Key Concepts

• To find the prime factorization of a number, use the factor tree method to divide away small prime numbers until only prime factors remain. The product of those prime factors is the prime factorization of the number.
• To find the least common multiple (LCM) of a list of numbers, write each number as a product of its prime factors, select the largest instance of each prime that appears in any one number, then multiply the selections together to obtain the LCM.
• Equivalent fractions property: If [latex]a,b,[/latex] and [latex]c[/latex] are real numbers where [latex]b \neq 0[/latex] and [latex]c \neq 0[/latex], then [latex]\frac{a \cdot c}{b \cdot c} = \frac{a}{c}[/latex].
• To simplify a fraction, rewrite the numerator and denominator as products of their prime factors, then remove common factors.
• To add or subtract fractions, first rewrite each fraction as an equivalent fraction, all having the same denominator, then add or subtract the numerators and place the result over the common denominator.
  • [latex]\dfrac{a}{b}\pm\dfrac{c}{d} = \dfrac{ad \pm bc}{bd}[/latex], where [latex]a,b,[/latex] and [latex]c[/latex] are real numbers, and [latex]b \neq 0[/latex].
• To multiply fractions, place the product of the numerators over the product of the denominator.
  • [latex]\dfrac{a}{b}\cdot\dfrac{c}{d} = \dfrac {ac}{bd}[/latex], where [latex]a,b,c,[/latex] and [latex]d[/latex] are real numbers, and [latex]b \neq 0[/latex] and [latex]d \neq 0[/latex].
• To divide fractions, multiply the first fraction by the reciprocal of the second
  • [latex]\dfrac{a}{b}\div\dfrac{c}{d}=\dfrac{a}{b}\cdot\dfrac{d}{c}=\dfrac{ad}{bc}[/latex], where [latex]a,b,c,[/latex] and [latex]d[/latex] are real numbers, and [latex]b \neq 0[/latex] and [latex]d \neq 0[/latex].

Glossary

categorical variables: variables that can have one of a limited number of values, or labels; ror example, a person’s eye color, gender, or home state; a vehicle’s body style (sedan, SUV, minivan, etc.); a dog’s breed (bulldog, greyhound, beagle, etc.).

composite number: a natural number that can be written as the product of other natural numbers. For example, the number 10 can be written as the product of 2 and 5.

denominator: the bottom part of a fraction; the denominator in the fraction [latex]\Large\frac{2}{3}[/latex] is [latex]3[/latex]

factor: a number being multiplied, for example, for  [latex]2 \cdot 5 = 10[/latex] , the numbers [latex]2[/latex] and [latex]5[/latex] are factors of [latex]10[/latex]

least common multiple, LCM: given two or more numbers, the least common multiple between them is the smallest number into which each of the given numbers divides evenly

numerator: the top part of a fraction; the numerator in the fraction [latex]\Large\frac{2}{3}[/latex] is [latex]2[/latex]

prime number: a number that is divisible only by itself and 1

prime factorization: the process of writing a number as a product of only prime factors

quantitative variables: variables with numeric values that can be averaged. A quantitative variable is frequently a measurement—for example, a person’s height in inches.

simplify, reduce: to write a mathematical statement in smallest terms