Compute the sum $1\frac{3}{4} + 1\frac{31}{32}$. Ans: $3\frac{23}{32}$

Show that the solution for $x$ in the equation $\frac{1}{x}=\frac{1}{a}+\frac{1}{b}$ is $x=\frac{ab}{a+b}$.

The golden ratio $\varphi$ is the positive solution to the equation $\varphi^2 - \varphi - 1=0$. Find the golden ratio.

Consider a right triangle in which the shorter sides are 8cm and 6cm. What is the length of the long side? Ans: 10cm.

An LCD television screen measures 26 inches on the diagonal. The screen height is 13 inches. How wide is the screen? Ans: $22.51$ inches.

Consider a right angle triangle in which the opposite size has length 1 and the adjacent side has length $\sqrt{\phi}$, where $\phi = \frac{\sqrt{5}+1}{2}$ is the golden ratio. Show that the hypotenuse will have length $\phi$.

The old iPad screen had a screen resolution of 768 pixels by 1024 pixels and its physical dimensions are 6[in] by 8[in]. The screen has the three-by-four aspect ratio. One might conclude that the best choice of paper size for a PDF for such a screen would be be 6[in] by 8[in]. At first I thought so too, but I forgot about the status bar which is 20 pixels tall. There is actually only an usable screen area of 768 pixels by 1004 pixels.

Assuming the width of the PDF is chosen to be 6[in], what height should the PDF have so that it fits perfectly in the content area of the iPad?

Ans: We want the document to have the $768/1004$ aspect ratio, so the height is going to be $6 \times \frac{1004}{768} = 7.84375$[in].

Find the range of values of the parameter $m$ (a real number) so that the equation $2x^2 - mx + m = 0$ has no real solutions.
Ans: The equation has no real solutions whenever $0 < m < 8$.