A set of five two-digit integers is given. The average of the first and last number is the middle number. The second number is half of the first number. The sum of the first three numbers is 127. The middle number is (A) and the average of the five numbers is (B). The fourth number is 62. What are the values of (A) and (B) respectively?

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**Solution:**

1. Let the first number be $2x$ and the fifth number be $2a$.

2. The middle number (third number), denoted as $A$, is the average of the first and last numbers:
   $$A = \frac{2x + 2a}{2} = x + a$$

3. The second number is half of the first number:
   $$	ext{Second number} = \frac{2x}{2} = x$$

4. The fourth number is given as 62.

5. The sum of the first three numbers is 127:
   $$2x + x + (x + a) = 127 \implies 4x + a = 127$$

6. From the options, try option (d) where $A = 64$ and average of five numbers $B = 55$.

Since $A = x + a = 64$, then $a = 64 - x$.

7. Substitute $a$ into the sum equation:
   $$4x + (64 - x) = 127 \implies 3x + 64 = 127 \implies 3x = 63 \implies x = 21$$

8. Calculate $a$:
   $$a = 64 - 21 = 43$$

9. Now, the five numbers are:
   - First: $2x = 2 	imes 21 = 42$
   - Second: $x = 21$
   - Third: $A = 64$
   - Fourth: $62$
   - Fifth: $2a = 2 	imes 43 = 86$

10. Calculate the average of the five numbers:
    $$\frac{42 + 21 + 64 + 62 + 86}{5} = \frac{275}{5} = 55$$

This matches the given average $B = 55$.

**Answer:** (A, B) = (64, 55)

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