The average of the three numbers a, b and c is 20, and the average of the three numbers b, c and d is 25. If d = 30, then what is the value of a?

Introduction / Context:
This algebraic average problem relates the sums of overlapping triplets of numbers. We know the averages of (a, b, c) and (b, c, d) and the actual value of d. Using these, we can find the value of a.

Given Data / Assumptions:

• (a + b + c) / 3 = 20.
• (b + c + d) / 3 = 25.
• d = 30.
• We need to find a.

Concept / Approach:
From the given averages, we can find the sums:

• a + b + c = 3 * 20 = 60.
• b + c + d = 3 * 25 = 75.

We use the known value of d to find b + c from the second sum, then substitute into the first to find a.

Step-by-Step Solution:
Step 1: Convert averages to sums. a + b + c = 3 * 20 = 60. b + c + d = 3 * 25 = 75. Step 2: Use d = 30 in the second equation. b + c + 30 = 75. b + c = 75 - 30 = 45. Step 3: Substitute b + c into the first equation to find a. a + (b + c) = 60. a + 45 = 60. a = 60 - 45 = 15.

Verification / Alternative check:
Using a = 15, b + c = 45 and d = 30, we see that:

• a + b + c = 15 + 45 = 60, giving an average of 60 / 3 = 20.
• b + c + d = 45 + 30 = 75, giving an average of 75 / 3 = 25.

Both averages match the given data, confirming that a = 15 is correct.

Why Other Options Are Wrong:
If a were 25, then a + b + c would be 25 + 45 = 70, contradicting the required total of 60. Similarly, a = 45 or a = 30 gives sums that do not match 60. Only a = 15 makes both average equations consistent with the given value of d.

Common Pitfalls:
Some students attempt to average 20 and 25 or treat a and d as symmetric, which they are not. Others might miscalculate the sums when converting from averages. Always convert each average to its corresponding total sum and use substitution carefully to isolate the unknown variable.

Final Answer:
The value of a is 15.