Problem:
Out of a pair of complementary angles, one is 2/3 of the other. Find the angles.

Solution:

Step 1: Understand the problem
We are given a pair of complementary angles, and we need to find the measures of these angles. Additionally, we are provided with the information that one angle is 2/3 of the other.

Step 2: Define the problem
Let's assume one angle is x degrees. Since the other angle is complementary to the first angle, its measure can be expressed as (90 - x) degrees. According to the given information, we know that one angle is 2/3 of the other. Therefore, we can write the equation:

x = (2/3)(90 - x)

Step 3: Solve the equation
To solve the equation, we can simplify it by multiplying both sides by 3 to eliminate the fraction:

3x = 2(90 - x)

Next, we can distribute the 2 on the right side of the equation:

3x = 180 - 2x

Now, we can combine like terms by adding 2x to both sides:

3x + 2x = 180

This simplifies to:

5x = 180

Finally, we can solve for x by dividing both sides by 5:

x = 36

Step 4: Find the measures of the angles
Now that we know the value of x, we can substitute it back into the equation to find the measures of the angles:

One angle = x = 36 degrees
Other angle = 90 - x = 90 - 36 = 54 degrees

Step 5: Verify the solution
To verify our solution, we can check if the angles are indeed complementary and if one is 2/3 of the other.

Complementary angles add up to 90 degrees:
36 + 54 = 90

One angle is 2/3 of the other:
36 = (2/3)(54)

Both conditions are satisfied, confirming that our solution is correct.

Conclusion:
The pair of complementary angles is 36 degrees and 54 degrees.