There are two numbers such that the sum of twice the first number and thrice the second number is 100 and the sum of thrice the first number and twice the second number is 120. Which is the larger number?

## Concept & Logic When given a system of linear equations where the coefficients of the variables are swapped (i.e., $ax + by = m$ and $bx + ay = n$), the fastest way to solve them is to first add both equations together, and then subtract one from the other. This quickly yields the values of $(x + y)$ and $(x - y)$. ## Step-by-Step Solution * **Given:** Let the two numbers be $x$ and $y$. Equation 1: $2x + 3y = 100$ Equation 2: $3x + 2y = 120$ * **Calculation / Deduction:** * Add both equations together: * $(2x + 3y) + (3x + 2y) = 100 + 120$ * $5x + 5y = 220$ * Divide by 5 to simplify: $x + y = 44$ * Now, subtract Equation 1 from Equation 2: * $(3x + 2y) - (2x + 3y) = 120 - 100$ * $x - y = 20$ * Now we have a simple system: $x + y = 44$ and $x - y = 20$. * Add these new equations: $2x = 64 \Rightarrow x = 32$. * Substitute $x$ back: $32 - y = 20 \Rightarrow y = 12$. * The question asks for the larger number, which is $32$. ## Exam Strategy & Shortcut **Coefficient Symmetry Method:** Recognize the swapped coefficients ($2,3$ and $3,2$). Immediately mentally add them to get $5(x+y) = 220 \Rightarrow x+y = 44$. Mentally subtract them to get $1(x-y) = 20$. The larger number is simply half the sum of these new values: $(44 + 20) / 2 = 64 / 2 = 32$. This can be done in under 15 seconds without writing full equations. ## Common Pitfall Students often attempt to use the traditional substitution or elimination method by multiplying the first equation by 3 and the second by 2. While this works, it involves larger numbers ($300$ and $240$) and significantly increases the chance of a simple arithmetic error under time pressure. ## Final Answer **Therefore, the correct answer is 32.**