大联盟2

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

egin{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) $frac{a}{b}=acdot frac{1}{b}$, $frac{1}{ab}=frac{1}{a}cdot frac{1}{b}$.
end{proposition}

egin{ltbox}egin{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}

egin{ltbox}egin{example}
(2014-2015 Math League G6 Question 15) $1frac{1}{3} imes 1frac{1}{4} imes 1frac{1}{5}=$?
egin{tasks}(5)
ask $2$

ask $3$

ask $1frac{1}{6}$

ask $1frac{1}{60}$
end{tasks}
end{example}end{ltbox}

egin{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}

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

ask $5^{91}-5^{90}$

ask $5^{93}-5^{92}$

ask $5^{94}$
end{tasks}
end{example}end{ltbox}

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

ask $6$

ask $7$

ask $8$
end{tasks}
end{example}end{ltbox}

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

ask $1$

ask $1.5$

ask $2$
end{tasks}
end{example}end{ltbox}

section{Decimals}

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

ask $0.01$

ask $0.0011$

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

egin{ltbox}egin{example}

egin{tasks}(5)
ask $0.5$

ask $1$

ask $1.5$

ask $2$
end{tasks}
end{example}end{ltbox}

section{Percent}

egin{definition}
extbf{Percent:} is just a fraction with a hidden denominator of $100$.
end{definition}

egin{ltbox}egin{example}
egin{tasks}(5)
ask $0.5$

ask $1$

ask $1.5$

ask $2$
end{tasks}
end{example}end{ltbox}

egin{ltbox}egin{example}
egin{tasks}(5)
ask $0.5$

ask $1$

ask $1.5$

ask $2$
end{tasks}
end{example}end{ltbox}

egin{ltbox}egin{example}
(2019 AMC 8) What is the value of the product[left(frac{1cdot3}{2cdot2} ight)left(frac{2cdot4}{3cdot3} ight)left(frac{3cdot5}{4cdot4} ight)cdotsleft(frac{97cdot99}{98cdot98} ight)left(frac{98cdot100}{99cdot99} ight)?]

egin{tasks}(6)
ask $frac{1}{2}$

ask $frac{50}{99}$

ask $frac{9800}{9801}$

ask $frac{100}{99}$

ask $50$
end{tasks}
end{example}end{ltbox}
%B

egin{ltbox}egin{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?

egin{tasks}(3)
ask $frac{15}{11}< frac{17}{13}< frac{19}{15}$

ask $frac{15}{11}< frac{19}{15}<frac{17}{13}$

ask $frac{17}{13}<frac{19}{15}<frac{15}{11}$

ask $frac{19}{15}<frac{15}{11}<frac{17}{13}$

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

egin{ltbox}egin{example}
(2018 AMC 8) What is the value of the product[left(1+frac{1}{1} ight)cdotleft(1+frac{1}{2} ight)cdotleft(1+frac{1}{3} ight)cdotleft(1+frac{1}{4} ight)cdotleft(1+frac{1}{5} ight)cdotleft(1+frac{1}{6} ight)?]
egin{tasks}(6)
ask $frac{7}{6}$

ask $frac{4}{3}$

ask $frac{7}{2}$

ask $7$

ask $8$
end{tasks}
end{example}end{ltbox}
%D