Let's solve the problem step by step.

Step 1: Define the shares of $x$, $y$, and $z$.

• Let the share of $y$ be $y$.
• The share of $x$ is $\frac{3}{5}$ of $y$, so $x = \frac{3}{5} y$.
• The ratio of the share of $y$ to $z$ is 6:11, so $\frac{y}{z} = \frac{6}{11}$, or equivalently, $z = \frac{11}{6} y$.

Step 2: Write the total sum equation.

The total sum of the shares of $x$, $y$, and $z$ is 2060, so:

$x + y + z = 2060$

Substitute the expressions for $x$ and $z$:

$\frac{3}{5} y + y + \frac{11}{6} y = 2060$

Step 3: Solve for $y$.

To simplify the equation, get rid of the fractions by multiplying through by the least common denominator (LCD), which is 30:

$30 \left( \frac{3}{5} y \right) + 30(y) + 30 \left( \frac{11}{6} y \right) = 30(2060)$

This simplifies to:

$18y + 30y + 55y = 61800$

Combine the terms:

$103y = 61800$

Now, solve for $y$:

$y = \frac{61800}{103} = 600$

Step 4: Find $x$ and $z$.

Now that we have $y = 600$, we can find $x$ and $z$:

• $x = \frac{3}{5} \times 600 = 360$
• $z = \frac{11}{6} \times 600 = 1100$

Step 5: Verify the solution.

Check if the sum of $x$, $y$, and $z$ equals 2060:

$x + y + z = 360 + 600 + 1100 = 2060$

The solution is correct.

Final Answer:

• $x = 360$
• $y = 600$
• $z = 1100$

Would you like further details or have any questions?

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Here are 5 related questions for practice:

1. Divide 1800 between three people such that the first gets $\frac{4}{7}$ of the second, and the ratio of the second to third is 5:9.
2. If $x = \frac{4}{9} y$ and $\frac{y}{z} = 7:12$, divide 2400 between them.
3. Find the shares if 3200 is divided such that $x = \frac{5}{8} y$ and the ratio of $y$ to $z$ is 2:7.
4. Divide 1540 between $x$, $y$, and $z$, where $x = \frac{2}{3} y$ and $y:z = 3:5$.
5. In a sum of 5000, the ratio of $x$ to $y$ is 2:5 and the ratio of $y$ to $z$ is 3:4.

Tip: Always check your final result by verifying that the sum of all parts matches the given total!