Articles of ratio

What will be quantity of wine at the end in the vessel?

There are two vessels containing 12 liter of water the other contains 6 liters of wine respectively.One liter of mixture was taken out from each vessel and poured into the other and if this process is to be done four times. Find how much liters of wine will be at the end in second vessel?

Find ratio / division between two numbers

I am reverse engineering custom software for a stepper motor. The original software eases in and out of any motion, and the duration of the ramping up to speed is directly related to the speed that the motor is ramping up to. In other words, when the motor is ramping up to a high speed, […]

Math Behind Creating a “Perfect” Star

I am busy looking to create star paths in my app, and I was wondering how to determine the ratio between the inner radius and the outer radius of the points of a star so that the star has “straight” lines across. I have a function that takes 3 parameters: pointCount = 5 outerRadius = […]

How is the Radius of Convergence of a Series determined?

Consider $$\sum_{n=0}^{\infty}\frac{(-1)^nx^n}{(n+1)^2}$$ which by the ratio test the ratio of two consecutive terms converges to $|x|$ as $n\rightarrow \infty$ and has a radius of convergence equal to $1$. Now consider $$\sum_{n=0}^{\infty}(-1)^n(2^n+n^2)x^n$$ which by the ratio test the ratio of two consecutive terms converges to $2|x|$ as $n\rightarrow \infty$ and has a radius of convergence equal […]

How to get ratio of a,b,c from 2 equations in a,b,c

I have 2 equation in terms of $a,b$ and $c$ . $3a + 10b + 5c =0$ and $4a + 6b + 2c =0$ I need to find a:b:c and answer is $\dfrac a5 = \dfrac {b}{-7} = \dfrac {c}{11}$ I want to know how to get that? My attempt: Given equations can be written […]

Ratios of Four Numbers That Total 1285

The sum of 4 numbers is 1285. The ratio of the first to second is 2:3. The ratio of the second to third is 4:5. The ratio of the third to fourth is 5:6. Find the third number. I had absolutely no idea even how to start this. Your advice would be much appreciated. I […]

Ratio test with limsup vs lim

Could I prove that the ratio test still works using $\limsup(\frac{a_{n+1}}{a_n})$ instead of $\lim(\frac{a_{n+1}}{a_n})$? I think for $\limsup<1$ I could show that for $\epsilon>0, N>1 \limsup(\frac{a_{n+1}}{a_n})<1-\epsilon.$ From there I can solve that $\lvert a_k\rvert<(1-\epsilon)^{k-N}\lvert a_N\rvert$ for $k>N$. Thus, by comparison test the left-hand side will converge absolutely. Is this enough to show I can use […]

Is this a correct/good way to think interpret differentials for the beginning calculus student?

I was reading the answers to this question, and I came across the following answer which seems intuitive, but too good to be true: Typically, the $\frac{dy}{dx}$ notation is used to denote the derivative, which is defined as the limit we all know and love (see Arturo Magidin’s answer). However, when working with differentials, one […]

Alligation or mixture

A vessel is filled with liquid,3 parts of which are water and 5 parts syrup.How much of the mixture must be drawn off and replaced with water so that the mixture may be half water and half syrup? What is the best possible way to solve this question?

Relationship of aspect ratio to the homography matrices between a rectangle and an arbitrary quadrilateral

I’ve been reading everything I can on the perspective mapping between a 2D rectangle and the projection onto the plane in 3D space of a rectangle. I’ve learned that any such quadrilateral resulting from the projection can be mapped to any rectangle. I’ve learned that the only constraints are that the quadrilateral must be convex. […]