Difficulty: Medium
Correct Answer: 32
Explanation:
Introduction / Context:
This is a linear equations problem in two variables. It tests your ability to translate word statements into equations and then solve the simultaneous equations to determine the values of the unknown numbers.
Given Data / Assumptions:
Concept / Approach:
We solve the pair of linear equations using elimination or substitution. By eliminating one variable, we can quickly find the other. Once both x and y are known, we simply compare them to see which is larger.
Step-by-Step Solution:
Step 1: Write the equations clearly: 2x + 3y = 100 (1) and 3x + 2y = 120 (2).Step 2: Multiply equation (1) by 3: 6x + 9y = 300.Step 3: Multiply equation (2) by 2: 6x + 4y = 240.Step 4: Subtract the new equation (2) from the new equation (1): (6x + 9y) - (6x + 4y) = 300 - 240.Step 5: This simplifies to 5y = 60, so y = 60 / 5 = 12.Step 6: Substitute y = 12 into 2x + 3y = 100: 2x + 3 * 12 = 100 ⇒ 2x + 36 = 100 ⇒ 2x = 64 ⇒ x = 32.Step 7: Therefore, the first number is 32 and the second number is 12, so the larger number is 32.
Verification / Alternative check:
Check equation (2): 3x + 2y = 3 * 32 + 2 * 12 = 96 + 24 = 120, which matches the given data.Both equations are satisfied, confirming that x = 32 and y = 12 are correct.
Why Other Options Are Wrong:
24, 26 and 38 are distractor values that may arise from solving only one equation or from arithmetic mistakes in elimination.Substituting any of these values in place of 32 fails to satisfy both equations simultaneously.
Common Pitfalls:
Errors often occur in multiplying or subtracting equations, especially when handling coefficients.Some learners may incorrectly assume a value for one variable and not verify it in both original equations.Using a systematic elimination method and always checking the solution in both equations avoids these mistakes.
Final Answer:
The larger number is 32.
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