$1222 is divided among $2X$, $6Y$, and $8Z$ such that the shares of $X$, $Y$, and $Z$ are in the ratio $3:2:1$. We need to find how much $X$ gets.

1. Problem Description

$1222 is divided among$2X$,$6Y$, and$8Z$such that the shares of$X$,$Y$, and$Z$are in the ratio$3:2:1$. We need to find how much$X$ gets.

2. Solution Steps

Let $x$, $y$, and $z$ be the amounts received by $2X$, $6Y$, and $8Z$ respectively.

We are given that the ratio of the shares of $X$, $Y$, and $Z$ is $3:2:1$. Thus, we have

$X:Y:Z = 3:2:1$.

We can represent this as $X = 3k$, $Y = 2k$, and $Z = k$ for some constant $k$.

Then we know that $2X$ receives $x$, $6Y$ receives $y$, and $8Z$ receives $z$.

We have $x = 2X = 2(3k) = 6k$, $y = 6Y = 6(2k) = 12k$, and $z = 8Z = 8(k) = 8k$.

The total amount is $1222$, so $x + y + z = 1222$.

Substituting the expressions for $x$, $y$, and $z$ in terms of $k$, we have

$6k + 12k + 8k = 1222$.

$26k = 1222$.

$k = \frac{1222}{26} = 47$.

Now we can find the amounts received by $2X$, $6Y$, and $8Z$.

$x = 6k = 6(47) = 282$.

$y = 12k = 12(47) = 564$.

$z = 8k = 8(47) = 376$.

We want to find how much $X$ gets, which is $2X = x = 282$. Since $X = 3k$, then $2X = 6k = 282$.

Since we are looking for the amount $X$ gets, and $X=3k$, we want to determine the value of $X$. We know $k=47$, so $X=3(47) = 141$.

3. Final Answer

141