I've been waiting for someone to ask this question. It's one of my favorite questions to answer.

percent means out of 100 so 1% = 1/100 and 40%=40/100 or 1/25
so for percent of a number, 'of' means multiply so 40% of 500= 40/100 times 500 or 20000/100 or 200

if you multiply a fraction by its recipricol (the number fliped upside down) the you will get 1 so if you multiply 1/2 by 2/1 then you will get 1

thus
percent is out of 100 so 1%=1/100 25%=25/100
if you represent 45 out of something as 45/x then you multiply it by the reciprocal of the fraction that is supposed to be equal (1/2 =2/4) then you will get 1
so 45/x times reciprocal of 100/25 then you will get 1/1 or 4500/25x
so you divide to find x and get 180
so 45 is 25% of 180

The following is extra
   as stated above 'of' can be taken as multiply so
   45 is 25% of 180 can be rewriten as
   45=25/100 times 180
   but we don't know that 180 is the answer so we replace it with x then       solve for x

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3 years ago

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Determine whether or not the vector field is conservative. If it is conservative, find a function f such that F = ∇f. (If the ve

djyliett [7]

We're looking for $f(x,y,z)$ such that $\nabla f(x,y,z)=\vec F(x,y,z)$ , which requires

$\dfrac{\partial f}{\partial x}=xyz^2$

$\dfrac{\partial f}{\partial y}=x^9yz^2$

$\dfrac{\partial f}{\partial z}=x^9y^2z$

Integrating both sides of the first PDE wrt $x$ gives

$f(x,y,z)=\dfrac12x^2yz^2+g(y,z)$

Differenting this wrt $y$ gives

$\dfrac{\partial f}{\partial y}=x^9yz^2=\dfrac12x^2z^2+\dfrac{\partial g}{\partial y}$

$\implies\dfrac{\partial g}{\partial y}=\dfrac12x^2z^2(2x^7y-1)$

but we're assuming $g(y,z)$ is a function that doesn't depend on $x$ , which is contradicted by this result, and so there is no such $f$ and $\vec F$ is not conservative.