User:Melikamp/elementary math

Notation

Here $\cdot$ is multiplication and $\div$ is division.

Integer Factorization

Recall that a whole number $n\geq 2$ is prime if and only if its only whole factors are one and $n$. Express the following numbers as products of primes.

$\displaystyle 4$

$\displaystyle 6$

$\displaystyle 12$

$\displaystyle 15$

$\displaystyle 20$

$\displaystyle 51$

$\displaystyle 60$

$\displaystyle 70$

$\displaystyle 81$

Fractions

Irreducible Fractions

Express given fractions in lowest terms or as integers.

$\displaystyle {\frac {24}{20}}$

$\displaystyle {\frac {42}{20}}$

$\displaystyle {\frac {33}{60}}$

$\displaystyle {\frac {77}{14}}$

$\displaystyle {\frac {30}{21}}$

Fraction Multiplication

Evaluate the following expressions and express the answers as fractions in lowest terms or integers.

$\displaystyle {\frac {3}{7}}\cdot 14$

$\displaystyle {\frac {3}{5}}\cdot {\frac {6}{7}}$

$\displaystyle {\frac {3}{4}}\cdot {\frac {2}{9}}$

$\displaystyle {\frac {3}{10}}\cdot {\frac {30}{21}}$

$\displaystyle {\frac {22}{10}}\cdot {\frac {5}{11}}$

Fraction Division

$\displaystyle {\frac {55}{3}}\div 11$

$\displaystyle {\frac {3}{5}}\div {\frac {6}{7}}$

$\displaystyle {\frac {3}{4}}\div {\frac {2}{9}}$

$\displaystyle {\frac {15}{7}}\div {\frac {30}{21}}$

$\displaystyle {\frac {22}{10}}\div {\frac {11}{2}}$

Fraction Addition and Subtraction

$\displaystyle {\frac {3}{5}}+7$

$\displaystyle {\frac {3}{5}}+{\frac {7}{5}}$

$\displaystyle {\frac {3}{5}}+{\frac {1}{2}}$

$\displaystyle {\frac {17}{6}}-{\frac {2}{3}}$

$\displaystyle {\frac {4}{15}}-{\frac {3}{12}}$

Expressions Involving Fractions

$\displaystyle \left({\frac {4}{5}}-{\frac {3}{2}}\right)\cdot {\frac {5}{2}}$

$\displaystyle \left({\frac {5}{4}}+{\frac {3}{2}}\right)\div {\frac {7}{2}}$

$\displaystyle \left({\frac {1}{2}}+{\frac {1}{3}}\right)\div \left({\frac {3}{4}}-2\right)$

Order of Operations

Evaluate given expressions.

$\displaystyle 8+5\cdot 3$

$\displaystyle 2\cdot 5^{2}$

$\displaystyle {\frac {15-|3-1|}{12-3\cdot 2}}$

$\displaystyle 3(4+3(6-4))$

$\displaystyle \left({\frac {3}{8}}\right)^{2}+{\frac {1}{4}}+{\frac {1}{8}}\cdot {\frac {3}{2}}$

Symbolic Expressions and Equations

Evaluate the following expressions if $x=1$, $y=-3$, and $z=2$.

$\displaystyle yz-x$

$\displaystyle {\frac {y^{2}+x}{x^{2}+3y}}$

Determine whether 9 is a solution to the equation $\displaystyle 3(x-1)=2x+6$