Divide 2060 between three people x, y and z such that x gets the three fifth of what y gets and the ratio of the share of y to z is 6:11

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!

Learn how to divide 2060 among three people x, y, and z, where x gets three-fifths of what y gets, and the ratio of y to z is 6:11. This problem covers ratio and proportion concepts and linear equations. Suitable for students in Grades 8-10.

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