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A pen costs p dollars and a notebook costs q dollars. What is the cost of 5 pens and 5 notebooks?
(1) The cost of 4 pens and 1 notebook is $13.
(2) The cost of 1 pen and 4 notebooks is $12.
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Correct answer
C
The question asks for 5p + 5q = 5(p + q); we need p + q, not p and q separately.
Statement (1): 4p + q = 13 is one equation in two unknowns. The sum p + q is free, (p, q) = (3, 1) gives p + q = 4; (2, 5) gives p + q = 7. Not sufficient.
Statement (2): p + 4q = 12 is again one equation in two unknowns, and the sum p + q is free here too, (p, q) = (8, 1) gives p + q = 9; (4, 2) gives p + q = 6. Both satisfy p + 4q = 12 (8 + 4 = 12 and 4 + 8 = 12) yet give different totals, so the asked quantity moves. Not sufficient.
Together: add the two equations: (4p + q) + (p + 4q) = 13 + 12, so 5p + 5q = 25. The asked quantity, 5p + 5q, is recovered directly as 25, without ever solving for p or q individually. Together sufficient; neither alone.
The "I must find p and q" over-solve, and the inverse "neither equation pins the total" trap (combined-sufficient illusion misjudged as never-sufficient). The lever is recognizing that the asked quantity is exactly the sum of the two left-hand sides, adding the asymmetric equations lands the symmetric target in one move. A solver who instead solves the 2×2 system finds p = 8/3 and q = 7/3, so p + q = 5, reaching the same 5(p + q) = 25 but spending longer.
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