base rule? [closed]

I have to show that $\frac{\ln 12}{\ln 18}$ is irrational by using change of base rule. At the beginning I have proved that $\ln r$ is irrational for any rational $r$, $r\ne 1$. Then using this we ...
0
votes
4answers
35 views

Mixed Fractional Equation?

$$3 \frac{3}{5} + \frac{2}{x} = 4\frac{4}{15}$$ I tried subtracting by both sides, etc, but it didn't come out right. I also tried multiplying by both sides, but, it didn't seem to work. what would ...
1
vote
4answers
32 views

Monotonicity of a fraction.

So I want to prove that the following fraction is monotone increasing, as a part of another proof, that's why I stumbled on: $$\frac{4^{n+1}}{2\sqrt{n+1}} \ge \frac{4^{n}}{2\sqrt{n}}$$ I know it's ...
0
votes
0answers
28 views

Long division for multipolynomial expression, little o notation

I have this expression: $$\mathrm{Exp}=\frac{d^3(-12a^4)+d^2(4a^4-16a^3)+d(4a^3-6a^2-a)}{d^3(-12a^4+12a^3)+d^2(4a^4-20a^3+16a^2)+d(4a^3-11a+7a)+(1-2a+a^2)}$$ Is there any way I can take the second ...
1
vote
2answers
87 views

Can we say that $\sqrt{2}=2/(2/(2/(2/\ldots)))$?

Can we say that $\sqrt{2}= \cfrac{2}{\cfrac{2}{\cfrac{2}{\cfrac{2}{\ldots}}}}$? We have ...
0
votes
1answer
32 views

Equivalent forms of expressions with complex numbers

Which expressions are equivalent to $ {1\over{(9i+z)^4}} + {1\over{(9i-z)^4}}$ Select all that apply. $ {18i\over{(81−z)^8}}$ $ {−18i\over{(81+z)^8}}$ $ {18i\over{(81+z)^8}}$ $ ...
1
vote
4answers
51 views

What fraction of her salary does Joan manage to save?

Last Month, Joan spent 1/3 of her monthly salary on food, 2/5 on her child's tuition fees and 3/4 of the remainder on transportation. If she then saved the rest, what fraction of her salary did she ...
-1
votes
1answer
30 views

Elementary Fractions

There are two identical water jugs, A and B. Jug A is 3/7 full of water and Jug B is 8/11 full. What fraction of the capacity of a jug should water be poured out from jug B to jug A so that they both ...
1
vote
3answers
63 views

How is $\frac{(10^{4})^{6}-1}{10^4-1} = 1 + 10^{4} + 10^{8} + 10^{12} + 10^{16} + 10^{20}$?

As the title states, how is: $$\frac{(10^{4})^{6}-1}{10^4-1} = 1 + 10^{4} + 10^{8} + 10^{12} + 10^{16} + 10^{20}$$ I can't see the pattern. Can someone please help? Thanks.
-1
votes
4answers
57 views

Multiplying whole number with fractions.

I'm looking at a solution to a math problem and there are obviously some rules regarding multiplication of fractions that I don't know. Can someone make any sense of this? $$s_n = 625 \cdot ...
3
votes
1answer
44 views

Finding all possible pairs of positive integer values

The ratio of the sum of two positive integers to their difference is $7:5$. If the the sum of the two numbers is at most $25$, find all possible values for the pair of numbers. Let $m$ be the first ...
0
votes
1answer
25 views

Fraction Transforms

Here's a number theory problem I'm having some difficulty with: Say we transform a fraction by the following rule: we start with some fraction $\frac{m}{n}$ with $m > n$ and then convert it to ...
1
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5answers
75 views

How to compute $\frac{t^2}{t+1}$ to the form $\frac{1}{t+1} +t -1$

One of my attempts would look like below. $\frac{t^2}{t+1}$ = $\frac{t \times t+1-1}{t+1}$ = 1+ $\frac{t-1}{t+1}$ = $\frac{t-1+1-1}{t+1} + 1$ = $\frac{-2}{t+1} +2$ Also, I put into t an arbitrary ...
0
votes
3answers
29 views

Adding fractions with exponents

$$3^5 + {1\over3^5}=?$$ My first instinct was to rewrite the second term as $3^{-5}$. Since the base is $3$, rewrite as $3^{5+-5}$. It simplifies to $3^0= 1$. Apparently this is incorrect. Can anyone ...
0
votes
2answers
42 views

Reciprocal over a summation [closed]

Is this statement true? Can we take reciprocal over a summation? $$\frac 1{\sum_{n=1}^\infty\frac 1{(n+1)^3}}=\sum_{n=1}^\infty (n+1)^3$$
0
votes
2answers
37 views

Adding drawnfractions (Very simple question)

I have been surprised not being able to solve this: ...
0
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1answer
33 views

Recursive formula for partial fraction decomposition of a specific kind of fractions

I need to make a partial fraction decomposition of the following fraction : $$ \frac{1}{(x-a)^2(x-b)^2(x-c)^2(x-d)^2(x-e)} $$ The problem is that Wolfram doesn't give any answer : ...
2
votes
3answers
83 views

If $\frac{a}{b}=\frac{x}{y}$, is $\frac{x-a}{y-b}=\frac{x}{y}$? [closed]

Does this hold? $b,y \neq 0$, $b \neq y$.
0
votes
4answers
29 views

Simplifying $\frac{3(a^{1/4}+4)}{2a-32a^{1/2}}$

I have a fraction $\frac{3a^{1/4}+12}{2a-32a^{1/2}}$ which I have factored out into $\frac{3\left(a^{\frac{1}{4}}+4\right)}{2a-32a^{\frac{1}{2}}}$, but checking out W|A I also get that there ought to ...
0
votes
4answers
79 views

What is the reciprocal of $(-1/2)^k$?

What is the reciprocal of $(-1/2)^k$? The answer is meant to be $2^{-k}$ as if you flip something upside down the power becomes negative. However, I am not sure what happens to the negative in front ...
0
votes
4answers
39 views

Algebraic fractions: addition

I have very elementary question about adding algebraic fractions. Now, I know the following: $$\frac{a}{c} + \frac{b}{d} = \frac{da + cb}{dc}$$ My question is however, how given this expression: ...
1
vote
0answers
32 views

How many elements are in the following set?

The set is $$\{ x \in Q:x^2 =64/25 \} $$ I thought the answer was $\{ \frac{8}{5}, -\frac{8}{5} \}$ but I am told there are in fact 4 distinct elements: $$\{ \frac{8}{5}, \frac{8}{-5}, \frac{-8}{5}, ...