The product of two positive integers is 546. If the first integer is 11 greater than twice the second integer,...

1. TRANSLATE the problem information

• Given information:
  • Product of two positive integers = 546
  • First integer = 11 greater than twice the second integer
  • Need to find: the smaller integer

• Let y = second integer, x = first integer
• This gives us: \(\mathrm{xy = 546}\) and \(\mathrm{x = 2y + 11}\)

2. INFER the solving strategy

• We have two equations with two unknowns
• Since the second equation expresses x in terms of y, substitute this into the first equation
• This will give us one equation with only y, which we can solve

3. SIMPLIFY by substitution and expansion

• Substitute \(\mathrm{x = 2y + 11}\) into \(\mathrm{xy = 546}\):
  \(\mathrm{(2y + 11)y = 546}\)
• Expand: \(\mathrm{2y^2 + 11y = 546}\)
• Rearrange: \(\mathrm{2y^2 + 11y - 546 = 0}\)

4. SIMPLIFY the quadratic equation

• Need to factor \(\mathrm{2y^2 + 11y - 546 = 0}\)
• Look for two numbers that multiply to \(\mathrm{(2)(-546) = -1092}\) and add to 11
• Those numbers are 39 and -28: \(\mathrm{(39)(-28) = -1092}\) and \(\mathrm{39 + (-28) = 11}\)
• Rewrite: \(\mathrm{2y^2 + 39y - 28y - 546 = 0}\)
• Factor by grouping: \(\mathrm{(y - 14)(2y + 39) = 0}\)

5. APPLY CONSTRAINTS to select valid solution

• From \(\mathrm{(y - 14)(2y + 39) = 0}\), we get \(\mathrm{y = 14}\) or \(\mathrm{y = -39/2}\)
• Since we need positive integers, \(\mathrm{y = 14}\)
• Find x: \(\mathrm{x = 2(14) + 11 = 39}\)
• Verify: \(\mathrm{39 \times 14 = 546}\) ✓

Answer: B. 14

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students struggle to correctly set up the relationship "11 greater than twice the second integer." They might write \(\mathrm{x = 11 + 2y}\) instead of \(\mathrm{x = 2y + 11}\), or confuse which integer is which.

This algebraic setup error propagates through the entire solution, leading to the wrong quadratic equation and ultimately confusion about which answer choice to select.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{2y^2 + 11y - 546 = 0}\) but make errors in the factoring process. They might struggle to find the correct factor pair for -1092, or make arithmetic mistakes during the grouping and factoring steps.

This may lead them to select Choice A (7) if they incorrectly solve the quadratic, or causes them to get stuck and guess.

The Bottom Line:

This problem tests whether students can correctly translate complex word relationships into algebra and then execute multi-step algebraic manipulations without computational errors. The factoring of a quadratic with large coefficients is particularly challenging.