Among three numbers, the first is twice the second and three times the third. If the average of the three numbers is 429, what is the difference between the first and the third number?

Introduction / Context:
This question checks your understanding of averages and linear relationships between numbers. Three numbers are linked by simple multiplier relationships: the first is twice the second and three times the third. The average of the three numbers is given, and you are asked to find the difference between the first and the third number. This is a typical algebraic word problem where forming the correct equations is the key step.

Given Data / Assumptions:

- There are three numbers: first, second and third. - First number = 2 * (second number). - First number = 3 * (third number). - The average of the three numbers is 429. - We need the value of (first number) minus (third number).

Concept / Approach:
Averages relate the sum of numbers to the number of terms. If the average of three numbers is known, their total sum is simply 3 times the average. Using the given multiplier relationships, we can express all three numbers in terms of a single variable representing the first number. Substituting into the average formula produces an equation that we solve for this first number. Then we compute the third number and finally their difference. Expressing everything in one variable keeps the algebra straightforward.

Step-by-Step Solution:
Step 1: Let the first number be a. Step 2: The first is twice the second, so the second number is a / 2. Step 3: The first is three times the third, so the third number is a / 3. Step 4: The average of the three numbers is 429, so (a + a / 2 + a / 3) / 3 = 429. Step 5: Combine terms in the numerator using a common denominator. The coefficients are 1, 1/2 and 1/3. Their sum is 1 + 1/2 + 1/3 = 6/6 + 3/6 + 2/6 = 11/6. Step 6: So the numerator is (11/6) * a, and the average is [(11/6) * a] / 3 = (11/18) * a. Step 7: Set this equal to 429: (11/18) * a = 429. Step 8: Solve for a: a = 429 * 18 / 11. Step 9: Compute 429 / 11 = 39, because 11 * 39 = 429. Step 10: Then a = 39 * 18 = 702. Step 11: The first number is 702. The third number is a / 3 = 702 / 3 = 234. Step 12: The required difference is first minus third = 702 - 234 = 468.

Verification / Alternative check:
Compute all three numbers explicitly. First = 702. Second = a / 2 = 351. Third = 234. Check the relationships: 702 is indeed twice 351 and three times 234. Now check the average: (702 + 351 + 234) / 3 = 1287 / 3 = 429, which matches the given average. This confirms that the numbers are correct and the difference 468 is accurate.

Why Other Options Are Wrong:
Values 412, 517 and 427 come from incorrect algebra or arithmetic. For example, a mistake in summing the fractional coefficients or mishandling division by 11 can easily produce such wrong differences. None of these values correspond to the difference between 702 and 234, which is the only pair that satisfies all conditions of the problem.

Common Pitfalls:
Students sometimes mix up which number is multiple of which, for example treating the second as twice the first instead of the other way around. Another frequent error is in handling the fractions when adding 1, 1/2 and 1/3, or forgetting to divide the sum by 3 when using the average formula. Writing each step clearly, especially the conversion to a single variable, greatly reduces the chance of such errors.

Final Answer:
The difference between the first and the third number is 468, which matches option B.