The sum of one-half, one-third and one-fourth of a positive number is greater than the number itself by 22. What is the value of this number?

Introduction / Context:
This problem involves forming and solving a linear equation using fractions of an unknown number. Such questions are standard in aptitude tests to check whether a learner can translate a verbal description involving fractions into an algebraic equation, and then solve it correctly. Here, three fractional parts of a number are added, and the result is said to exceed the original number by a fixed amount. Our goal is to find that number using basic algebra and fraction operations.

Given Data / Assumptions:

• Let the unknown positive number be represented by N.
• One-half of the number is N/2.
• One-third of the number is N/3.
• One-fourth of the number is N/4.
• The sum of these three fractional parts exceeds the original number N by 22.

Concept / Approach:
We convert the English statement into an equation. The phrase "sum of one-half, one-third and one-fourth of a number exceeds the number by 22" can be written as N/2 + N/3 + N/4 = N + 22. Then we simplify by finding a common denominator for the fractions and reducing the equation to a simple linear equation in N. Finally, we solve for N and verify that the result matches one of the provided options.

Step-by-Step Solution:
Step 1: Let the number be N. Step 2: Translate the condition into an equation: N/2 + N/3 + N/4 = N + 22. Step 3: Find the least common denominator of 2, 3 and 4, which is 12. Step 4: Express each term with denominator 12: N/2 = 6N/12, N/3 = 4N/12, and N/4 = 3N/12. Step 5: Add the fractions: N/2 + N/3 + N/4 = (6N + 4N + 3N) / 12 = 13N/12. Step 6: The equation becomes 13N/12 = N + 22. Step 7: Multiply both sides by 12 to clear the denominator: 13N = 12N + 264. Step 8: Subtract 12N from both sides: N = 264. Step 9: Therefore, the required number is 264.

Verification / Alternative check:
To verify, substitute N = 264 back into the original statement. One-half of 264 is 132, one-third is 88, and one-fourth is 66. Their sum is 132 + 88 + 66 = 286. The original number is 264, and 286 minus 264 equals 22. This matches the given condition that the sum exceeds the number by 22, so N = 264 is correct.

Why Other Options Are Wrong:
Option (b) 284: Fractions would sum to a value that does not exceed 284 by exactly 22.
Option (c) 215: The difference between the sum of fractions and 215 is not 22.
Option (d) 302: Again, the resulting difference is not exactly 22, so it does not satisfy the condition.

Common Pitfalls:
Learners sometimes misinterpret the phrase "exceeds the number by 22" and set up the equation with 22 on the wrong side or with the wrong sign. Others add the fractions incorrectly by not using a common denominator or by making arithmetic errors in the numerator. Writing and simplifying the equation carefully helps avoid these mistakes.

Final Answer:
The value of the number is 264.