The sum of three numbers is 2. The first number is one half of the second number, and the third number is one fourth of the second number. What is the value of the second number?

Introduction / Context:
This question is a classic example of solving a system of equations involving fractions and a total sum. You are given relationships between three numbers and their total, and you must find one of the numbers. Such problems are very common in quantitative aptitude tests because they check both your comfort with fractions and your ability to set up algebraic equations correctly.

Given Data / Assumptions:

• The sum of three numbers is 2.
• The first number is one half of the second number.
• The third number is one fourth of the second number.
• We are required to find the exact value of the second number.
• All numbers are assumed to be real, and fractional answers are allowed.

Concept / Approach:
The standard approach is to introduce a variable for the second number and then express the first and third numbers in terms of this same variable. Using the sum condition, we obtain a single equation in one variable, which can be solved by ordinary fraction manipulation. The key idea is to keep the fractions organized and combine like terms carefully to avoid arithmetic mistakes.

Step-by-Step Solution:
Step 1: Let the second number be y. Step 2: The first number is one half of the second, so the first number is (1 / 2) * y. Step 3: The third number is one fourth of the second, so the third number is (1 / 4) * y. Step 4: The sum of the three numbers is 2, so write the equation (1 / 2) * y + y + (1 / 4) * y = 2. Step 5: Combine like terms by adding the fractions: (1 / 2) + 1 + (1 / 4) = (2 / 4) + (4 / 4) + (1 / 4) = 7 / 4. Step 6: Therefore, the equation becomes (7 / 4) * y = 2. Step 7: Solve for y by multiplying both sides by 4 / 7, obtaining y = 2 * (4 / 7) = 8 / 7. Step 8: Hence, the second number is 8 / 7.

Verification / Alternative check:
Verify by computing all three numbers explicitly. The second number is 8 / 7. The first number is half of this, so it is (1 / 2) * (8 / 7) = 4 / 7. The third number is one fourth of the second, so it is (1 / 4) * (8 / 7) = 2 / 7. Add them: 4 / 7 + 8 / 7 + 2 / 7 = 14 / 7 = 2, which matches the given total. This confirms that 8 / 7 is correct.

Why Other Options Are Wrong:
If the second number were 7 / 6, the first and third numbers would be 7 / 12 and 7 / 24, and their sum would not equal 2. Similar mismatches occur for 9 / 8, 10 / 9, and 4 / 3, where the computed total sum of the three numbers will not be exactly 2. Only 8 / 7 leads to a consistent set of three numbers that satisfy both the relationship conditions and the sum condition.

Common Pitfalls:
Many learners make errors when adding fractions or when converting them to a common denominator. Another frequent issue is incorrectly interpreting phrases like one half of the second number or one fourth of the second number, which must be multiplied, not added. Being patient with fraction arithmetic and writing each step clearly helps prevent these mistakes. It is also good practice to verify the final result by substituting back into the original conditions, which can catch small calculation errors.

Final Answer:
The second number that satisfies all the conditions is 8/7.