大联盟2

$$\begin{definition} \textbf{Fraction:} The fraction $\frac{a}{b}$ equals $a\div b$. A line in the middle called a \textbf{fraction bar}. We call the top the \textbf{numerator}, and we call the bottom the \textbf{denominator}. \end{definition}$$

\begin{proposition}

(1) If $a$ is not zero, then $\frac{0}{a}=0$;

(2) If $a$ is not zero, then $\frac{a}{a}=1$; \quad $\frac{ab}{ac}=\frac{b}{c}$;

(3) $\frac{-a}{b}=\frac{a}{-b}=-\frac{a}{b}$, \quad $\frac{-a}{-b}=\frac{a}{b}$;

(4) $\frac{b}{a}-\frac{c}{a}=\frac{b-c}{a}$, $\frac{b}{a}+\frac{c}{a}=\frac{b+c}{a}$;

(5) $\frac{a}{b}=a\cdot \frac{1}{b}$, $\frac{1}{ab}=\frac{1}{a}\cdot \frac{1}{b}$.
\end{proposition}

$$\begin{ltbox}\begin{example} (2017 Math League Summer Tournament Speed Test Question 30) What is the value of $\frac{3^{2017}+3^{2018}}{3^{2015}+3^{2016}}$. \end{example}\end{ltbox}$$

\begin{ltbox}\begin{example}
(2014-2015 Math League G6 Question 15) $1\frac{1}{3}\times 1\frac{1}{4}\times 1\frac{1}{5}=$?
\begin{tasks}(5)
\task $2$

\task $3$

\task $1\frac{1}{6}$

\task $1\frac{1}{60}$
\end{tasks}
\end{example}\end{ltbox}

\begin{proposition}

(1) $\frac{1}{1/a}=a$, $\frac{1}{a/b}=\frac{b}{a}$.

(2) If $b,c$ and $d$ are nonzero,then
$$\frac{a}{b}\div \frac{c}{d}=\frac{a}{b}\cdot \frac{d}{c}=\frac{ad}{bc}.$$
\end{proposition}

 

\begin{ltbox}\begin{example}
(2013-2014 Math League G5-6 Question 29) $\frac{5^{95}-5^{94}}{2^2}=$?
\begin{tasks}(5)
\task $1.25$

\task $5^{91}-5^{90}$

\task $5^{93}-5^{92}$

\task $5^{94}$
\end{tasks}
\end{example}\end{ltbox}

\begin{ltbox}\begin{example}
(2016-2017 Math League G6 Question 38) For how many positive integers $k$ is $\frac{k+2016}{k+3}$
a positive integer?
\begin{tasks}(5)
\task $5$

\task $6$

\task $7$

\task $8$
\end{tasks}
\end{example}\end{ltbox}

\begin{ltbox}\begin{example}
(2016-2017 Math League G5 Question 32) Write, in simplest reduced form, the value of
$$\frac{(33\, 333\, 333)^2-(33\, 333\, 333)(16\, 666\, 667) + (16\, 666\, 667)^2}{(33\, 333\, 333)^2-(33\, 333\, 333)(16\, 666\, 666) + (16\, 666\, 666)^2}$$
\begin{tasks}(5)
\task $0.5$

\task $1$

\task $1.5$

\task $2$
\end{tasks}
\end{example}\end{ltbox}

\section{Decimals}

\begin{ltbox}\begin{example}
(2018-2019 Math League G6 Question 25) Which of the following has the least value?
\begin{tasks}(5)
\task $0.1$

\task $0.01$

\task $0.0011$

\task $(0.01)^2$
\end{tasks}
\end{example}\end{ltbox}

\begin{ltbox}\begin{example}

\begin{tasks}(5)
\task $0.5$

\task $1$

\task $1.5$

\task $2$
\end{tasks}
\end{example}\end{ltbox}

\section{Percent}

$$\begin{definition} \textbf{Percent:} is just a fraction with a hidden denominator of $100$. \end{definition}$$

\begin{ltbox}\begin{example}
\begin{tasks}(5)
\task $0.5$

\task $1$

\task $1.5$

\task $2$
\end{tasks}
\end{example}\end{ltbox}

\begin{ltbox}\begin{example}
\begin{tasks}(5)
\task $0.5$

\task $1$

\task $1.5$

\task $2$
\end{tasks}
\end{example}\end{ltbox}

\begin{ltbox}\begin{example}
(2019 AMC 8) What is the value of the product $$\left(\frac{1\cdot3}{2\cdot2}\right)\left(\frac{2\cdot4}{3\cdot3}\right)\left(\frac{3\cdot5}{4\cdot4}\right)\cdots\left(\frac{97\cdot99}{98\cdot98}\right)\left(\frac{98\cdot100}{99\cdot99}\right)?$$

\begin{tasks}(6)
\task $\frac{1}{2}$

\task $\frac{50}{99}$

\task $\frac{9800}{9801}$

\task $\frac{100}{99}$

\task $50$
\end{tasks}
\end{example}\end{ltbox}
%B

\begin{ltbox}\begin{example}
(2019 AMC 8) Which of the following is the correct order of the fractions $\frac{15}{11},\frac{19}{15},$ and $\frac{17}{13},$ from least to greatest?

\begin{tasks}(3)
\task $\frac{15}{11}< \frac{17}{13}< \frac{19}{15}$

\task $\frac{15}{11}< \frac{19}{15}<\frac{17}{13}$

\task $\frac{17}{13}<\frac{19}{15}<\frac{15}{11}$

\task $\frac{19}{15}<\frac{15}{11}<\frac{17}{13}$

\task $\frac{19}{15}<\frac{17}{13}<\frac{15}{11}$
\end{tasks}
\end{example}\end{ltbox}
%E

\begin{ltbox}\begin{example}
(2018 AMC 8) What is the value of the product $$\left(1+\frac{1}{1}\right)\cdot\left(1+\frac{1}{2}\right)\cdot\left(1+\frac{1}{3}\right)\cdot\left(1+\frac{1}{4}\right)\cdot\left(1+\frac{1}{5}\right)\cdot\left(1+\frac{1}{6}\right)?$$
\begin{tasks}(6)
\task $\frac{7}{6}$

\task $\frac{4}{3}$

\task $\frac{7}{2}$

\task $7$

\task $8$
\end{tasks}
\end{example}\end{ltbox}
%D