Key Concepts

• If $a,b$, and $c$ are numbers where $c\ne 0$ , then ${\Large\frac{a}{c}}+{\Large\frac{b}{c}}={\Large\frac{a+c}{c}}$ .
• To add fractions, add the numerators and place the sum over the common denominator.
• Fraction Subtraction
• If $a,b$, and $c$ are numbers where $c\ne 0$ , then ${\Large\frac{a}{c}}-{\Large\frac{b}{c}}={\Large\frac{a-b}{c}}$ .
• 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.
1. Find any factor pair of the given number, and use these numbers to create two branches.
2. If a factor is prime, that branch is complete. Circle the prime.
3. If a factor is not prime, write it as the product of a factor pair and continue the process.
4. Write the composite number as the product of all the circled primes.
• Find the prime factorization of a composite number using the ladder method.
1. Divide the number by the smallest prime.
2. Continue dividing by that prime until it no longer divides evenly.
3. Divide by the next prime until it no longer divides evenly.
4. Continue until the quotient is a prime.
5. 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.
1. Find the prime factorization of each number.
2. Write each number as a product of primes, matching primes vertically when possible.
3. Bring down the primes in each column.
4. Multiply the factors to get the LCM.
• Find the LCM using the prime factors method.
1. Find the prime factorization of each number.
2. Write each number as a product of primes, matching primes vertically when possible.
3. Bring down the primes in each column.
4. Multiply the factors to get the LCM.
• Find the least common denominator (LCD) of two fractions.
1. Factor each denominator into its primes.
2. List the primes, matching primes in columns when possible.
3. Bring down the columns.
4. Multiply the factors. The product is the LCM of the denominators.
5. The LCM of the denominators is the LCD of the fractions.
• Equivalent Fractions Property
• If $a,b$ , and $c$ are whole numbers where $b\ne 0$ , $c\ne 0$ then$\Large\frac{a}{b}=\Large\frac{a\cdot c}{b\cdot c}$
and $\Large\frac{a\cdot c}{b\cdot c}=\Large\frac{a}{b}$
• Convert two fractions to equivalent fractions with their LCD as the common denominator.
1. Find the LCD.
2. For each fraction, determine the number needed to multiply the denominator to get the LCD.
3. Use the Equivalent Fractions Property to multiply the numerator and denominator by the number from Step 2.
4. Simplify the numerator and denominator.
• Add or subtract fractions with different denominators.
1. Find the LCD.
2. Convert each fraction to an equivalent form with the LCD as the denominator.
3. Add or subtract the fractions.
4. Write the result in simplified form.

Glossary

composite number
A composite number is a counting number that is not prime
divisibility
If a number $m$ is a multiple of $n$ , then we say that $m$ is divisible by $n$
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 $n$ if it is the product of a counting number and $n$
ratio
A ratio compares two numbers or two quantities that are measured with the same unit. The ratio of $a$ to $b$ is written $a$ to $b$ , $\Large\frac{a}{b}$ , or $a:b$
prime number
A prime number is a counting number greater than 1 whose only factors are 1 and itself