Problem of the Day: 1/15/13

How many four-digit numbers are divisible by 11 and have a sum of digits of 10?

Solution to yesterday’s problem:

Let’s designate the square’s side length a. Then the area of the square is $a^2$. To find the area of the triangle, we need to find its height. First we’ll draw the altitude to make two 30-60-90 right triangles (whose side lengths are s, $s\sqrt{3}$, and 2s). The short side of each triangle is $\frac{1}{2}a$, so the vertical side is $\frac{1}{2}a\sqrt{3}$. Using the triangle area formula,

$A=\frac{1}{2}a(\frac{1}{2}a\sqrt{3})$

$A=\frac{1}{4}a^2\sqrt{3}$

Now we can write our two areas as a ratio:

$\frac{\frac{1}{4}a^2\sqrt{3}}{a^2}$

Canceling and multiplying both sides by four, we get the final answer $\frac{\sqrt{3}}{4}$.