Difference between revisions of "2011 AMC 10B Problems/Problem 24"
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<math> \textbf{(A)}\ \frac{51}{101} \qquad\textbf{(B)}\ \frac{50}{99} \qquad\textbf{(C)}\ \frac{51}{100} \qquad\textbf{(D)}\ \frac{52}{101} \qquad\textbf{(E)}\ \frac{13}{25}</math> | <math> \textbf{(A)}\ \frac{51}{101} \qquad\textbf{(B)}\ \frac{50}{99} \qquad\textbf{(C)}\ \frac{51}{100} \qquad\textbf{(D)}\ \frac{52}{101} \qquad\textbf{(E)}\ \frac{13}{25}</math> | ||
− | ==Solution== | + | ==Solution 1== |
We see that for the graph of <math>y=mx+2</math> to not pass through any lattice points, the denominator of <math>m</math> must be greater than <math>100</math>, or else it would be canceled by some <math>0<x\le100</math> which would make <math>y</math> an integer. By using common denominators, we find that the order of the fractions from smallest to largest is <math>(A), (B), (C), (D), (E)</math>. We can see that when <math>m=\frac{50}{99}</math>, <math>y</math> would be an integer, so therefore any fraction greater than <math>\frac{50}{99}</math> would not work, as substituting our fraction <math>\frac{50}{99}</math> for <math>m</math> would produce an integer for <math>y</math>. So now we are left with only <math>\frac{51}{101}</math> and <math>\frac{50}{99}</math>. But since <math>\frac{51}{101}=\frac{5049}{9999}</math> and <math>\frac{50}{99}=\frac{5050}{9999}</math>, we can be absolutely certain that there isn't a number between <math>\frac{51}{101}</math> and <math>\frac{50}{99}</math> that can reduce to a fraction whose denominator is less than or equal to <math>100</math>. Since we are looking for the maximum value of <math>a</math>, we take the larger of <math>\frac{51}{101}</math> and <math>\frac{50}{99}</math>, which is <math>\boxed{\textbf{(B)}\frac{50}{99}}</math>. | We see that for the graph of <math>y=mx+2</math> to not pass through any lattice points, the denominator of <math>m</math> must be greater than <math>100</math>, or else it would be canceled by some <math>0<x\le100</math> which would make <math>y</math> an integer. By using common denominators, we find that the order of the fractions from smallest to largest is <math>(A), (B), (C), (D), (E)</math>. We can see that when <math>m=\frac{50}{99}</math>, <math>y</math> would be an integer, so therefore any fraction greater than <math>\frac{50}{99}</math> would not work, as substituting our fraction <math>\frac{50}{99}</math> for <math>m</math> would produce an integer for <math>y</math>. So now we are left with only <math>\frac{51}{101}</math> and <math>\frac{50}{99}</math>. But since <math>\frac{51}{101}=\frac{5049}{9999}</math> and <math>\frac{50}{99}=\frac{5050}{9999}</math>, we can be absolutely certain that there isn't a number between <math>\frac{51}{101}</math> and <math>\frac{50}{99}</math> that can reduce to a fraction whose denominator is less than or equal to <math>100</math>. Since we are looking for the maximum value of <math>a</math>, we take the larger of <math>\frac{51}{101}</math> and <math>\frac{50}{99}</math>, which is <math>\boxed{\textbf{(B)}\frac{50}{99}}</math>. | ||
+ | |||
+ | ==Solution 2== | ||
+ | We want to find the smallest <math>m</math> such that there will be an integral solution to <math>y=mx+2</math> with <math>0<x\le100</math>. We first test A, but since the denominator has a <math>101</math>, <math>x</math> must be a nonzero multiple of <math>101</math>, but it then will be greater than <math>100</math>. We then test B. <math>y=\frac{50}{99}x+2</math> yields the solution <math>(99,52)</math> which satisfies <math>0<x\le100</math>. We know that MAA orders the answers in ascending order, so therefore we know that the smallest possible <math>a</math> must be <math>\frac{50}{99}\implies\boxed{\textbf{(B)}}</math> | ||
==See Also== | ==See Also== | ||
{{AMC10 box|year=2011|ab=B|num-a=25|num-b=23}} | {{AMC10 box|year=2011|ab=B|num-a=25|num-b=23}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 01:30, 26 June 2018
Contents
[hide]Problem
A lattice point in an -coordinate system is any point
where both
and
are integers. The graph of
passes through no lattice point with
for all
such that
. What is the maximum possible value of
?
Solution 1
We see that for the graph of to not pass through any lattice points, the denominator of
must be greater than
, or else it would be canceled by some
which would make
an integer. By using common denominators, we find that the order of the fractions from smallest to largest is
. We can see that when
,
would be an integer, so therefore any fraction greater than
would not work, as substituting our fraction
for
would produce an integer for
. So now we are left with only
and
. But since
and
, we can be absolutely certain that there isn't a number between
and
that can reduce to a fraction whose denominator is less than or equal to
. Since we are looking for the maximum value of
, we take the larger of
and
, which is
.
Solution 2
We want to find the smallest such that there will be an integral solution to
with
. We first test A, but since the denominator has a
,
must be a nonzero multiple of
, but it then will be greater than
. We then test B.
yields the solution
which satisfies
. We know that MAA orders the answers in ascending order, so therefore we know that the smallest possible
must be
See Also
2011 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 23 |
Followed by Problem 25 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
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