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Test
Math
Domain
Advanced Math
Skill
Nonlinear functions
Difficulty
Medium
ID: 203774bc
Modded SAT Question Bank by Abdullah Mallik

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

  1. 7

  2. 14

  3. 39

  4. 78


Tip: Press CTRL/Command to toggle answer
Correct Answer: B
Rationale

Choice B is correct. Let x be the first integer and let y be the second integer. If the first integer is 11 greater than twice the second integer, then x=2y+11. If the product of the two integers is 546, then xy=546. Substituting 2y+11 for x in this equation results in 2y+11y=546. Distributing the y to both terms in the parentheses results in 2y2+11y=546. Subtracting 546 from both sides of this equation results in 2y2+11y-546=0. The left-hand side of this equation can be factored by finding two values whose product is 2-546, or -1,092, and whose sum is 11. The two values whose product is -1,092 and whose sum is 11 are 39 and -28. Thus, the equation 2y2+11y-546=0 can be rewritten as 2y2+28y-39y-546=0, which is equivalent to 2yy-14+39y-14=0, or 2y+39y-14=0. By the zero product property, it follows that 2y+39=0 and y-14=0. Subtracting 39 from both sides of the equation 2y+39=0 yields 2y=-39. Dividing both sides of this equation by 2 yields y=-392. Since y is a positive integer, the value of y is not -392. Adding 14 to both sides of the equation y-14=0 yields y=14. Substituting 14 for y in the equation xy=546 yields x14=546. Dividing both sides of this equation by 14 results in x=39. Therefore, the two integers are 14 and 39, so the smaller of the two integers is 14.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect. This is the larger of the two integers.

Choice D is incorrect and may result from conceptual or calculation errors.

Question Difficulty: Medium
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