Master Fraction Math Problems with Expert Help

Irreducible fraction of a given rational
Given a rational $r \in Q$ , how to find the irreducible fraction $\frac{a}{b} = r$ ? Any direct formula based on the digits of $r$ , instead of successive approximations by increasing numerator and denominator alternatively?

Lorena Beard 2022-07-13

Is 1.0000... ( 1 with infinite zeros) greater than 1.0?
Given that $0.3333...$ is greater than $0.3$ and similarly $0.777...$ is greater than $0.7$ , does it follow that the sum of $0.33...$ and $0.77...$ is greater than sum of $0.3$ and $0.7$ ?

Audrina Jackson 2022-07-13

How to make sense of fractions concretely
I can solve fractions abstractly, for example, $\frac{5}{2}$ divided by $\frac{3}{2}$ , you can flip $\frac{3}{2}$ so that $\frac{5}{2}$ multiplied by $\frac{2}{3}$ . Specifically, math makes sense abstractly, but concretely it just won't make sense, like in word problems. I understand the concept of complex fractions I know how to solve them, but by applying it on practical use such as a shape it does not make sense. How to make sense of fractions concretely?? or perhaps there is a book that you can advice me that help solve this problem

aggierabz2006zw 2022-07-13

Finding $lim_{n \to \infty} \frac{1 + 2^{2} + 3^{3} + 4^{4} + \dots + n^{n}}{n^{n}}$
Finding
$lim_{n \to \infty} \frac{1 + 2^{2} + 3^{3} + 4^{4} + \dots + n^{n}}{n^{n}}$
Attempt:
$lim_{n \to \infty} [\frac{1}{n^{n}} + \frac{2^{2}}{n^{n}} + \frac{3^{3}}{n^{n}} + \dots \dots + \frac{n^{n}}{n^{n}}] = 1$
because all terms are approaching to zero except last terms
but answer is not $1$ , could some help me to solve it , thanks

Agostarawz 2022-07-13

algebra with fractions
While studying math, I have come across this problem:
$x < \frac{x}{3} + \frac{2}{3}$
Now according to the textbook, the answer is $x < 1$ , but I can't see how they came up with that solution, as I could only come up with the answer $x < x + 2$ . So I'd appreciate it if someone could explain this to me.

skynugurq7 2022-07-13

Simplifying fraction with square root as denominator
I'm trying to find the integral of:
$\frac{2 \sqrt{x} - 3 x + x^{2}}{\sqrt{x}}$
but I first need to simplify it so I tried dividing by the $\sqrt{x}$ for each of the numbers on the top like so:
$\frac{2 \sqrt{x}}{\sqrt{x}}$
and did the same for the others. For the one above it was easy to see that it just simplifies to 2. But I am unsure how to do the same for the others for instance $\frac{- 3 x}{\sqrt{x}}$ I know to $- \sqrt{x}$ but i don't know what $- 3 x - \sqrt{x}$ would come out with?

Cameron Mercado2022-07-13

2.8 repeating as a fraction

Janessa Olson 2022-07-12

Prove a sum of fractions less than a value
I happened to see this problem from an elementary school textbook, but cannot solve it:
$C = \frac{1}{5} + \frac{1}{6} + \frac{1}{7} + . . . + \frac{1}{15} + \frac{1}{16} + \frac{1}{17}$
Prove
$C < 2$
Very much appreciate someone to enlighten me, especially with an elementary solution.

pouzdrotf 2022-07-12

How to simply this fraction with irrational denominators?
$\frac{1}{1 + \sqrt{3}} + \frac{1}{\sqrt{3} + \sqrt{5}} + \frac{1}{\sqrt{5} + \sqrt{7}} \frac{1}{\sqrt{7} + 3}$

orlovskihmw 2022-07-11

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 be the proper way to solve this? Thanks!

letumsnemesislh 2022-07-11

Fractions and Largest Common Multiple, Algebra, Numerator and Denominator Identical Numbers?
This is the question find x of equation:
$\frac{5 x - 2}{5} - \frac{2 x + 3}{2} = 3$
I tried multiplying this all by 10, the LCM. It ended with:
$x - x = 49.$
How do you solve this without cancelling the x out of the equation?

Michelle Mendoza 2022-07-10

Show that $\frac{(b + c)^{2}}{3 b c} + \frac{(c + a)^{2}}{3 a c} + \frac{(a + b)^{2}}{3 a b} = 1$
If $a^{3} + b^{3} + c^{3} = 3 a b c$ and $a + b + c = 0$ show that $\frac{(b + c)^{2}}{3 b c} + \frac{(c + a)^{2}}{3 a c} + \frac{(a + b)^{2}}{3 a b} = 1$

ban1ka1u 2022-07-10

What is $\frac{9}{3} - \frac{1}{2}$ ?

skynugurq7 2022-07-10

Fraction involving Surds
Fraction involving Surds. Can anyone please show me the working out?
$\frac{(\sqrt{6} - 1)}{\sqrt{3}} + \frac{(\sqrt{6} + 2)}{2 \sqrt{3}}$
I did this and it was incorrect:
$2 \sqrt{3} (\sqrt{6} - 1) + \frac{\sqrt{3} (\sqrt{6} + 2)}{\sqrt{3} \times 2 \sqrt{3}}$
$\frac{2 s q r t 3 (\sqrt{6} - 1)}{\sqrt{3}} + \frac{s q r t 3 (\sqrt{6} + 2)}{2 \sqrt{3}}$

doturitip9 2022-07-10

How to simplify an expression?
I have tried to simplify this expression for quite a long time now but I can't find how to do it.
$(\frac{1}{2 + 4 m} - \frac{1 - m}{8 m^{3} + 1} : \frac{1 - 2 m}{2 m^{2} - 2 m + 1}) \cdot \frac{4 m + 2}{2 m - 1} - \frac{1}{1 - 4 m + 4 m^{2}}$
Can someone help me with it?

Kaeden Hoffman 2022-07-10

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.
$\frac{t^{2}}{t + 1}$ = $\frac{t \times t + 1 - 1}{t + 1}$ = $\frac{t - 1}{t + 1}$ = $\frac{t - 1 + 1 - 1}{t + 1} + 1$ = $\frac{- 2}{t + 1} + 2$
Also, I put into t an arbitrary number in both equations and have gotten the same answer. The first number was 18785 to which both contained the answer $\frac{352876225}{18786}$

Raul Walker 2022-07-10

Inequality relationship when additing a constant to the denominators
When $\frac{a}{b} > \frac{c}{d}$ where $a, b, c,$ and $d$ are positive real numbers, is $\frac{a}{b + 1} > \frac{c}{d + 1}$ true?

Nylah Hendrix 2022-07-10

Order approximation for rational polynomial
I have this fraction: $\frac{(- 12 a^{3}) d^{3} + (4 w a^{3} - 16 a^{2}) d^{2} + (5 w a^{2} - 8 a) d - a^{2} w^{2} + 2 a w - 1}{(- 12 w a^{4} + 12 a^{3}) d^{3} + (4 a^{4} w^{2} - 20 a^{3} w + 16 a^{2}) d^{2} + (4 a^{3} w^{2} - 11 a^{2} w + 7 a) d + a^{2} w^{2} - 2 a w + 1}$
How can I approximate it so it may be written as a function of increasing order of d? As in $f (d) + f (d^{2}) + H . O . T$
I have tried using Taylor series but that is centered around a point (which I don't want). I am looking into Pade approximation but am utterly confused. Can someone help me?

Logan Wyatt 2022-07-10

How to divide the fraction $1 / 1 + 1$
This has to do with re-calculating the sigmoid function in ai. It isn't really important, but the simplest way to put it is I need a math guru to help my monkey brain do this:
$\frac{1}{1 + e}$
to like
$\frac{1}{s o m e t h i n g} + \frac{1}{e}$
Please help me remember my math from high-school if this was ever taught to us.

lilmoore11p8 2022-07-10

Determine if $\frac{k - 1}{k} + \frac{1}{k (k + 1)} = \frac{k}{k + 1}$ holds
How to prove if the following equality holds?
$\frac{k - 1}{k} + \frac{1}{k (k + 1)} = \frac{k}{k + 1}$
Maybe finding a common denominator would work, but I have no idea how to do it in this example.
I see that it holds for $k = 1$
$0 + \frac{1}{2} = \frac{1}{2} = \frac{1}{1 + 1} .$
It also holds for $k = 2$
$\frac{1}{2} + \frac{1}{6} = \frac{3}{6} + \frac{1}{6} = \frac{4}{6} = \frac{2}{3} = \frac{2}{2 + 1}$
It also works for $k = 3$
$\frac{2}{3} + \frac{1}{12} = \frac{8}{12} + \frac{1}{12} = \frac{9}{12} = \frac{3}{4} = \frac{3}{3 + 1}$
But I am not sure how to proceed if I am not working with numbers but with expressions.

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