Maths Olympiad Prep

Track / Stage 3 / 128 of 260 #128 of 1964

Problem 128

AMC 10/12, early questions
Number theory Difficulty 3.5 Find the answer

The fraction 13\frac{1}{3}:
(A) equals 0.33333333(B) is less than 0.33333333 by 13⋅108(C) is less than 0.33333333 by 13⋅109(D) is greater than 0.33333333 by 13⋅108(E) is greater than 0.33333333 by 13⋅109\textbf{(A)}\ \text{equals 0.33333333}\qquad\textbf{(B)}\ \text{is less than 0.33333333 by }\frac{1}{3\cdot 10^8}\\ \textbf{(C)}\ \text{is less than 0.33333333 by }\frac{1}{3\cdot 10^9}\\ \textbf{(D)}\ \text{is greater than 0.33333333 by }\frac{1}{3\cdot 10^8}\\ \textbf{(E)}\ \text{is greater than 0.33333333 by }\frac{1}{3\cdot 10^9}

This was a multiple-choice question, but the options didn't survive into the source we have, so there is nothing here to pick from. Work it on paper and mark yourself against the solution below.

Official solution

333333333108−13  ⟹  3⋅107+3⋅106+⋯+3⋅100108−13  ⟹  3(108−1)9⋅108−13  ⟹  108−13⋅108−13  ⟹  1083⋅108−13⋅108−13  ⟹  −13⋅108\frac{333333333}{10^8}-\frac{1}{3}\implies \frac{3\cdot 10^7+3\cdot 10^6+\dots+3\cdot 10^0}{10^8}-\frac{1}{3}\implies\frac{3(10^{8}-1)}{9\cdot 10^{8}}-\frac{1}{3}\implies\frac{10^8-1}{3\cdot 10^8}-\frac{1}{3}\implies\frac{10^8}{3\cdot 10^8}-\frac{1}{3\cdot 10^8}-\frac{1}{3}\implies\frac{-1}{3\cdot 10^8}
Because we did .333333333−13.333333333-\frac{1}{3}, it is B\fbox{B}

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.