2002 AMC 10A Problem 16 — Solution | LIVE by Po-Shen Loh

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2002 AMC 10A Problem 16

Attempt Problem 16 of the 2002 AMC 10A below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2002 AMC 10A solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

16.

If $a+1$ $=b+2$ $=c+3$ $=d+4$ $=a+b+c+d+5,$ then $a+b+c+d$ is

$-5$

$-\dfrac{10}{3}$

$-\dfrac{7}{3}$

$\dfrac{5}{3}$

$5$

Answer: B

Concepts:system of equations substitution

Difficulty rating: 1330

Solution:

Let the common value be $k.$ Then $a=k-1,$ $b=k-2,$ $c=k-3,$ $d=k-4,$ so $a+b+c+d=4k-10.$

Since $a+b+c+d+5=k,$ we get $4k-10+5=k,$ so $3k=5$ and $k=\dfrac{5}{3}.$ Then $a+b+c+d=k-5$ $=\dfrac{5}{3}-5=-\dfrac{10}{3}.$

Thus, the correct answer is B.

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