1991 AMC 8 Problem 1

Below is the professionally curated solution for Problem 1 of the 1991 AMC 8, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1991 AMC 8 solutions, or check the answer key.

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

Concepts:place value

Difficulty rating: 450

1.

What is the value of 1,000,000,000,000777,777,777,777? \begin{aligned} &1{,}000{,}000{,}000{,}000 \\ &\quad {}- 777{,}777{,}777{,}777? \end{aligned}

222,222,222,222222{,}222{,}222{,}222

222,222,222,223222{,}222{,}222{,}223

233,333,333,333233{,}333{,}333{,}333

322,222,222,223322{,}222{,}222{,}223

333,333,333,333333{,}333{,}333{,}333

Solution:

Lining up the subtraction gives 1,000,000,000,000777,777,777,777=222,222,222,223. \begin{aligned} &1{,}000{,}000{,}000{,}000 \\ &\quad {}- 777{,}777{,}777{,}777 \\ &= 222{,}222{,}222{,}223. \end{aligned}

Equivalently, to climb from 777,777,777,777777{,}777{,}777{,}777 up to 1,000,000,000,0001{,}000{,}000{,}000{,}000 you add 33 in the units place and 22 in each of the other eleven places.

Thus, the correct answer is B .

Full ExamProblem 2#2 →

Problem 1 in Other Years

1985 AMC 8 · 1986 AMC 8 · 1987 AMC 8 · 1988 AMC 8 · 1989 AMC 8 · 1990 AMC 8 · 1992 AMC 8 · 1993 AMC 8 · 1994 AMC 8 · 1995 AMC 8 · 1996 AMC 8 · 1997 AMC 8 · 1998 AMC 8 · 1999 AMC 8 · 2000 AMC 8 · 2001 AMC 8 · 2002 AMC 8 · 2003 AMC 8 · 2004 AMC 8 · 2005 AMC 8 · 2006 AMC 8 · 2007 AMC 8 · 2008 AMC 8 · 2009 AMC 8 · 2010 AMC 8 · 2011 AMC 8 · 2012 AMC 8 · 2013 AMC 8 · 2014 AMC 8 · 2015 AMC 8 · 2016 AMC 8 · 2017 AMC 8 · 2018 AMC 8 · 2019 AMC 8 · 2020 AMC 8 · 2022 AMC 8 · 2023 AMC 8 · 2024 AMC 8 · 2025 AMC 8 · 2026 AMC 8