2006 AMC 12A Problems/Problem 9

Problem

Oscar buys $13$ pencils and $3$ erasers for $$1.00$. A pencil costs more than an eraser, and both items cost a [[whole number]] of cents. What is the total cost, in cents, of one pencil and one eraser?$ \mathrm{(A) \ } 10\qquad \mathrm{(B) \ } 12\qquad \mathrm{(C) \ } 15\qquad \mathrm{(D) \ } 18$$ (Error compiling LaTeX. ! Missing $ inserted.)\mathrm{(E) \ } 20$== Solution == Let the price of a pencil be$p$and an eraser$e$.  Then$13p + 3e = 100$with$p > e > 0$.  Since$p$and$e$are [[positive integer]]s, we must have$e \geq 1$and$p \geq 2$.

Considering the [[equation]]$ (Error compiling LaTeX. ! Missing $ inserted.)13p + 3e = 100$[[modulo]] 3 (that is, comparing the [[remainder]]s when both sides are divided by 3) we have$p + 0e \equiv 1 \pmod 3$so$p$leaves a remainder of 1 on division by 3.

Since$ (Error compiling LaTeX. ! Missing $ inserted.)p \geq 2$, possible values for$p$are 4, 7, 10 ....

Since 13 pencils cost less than 100 cents,$ (Error compiling LaTeX. ! Missing $ inserted.)13p < 100$.$13 \times 10 = 130$is too high, so$p$must be 4 or 7.

If$ (Error compiling LaTeX. ! Missing $ inserted.)p = 4$then$13p = 52$and so$3e = 48$giving$e = 16$.  This contradicts the pencil being more expensive.  The only remaining value for$p$is 7; then the 13 pencils cost$7 \times 13= 91$cents and so the 3 erasers together cost 9 cents and each eraser costs$\frac{9}{3} = 3$cents.

Thus one pencil plus one eraser cost$ (Error compiling LaTeX. ! Missing $ inserted.)7 + 3 = 10$cents, which is answer choice$\mathrm{(A) \ }$.

See also

2006 AMC 12A (ProblemsAnswer KeyResources)
Preceded by
Problem 8
Followed by
Problem 10
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All AMC 12 Problems and Solutions
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