The sum of the one-third, one-fourth, and one-fifth parts of a number exceeds half the number by 34. Find the number.

Introduction: This is a fractional linear equation in a single variable. We will convert each verbal fraction into an algebraic expression in terms of the unknown number and solve.

Given Data / Assumptions:

• Let the number be N.
• (1/3)N + (1/4)N + (1/5)N exceeds (1/2)N by 34.
• N is real; the equation will produce a clean integer.

Concept / Approach: Bring all N-terms to one side to isolate the net fraction of N equal to 34. Use a common denominator to combine terms exactly.

Step-by-Step Solution:

Verification / Alternative check: Compute both sides with N = 120: LHS partials = 40 + 30 + 24 = 94; half of N = 60; 94 exceeds 60 by 34, correct.

Why Other Options Are Wrong: 60, 30, 90 do not satisfy the equation; “None of these” is unnecessary because a valid integer solution exists.

Common Pitfalls: Arithmetic with fractions, especially signs when moving 1/2N to the left; ensure exact common denominator.

Final Answer: 120