## Key Concepts

**Fraction Addition**- If [latex]a,b[/latex], and [latex]c[/latex] are numbers where [latex]c\ne 0[/latex] , then [latex]{\Large\frac{a}{c}}+{\Large\frac{b}{c}}={\Large\frac{a+c}{c}}[/latex] .
- To add fractions, add the numerators and place the sum over the common denominator.

**Fraction Subtraction**- If [latex]a,b[/latex], and [latex]c[/latex] are numbers where [latex]c\ne 0[/latex] , then [latex]{\Large\frac{a}{c}}-{\Large\frac{b}{c}}={\Large\frac{a-b}{c}}[/latex] .
- To subtract fractions, subtract the numerators and place the difference over the common denominator.

**Find the prime factorization of a composite number using the tree method.**- Find any factor pair of the given number, and use these numbers to create two branches.
- If a factor is prime, that branch is complete. Circle the prime.
- If a factor is not prime, write it as the product of a factor pair and continue the process.
- Write the composite number as the product of all the circled primes.

**Find the prime factorization of a composite number using the ladder method.**- Divide the number by the smallest prime.
- Continue dividing by that prime until it no longer divides evenly.
- Divide by the next prime until it no longer divides evenly.
- Continue until the quotient is a prime.
- Write the composite number as the product of all the primes on the sides and top of the ladder.

**Find the LCM using the prime factors method.**- Find the prime factorization of each number.
- Write each number as a product of primes, matching primes vertically when possible.
- Bring down the primes in each column.
- Multiply the factors to get the LCM.

**Find the LCM using the prime factors method.**- Find the prime factorization of each number.
- Write each number as a product of primes, matching primes vertically when possible.
- Bring down the primes in each column.
- Multiply the factors to get the LCM.

**Find the least common denominator (LCD) of two fractions.**- Factor each denominator into its primes.
- List the primes, matching primes in columns when possible.
- Bring down the columns.
- Multiply the factors. The product is the LCM of the denominators.
- The LCM of the denominators is the LCD of the fractions.

**Equivalent Fractions Property**- If [latex]a,b[/latex] , and [latex]c[/latex] are whole numbers where [latex]b\ne 0[/latex] , [latex]c\ne 0[/latex] then[latex]\Large\frac{a}{b}=\Large\frac{a\cdot c}{b\cdot c}[/latex]

and [latex]\Large\frac{a\cdot c}{b\cdot c}=\Large\frac{a}{b}[/latex]

- If [latex]a,b[/latex] , and [latex]c[/latex] are whole numbers where [latex]b\ne 0[/latex] , [latex]c\ne 0[/latex] then[latex]\Large\frac{a}{b}=\Large\frac{a\cdot c}{b\cdot c}[/latex]
**Convert two fractions to equivalent fractions with their LCD as the common denominator.**- Find the LCD.
- For each fraction, determine the number needed to multiply the denominator to get the LCD.
- Use the Equivalent Fractions Property to multiply the numerator and denominator by the number from Step 2.
- Simplify the numerator and denominator.

**Add or subtract fractions with different denominators.**- Find the LCD.
- Convert each fraction to an equivalent form with the LCD as the denominator.
- Add or subtract the fractions.
- Write the result in simplified form.

## Glossary

**composite number**- A composite number is a counting number that is not prime

**divisibility**- If a number [latex]m[/latex] is a multiple of [latex]n[/latex] , then we say that [latex]m[/latex] is divisible by [latex]n[/latex]

**least common denominator (LCD)**- The least common denominator (LCD) of two fractions is the least common multiple (LCM) of their denominators

**multiple of a number**- A number is a multiple of [latex]n[/latex] if it is the product of a counting number and [latex]n[/latex]

**ratio**- A ratio compares two numbers or two quantities that are measured with the same unit. The ratio of [latex]a[/latex] to [latex]b[/latex] is written [latex]a[/latex] to [latex]b[/latex] , [latex]\Large\frac{a}{b}[/latex] , or [latex]a:b[/latex]

**prime number**- A prime number is a counting number greater than 1 whose only factors are 1 and itself