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We are given the following information:

1. The sum of two numbers is 132.
2. One-third of the smaller number exceeds one-sixth of the larger number by 8.

Let:
- \( x \) be the smaller number.
- \( y \) be the larger number.

From the given information, we can write two equations:

### Equation 1:
\[
x + y = 132
\]

### Equation 2:
\[
\frac{x}{3} = \frac{y}{6} + 8
\]

We can solve this system of equations step-by-step.

### Step 1: Solve Equation 1 for \( y \)
\[
y = 132 - x
\]

### Step 2: Substitute \( y = 132 - x \) into Equation 2
\[
\frac{x}{3} = \frac{132 - x}{6} + 8
\]

### Step 3: Multiply through by 6 to eliminate the denominators
\[
2x = 132 - x + 48
\]

### Step 4: Simplify and solve for \( x \)
\[
2x + x = 132 + 48
\]
\[
3x = 180
\]
\[
x = \frac{180}{3} = 60
\]

### Step 5: Substitute \( x = 60 \) back into Equation 1 to find \( y \)
\[
60 + y = 132
\]
\[
y = 132 - 60 = 72
\]

Thus, the two numbers are:
- The smaller number \( x = 60 \)
- The larger number \( y = 72 \)

Would you like more details, or do you have any questions?

### Related Questions:
1. Can you derive other possible systems of equations from a similar problem?
2. How would the solution change if one-third of the smaller number exceeded one-sixth of the larger by 10 instead of 8?
3. What if the sum of the two numbers was a different value, say 150? How would you approach the problem?
4. Can you verify the solution by substituting the values back into both equations?
5. How does solving a system of equations graphically differ from solving it algebraically?

### Tip:
When solving systems of equations, always check your solutions by substituting the values back into the original equations to verify accuracy.

The sum of two numbers is 132. If one-third of the smaller exceeds one-sixth of the larger by 8, find the numbers.

Learn how to solve the system of equations where the sum of two numbers equals 132, and one-third of the smaller exceeds one-sixth of the larger by 8. This problem covers key algebraic concepts and methods suitable for students in Grades 7-9.

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