Divide the number 2200 into three parts $a, b,$ and $c$ such that $a, b,$ and $c$ are directly proportional to 2, 4, and 16, respectively.

1. Problem Description

Divide the number 2200 into three parts $a, b,$ and $c$ such that $a, b,$ and $c$ are directly proportional to 2, 4, and 16, respectively.

2. Solution Steps

Let $a, b,$ and $c$ be the three parts such that

$a + b + c = 2200$.

Since $a, b,$ and $c$ are directly proportional to 2, 4, and 16, we can write

$\frac{a}{2} = \frac{b}{4} = \frac{c}{16} = k$, where $k$ is a constant of proportionality.

Then we have:

$a = 2k$

$b = 4k$

$c = 16k$

Substituting these values into the equation $a + b + c = 2200$, we get:

$2k + 4k + 16k = 2200$

$22k = 2200$

$k = \frac{2200}{22}$

$k = 100$

Now we can find the values of $a, b,$ and $c$:

$a = 2k = 2(100) = 200$

$b = 4k = 4(100) = 400$

$c = 16k = 16(100) = 1600$

3. Final Answer

The three parts are $a = 200$, $b = 400$, and $c = 1600$.