2021 AMC 12A Spring Problem 3

Below is the professionally curated solution for Problem 3 of the 2021 AMC 12A Spring, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2021 AMC 12A Spring solutions, or check the answer key.

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Concepts:place value linear equation

Difficulty rating: 1120

The sum of two natural numbers is $17{,}402.$ One of the two numbers is divisible by $10.$ If the units digit of that number is erased, the other number is obtained. What is the difference of these two numbers?

$10{,}272$

$11{,}700$

$13{,}362$

$14{,}238$

$15{,}426$

Solution:

The larger number ends in $0,$ and erasing that digit divides it by $10$ to give the smaller number. So the larger number is $10$ times the smaller. Writing the smaller number as $x,$ the sum is $x + 10x = 11x = 17{,}402,$ giving $x = 1{,}582.$

The two numbers are $1{,}582$ and $15{,}820,$ whose difference is $15{,}820 - 1{,}582 = 14{,}238.$

Thus, the correct answer is D.

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