There are two real numbers such that the sum of twice the first number and three times the second number is 39, while the sum of three times the first number and twice the second number is 36. What is the value of the larger of these two numbers?

Introduction:
This is a classic pair of simultaneous linear equations in two variables. The question gives two different weighted sums of the same two unknown numbers and asks for the larger number. Such problems are very common in quantitative aptitude and test your ability to form equations, solve them systematically, and correctly identify which number is larger.

Given Data / Assumptions:

• Let the first number be x.
• Let the second number be y.
• Equation 1: 2x + 3y = 39.
• Equation 2: 3x + 2y = 36.
• We must find the larger of x and y.

Concept / Approach:
We use standard methods for solving simultaneous equations, such as elimination or substitution. Elimination is usually faster here: multiply and subtract the equations to remove one variable, find the other, then back-substitute to find the remaining variable. After computing both values, compare them to identify the larger number.

Step-by-Step Solution:
Start with 2x + 3y = 39 … (1)And 3x + 2y = 36 … (2)Multiply (1) by 3: 6x + 9y = 117.Multiply (2) by 2: 6x + 4y = 72.Subtract second from first: (6x + 9y) - (6x + 4y) = 117 - 72.5y = 45 => y = 9.Substitute into (2): 3x + 2*9 = 36 => 3x + 18 = 36 => 3x = 18 => x = 6.Thus, the larger number is 9.

Verification / Alternative check:
Check equation (1): 2*6 + 3*9 = 12 + 27 = 39, correct. Check equation (2): 3*6 + 2*9 = 18 + 18 = 36, also correct. So (x, y) = (6, 9) satisfies both equations.

Why Other Options Are Wrong:
3 and 6: These are the individual numbers or less than them but do not represent the larger one.12 and 15: Substituting these into the equations fails to satisfy both equations simultaneously.

Common Pitfalls:
Making arithmetic mistakes while multiplying or subtracting equations.Stopping after finding only one variable and not checking the second equation.Accidentally reporting the smaller number instead of the larger one.

Final Answer:
9