Brian had $$\frac{2}{5}$$ of a spool of yarn. He used $$\frac{4}{5}$$ of his yarn for a project. What fraction of the spool was used for the project?

Raina Gomez
2022-09-17
Answered

Brian had $$\frac{2}{5}$$ of a spool of yarn. He used $$\frac{4}{5}$$ of his yarn for a project. What fraction of the spool was used for the project?

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asked 2022-08-21

Prove inequality $\sum \frac{{a}^{3}-{b}^{3}}{{\left(a-b\right)}^{3}}\ge \frac{9}{4}$

Given $a,b,c$ are positive number. Prove that

$\frac{{a}^{3}-{b}^{3}}{{(a-b)}^{3}}+\frac{{b}^{3}-{c}^{3}}{{(b-c)}^{3}}+\frac{{c}^{3}-{a}^{3}}{{(c-a)}^{3}}\ge \frac{9}{4}$

$\iff \sum \frac{3(a+b{)}^{2}+(a-b{)}^{2}}{(a-b{)}^{2}}\ge 9$

$\iff \frac{(a+b{)}^{2}}{(a-b{)}^{2}}+\frac{(b+c{)}^{2}}{(b-c{)}^{2}}+\frac{(c+a{)}^{2}}{(c-a{)}^{2}}\ge 2$

Which

$\frac{a+b}{a-b}.\frac{b+c}{b-c}+\frac{b+c}{b-c}.\frac{c+a}{c-a}+\frac{c+a}{c-a}.\frac{a+b}{a-b}=-1$

Use ${x}^{2}+{y}^{2}+{z}^{2}\ge -2(xy+yz+zx)$ then inequality right

I don't know why use ${x}^{2}+{y}^{2}+{z}^{2}\ge -2(xy+yz+zx)$ then inequality right?

P/s: Sorry i knowed, let $x=\frac{a+b}{a-b}$

We have $(x+1)(y+1)(z+1)=(x-1)(y-1)(z-1)=>xy+yz+zx=-1$

Given $a,b,c$ are positive number. Prove that

$\frac{{a}^{3}-{b}^{3}}{{(a-b)}^{3}}+\frac{{b}^{3}-{c}^{3}}{{(b-c)}^{3}}+\frac{{c}^{3}-{a}^{3}}{{(c-a)}^{3}}\ge \frac{9}{4}$

$\iff \sum \frac{3(a+b{)}^{2}+(a-b{)}^{2}}{(a-b{)}^{2}}\ge 9$

$\iff \frac{(a+b{)}^{2}}{(a-b{)}^{2}}+\frac{(b+c{)}^{2}}{(b-c{)}^{2}}+\frac{(c+a{)}^{2}}{(c-a{)}^{2}}\ge 2$

Which

$\frac{a+b}{a-b}.\frac{b+c}{b-c}+\frac{b+c}{b-c}.\frac{c+a}{c-a}+\frac{c+a}{c-a}.\frac{a+b}{a-b}=-1$

Use ${x}^{2}+{y}^{2}+{z}^{2}\ge -2(xy+yz+zx)$ then inequality right

I don't know why use ${x}^{2}+{y}^{2}+{z}^{2}\ge -2(xy+yz+zx)$ then inequality right?

P/s: Sorry i knowed, let $x=\frac{a+b}{a-b}$

We have $(x+1)(y+1)(z+1)=(x-1)(y-1)(z-1)=>xy+yz+zx=-1$

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Simplify: $\frac{-4}{8}\xf7\frac{16}{64}$

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What is half of a half? What is half of that?

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Find $2\left(\frac{5}{7}\right)+1\left(\frac{1}{4}\right)$

asked 2022-05-21

Expression for binomial coefficient denominator

I'm trying to find an analytical expression for the denominator of $\left(\begin{array}{c}-1/2\\ k\end{array}\right)$ in terms of k when the fraction is fully reduced.

E.g., the first several such denominators, starting with $k=0$, are

$1,2,8,16,128,256,1024,2048,32768$ so there are various power-of-2 jumps, but I haven't been able to figure out the overall pattern so that I can nail down the expression.

Does anyone know of such an expression, or know of a good place to look to try to figure this out? If not, does anyone know if this is a fool's errand?

Thanks for any help.

I'm trying to find an analytical expression for the denominator of $\left(\begin{array}{c}-1/2\\ k\end{array}\right)$ in terms of k when the fraction is fully reduced.

E.g., the first several such denominators, starting with $k=0$, are

$1,2,8,16,128,256,1024,2048,32768$ so there are various power-of-2 jumps, but I haven't been able to figure out the overall pattern so that I can nail down the expression.

Does anyone know of such an expression, or know of a good place to look to try to figure this out? If not, does anyone know if this is a fool's errand?

Thanks for any help.

asked 2022-06-20

Simplify complex fraction

This is a really low level question. I'm trying to simplify

$f(x)=\frac{36{x}^{-2}-3{x}^{-1}-18}{12{x}^{-2}-25{x}^{-1}+12}$

After factoring, removing negative exponents, and flipping the second fraction I get

$f(x)=\frac{1}{3(3x+2)(4x-3)}\frac{(3x-4)(4x-3)}{1}$

Then $(4x-3)$ cancels leaving

$f(x)=-\frac{3x-4}{3(3x+2)}$

as my final answer. However the book I have says the correct answer is

$f(x)=-\frac{3(2x+3)}{4x-3}$

I've checked my work many times and I don't know how they get this answer. Could someone please help me solve this?

This is a really low level question. I'm trying to simplify

$f(x)=\frac{36{x}^{-2}-3{x}^{-1}-18}{12{x}^{-2}-25{x}^{-1}+12}$

After factoring, removing negative exponents, and flipping the second fraction I get

$f(x)=\frac{1}{3(3x+2)(4x-3)}\frac{(3x-4)(4x-3)}{1}$

Then $(4x-3)$ cancels leaving

$f(x)=-\frac{3x-4}{3(3x+2)}$

as my final answer. However the book I have says the correct answer is

$f(x)=-\frac{3(2x+3)}{4x-3}$

I've checked my work many times and I don't know how they get this answer. Could someone please help me solve this?

asked 2022-05-21

What is the smallest possible value of $a+b$

If $\frac{a}{b}$ rounded to the nearest trillionth is $0.008012018027$, where $a$ and $b$ are positive integers, what is the smallest possible value of $a+b$?

I don't see any strategies here for solving this problem, any help? Thanks in advance!

If $\frac{a}{b}$ rounded to the nearest trillionth is $0.008012018027$, where $a$ and $b$ are positive integers, what is the smallest possible value of $a+b$?

I don't see any strategies here for solving this problem, any help? Thanks in advance!