2022 AMC 12A Problem 2

Below is the professionally curated solution for Problem 2 of the 2022 AMC 12A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2022 AMC 12A solutions, or check the answer key.

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Concepts:system of equations substitution

Difficulty rating: 1020

The sum of three numbers is $96.$ The first number is $6$ times the third number, and the third number is $40$ less than the second number. What is the absolute value of the difference between the first and second numbers?

$1$

$2$

$3$

$4$

$5$

Solution:

Let the third number be $t.$ Then the first is $6t$ and the second is $t+40.$ Their sum is $6t+(t+40)+t=8t+40=96,$ so $t=7.$

The first number is $42$ and the second is $47,$ so the difference has absolute value $5.$

Thus, the correct answer is E.

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