An Annville Junior High School, \(30\%\) of the students in the Math Club are in the Science Club, and \(80\%\) of the students in the Science Club are in the Math Club. There are \(15\) students in the Science Club. How many students are in the Math Club? $$ \text{(A)}\ 12\qquad\text{(B)}\ 15\qquad\text{(C)}\ 30\qquad\text{(D)}\ 36\qquad\text{(E)}\ 40 $$
Solution: E
Tags: number,statistics,stats,counting,venn,pie
Elo: 1517.4358728782763
Source: AMC 8 1998 A Q14
EditHow many whole numbers from \(1\) through \(46\) are divisible by either \(3\) or \(5\) or both? $$\text{(A)}\ 18 \qquad \text{(B)}\ 21 \qquad \text{(C)}\ 24 \qquad \text{(D)}\ 25 \qquad \text{(E)}\ 27$$
Solution: B
Tags: number,divisibility,counting,pie
Elo: 1513.8278203282016
Source: AMC 8 1991 A Q9
EditLet \( S\) be the set of the \( 2005\) smallest multiples of \( 4\), and let \( T\) be the set of the \( 2005\) smallest positive multiples of \( 6\). How many elements are common to \( S\) and \( T\)? $$ \textbf{(A)}\ 166\qquad \textbf{(B)}\ 333\qquad \textbf{(C)}\ 500\qquad \textbf{(D)}\ 668\qquad \textbf{(E)}\ 1001$$
Solution: D
Tags: combinatorics,counting,divisibility,factors,pie
Elo: 1566.9264427538837
Source: AMC 10 2005 A Q22
EditEach principal of Lincoln High School serves exactly one \(3\)-year term. What is the maximum number of principals this school could have during an \(8\)-year period? $$\text{(A)}\ 2 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 5 \qquad \text{(E)}\ 8$$
Solution: C
Tags: combinatorics,counting,off-by-one,pie
Elo: 1471.1856554821084
Source: AMC 8 2000 A Q5
EditMientka Publishing Company prices its bestseller Where's Walter? as follows: \[C(n) = \begin{cases} 12n, &\text{if } 1 \le n \le 24\\ 11n, &\text{if } 25 \le n \le 48\\ 10n, &\text{if } 49 \le n \end{cases}\] where \( n\) is the number of books ordered, and \( C(n)\) is the cost in dollars of \( n\) books. Notice that \( 25\) books cost less than \( 24\) books. For how many values of \( n\) is it cheaper to buy more than \( n\) books than to buy exactly \( n\) books? $$\textbf{(A)}\ 3\qquad \textbf{(B)}\ 4\qquad \textbf{(C)}\ 5\qquad \textbf{(D)}\ 6\qquad \textbf{(E)}\ 8$$
Solution: D
Tags: case-work,discount,piece-wise-functions,piecewise-functions
Elo: 1469.4387739660467
Source: AMC 12 1997 A Q8
EditFor each \( x\) in \( [0,1]\), define \[ f(x)=\begin{cases}2x, &\text { if } 0 \leq x \leq \frac {1}{2}; \\ 2 - 2x, &\text { if } \frac {1}{2} < x \leq 1. \end{cases} \]Let \( f^{[2]}(x) = f(f(x))\), and \( f^{[n + 1]}(x) = f^{[n]}(f(x))\) for each integer \( n \geq 2\). For how many values of \( x\) in \( [0,1]\) is \( f^{[2005]}(x) = \frac {1}{2}\)? $$ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 2005 \qquad \textbf{(C)}\ 4010 \qquad \textbf{(D)}\ 2005^2 \qquad \textbf{(E)}\ 2^{2005}$$
Solution: E
Tags: functions,composition-of-functions,case-work,piecewise-functions
Elo: 1515.2975328274754
Source: AMC 12 2005 A Q20
EditIn a room, \(2/5\) of the people are wearing gloves, and \(3/4\) of the people are wearing hats. What is the minimum number of people in the room wearing both a hat and a glove? $$ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 15\qquad\textbf{(E)}\ 20 $$
Solution: A
Tags: counting,fractions,pie
Elo: 1545.9575825836287
Source: AMC 8 2010 A Q20
EditSets A and B, shown in the venn diagram, have the same number of elements. Thier union has 2007 elements and their intersection has 1001 elements. Find the number of elements in A. [asy] defaultpen(linewidth(0.7)); draw(Circle(origin, 5)); draw(Circle((5,0), 5)); label("\(A\)", (0,5), N); label("\(B\)", (5,5), N); label("\(1001\)", (2.5, -0.5), N);[/asy] $$ \textbf{(A)}\: 503\qquad \textbf{(B)}\: 1006\qquad \textbf{(C)}\: 1504\qquad \textbf{(D)}\: 1507\qquad \textbf{(E)}\: 1510\qquad $$
Solution: C
Tags: counting,venn,combinatorics,pie
Elo: 1514.5740753236228
Source: AMC 8 2007 A Q13
EditA flower bouquet contains pink roses, red roses, pink carnations, and red carnations. One third of the pink flowers are roses, three fourths of the red flowers are carnations, and six tenths of the flowers are pink. What percent of the flowers are carnations? $$ \textbf{(A)}\ 15\qquad\textbf{(B)}\ 30\qquad\textbf{(C)}\ 40\qquad\textbf{(D)}\ 60\qquad\textbf{(E)}\ 70 $$
Solution: E
Tags: venn,combinatorics,counting,pie
Elo: 1469.7492508522553
Source: AMC 10 2013 A Q10
EditIn a town of 351 adults, every adult owns a car, motorcycle, or both. If 331 adults own cars and 45 adults own motorcycles, how many of the car owners do not own a motorcycle? $$ \textbf{(A)} 20 \qquad\textbf{(B)} 25 \qquad\textbf{(C)} 45 \qquad\textbf{(D)} 306 \qquad\textbf{(E)} 351$$
Solution: D
Tags: counting,combinatorics,pie
Elo: 1446.9203557927906
Source: AMC 8 2011 A Q6
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