Of the three numbers, the sum of the first two is 73; the sum of the second and the third is 77 and the sum of the third and thrice the first is 104. The third number is

### Concept & Strategy This requires solving a system of three linear equations. The fastest approach is to eliminate variables by subtracting equations that share a common term. ### Step-by-Step Solution * **Given:** Let the numbers be $x$, $y$, and $z$. Eq 1: $x + y = 73$ Eq 2: $y + z = 77$ Eq 3: $z + 3x = 104$ * **Calculation:** Subtract Eq 1 from Eq 2 to eliminate $y$. $(y + z) - (x + y) = 77 - 73$ $z - x = 4 \implies z = x + 4$ * Substitute this expression for $z$ into Eq 3. $(x + 4) + 3x = 104$ $4x + 4 = 104$ $4x = 100 \implies x = 25$ * **Deduction:** Now, substitute $x = 25$ back into our expression for $z$. $z = 25 + 4 = 29$ The first number is $25$, the second is $48$, and the third is $29$. Since $29$ is not listed among the first three options, the correct choice is "None of these". ### Exam Strategy & Shortcut Instead of finding all three variables, strictly focus on the target. You need $z$. From $x + y = 73$ and $y + z = 77$, the difference is clearly $z - x = 4$. Rearranging gives $x = z - 4$. Substitute into $z + 3x = 104$: $z + 3(z - 4) = 104 \implies 4z - 12 = 104 \implies 4z = 116 \implies z = 29$. This prevents calculating unneeded variables. ### Common Pitfall A common trap is solving for the first variable ($x = 25$) and accidentally selecting option (a) because it matches, completely forgetting that the question specifically asks for the *third* number. Always reread the final prompt. ### Final Answer **Therefore, the correct answer is None of these.**