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2002 AMC 10A Problems/Problem 1: Difference between revisions

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New page: ==Problem== The ratio <math>\frac{10^{2000}+10^{2002}}{10^{2001}+10^{2001}}</math> is closest to which of the following numbers? <math>\text{(A)}\ 0.1 \qquad \text{(B)}\ 0.2 \qquad \text{...
 
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The ratio <math>\frac{10^{2000}+10^{2002}}{10^{2001}+10^{2001}}</math> is closest to which of the following numbers?
The ratio <math>\frac{10^{2000}+10^{2002}}{10^{2001}+10^{2001}}</math> is closest to which of the following numbers?


<math>\text{(A)}\ 0.1 \qquad \text{(B)}\ 0.2 \qquad \text{(C)}\ 1 \qquad \text{(D)}\ 5 \qquad \text{(E)}\ 10</math>
<math>\textbf{(A)}\ 0.1 \qquad \textbf{(B)}\ 0.2 \qquad \textbf{(C)}\ 1 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 10</math>


==Solution==
==Solution==
We factor <math>\frac{10^{2000}+10^{2002}}{10^{2001}+10^{2001}}</math> as <math>\frac{10^{2000}(1+100)}{10^{2001}(1+1)}=\frac{101}{20}</math>. As <math>\frac{101}{20}=5.05</math>, our answer is <math>\text{(D)}\ 5 \qquad</math>.
We factor <math>\frac{10^{2000}+10^{2002}}{10^{2001}+10^{2001}}</math> as <math>\frac{10^{2000}(1+100)}{10^{2001}(1+1)}=\frac{101}{20}</math>. As <math>\frac{101}{20}=5.05</math>, our answer is <math>\boxed{\textbf{(D)}\ 5 }</math>.
 
==Video Solution by Daily Dose of Math==
 
https://youtu.be/VEW3sYvitGI
 
~Thesmartgreekmathdude
 
==See Also==
{{AMC10 box|year=2002|ab=A|before=First question|num-a=2}}
 
[[Category:Introductory Number Theory Problems]]
{{MAA Notice}}

Latest revision as of 23:38, 18 July 2024

Problem

The ratio $\frac{10^{2000}+10^{2002}}{10^{2001}+10^{2001}}$ is closest to which of the following numbers?

$\textbf{(A)}\ 0.1 \qquad \textbf{(B)}\ 0.2 \qquad \textbf{(C)}\ 1 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 10$

Solution

We factor $\frac{10^{2000}+10^{2002}}{10^{2001}+10^{2001}}$ as $\frac{10^{2000}(1+100)}{10^{2001}(1+1)}=\frac{101}{20}$. As $\frac{101}{20}=5.05$, our answer is $\boxed{\textbf{(D)}\ 5 }$.

Video Solution by Daily Dose of Math

https://youtu.be/VEW3sYvitGI

~Thesmartgreekmathdude

See Also

2002 AMC 10A (Problems • Answer Key • Resources)
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First question 		Followed by
Problem 2
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All AMC 10 Problems and Solutions

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