We are given that $a$, $b$, and $c$ are three real numbers such that $4a - b + c = 112$. Also, $a$, $b$, and $c$ are directly proportional to 1, 2, and 5, respectively. We need to find the values of $a$, $b$, and $c$.

1. Problem Description

We are given that $a$, $b$, and $c$ are three real numbers such that $4a - b + c = 112$. Also, $a$, $b$, and $c$ are directly proportional to 1, 2, and 5, respectively. We need to find the values of $a$, $b$, and $c$.

2. Solution Steps

Since $a$, $b$, and $c$ are directly proportional to 1, 2, and 5, we can write:

$a = 1k$

$b = 2k$

$c = 5k$

where $k$ is the constant of proportionality.

Substitute these values into the given equation:

$4a - b + c = 112$

$4(1k) - (2k) + (5k) = 112$

$4k - 2k + 5k = 112$

$7k = 112$

$k = \frac{112}{7}$

$k = 16$

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

$a = 1k = 1(16) = 16$

$b = 2k = 2(16) = 32$

$c = 5k = 5(16) = 80$

3. Final Answer

$a = 16$, $b = 32$, $c = 80$