Polynomial and Rational Expressions: Background You’ll Need 1

  • Learn how to find the smallest shared denominator for two fractions and adjust them to have the same denominator.

Fractions are an essential part of mathematics, representing parts of a whole. When working with fractions, we often need to compare or combine them. However, this can be challenging when the fractions have different denominators. That’s where the concept of the least common denominator (LCD) comes in handy.

Before we dive into finding the least common denominator, let’s quickly review the components of a fraction:

  • Numerator: The number above the fraction line, representing the number of parts we have.
  • Denominator: The number below the fraction line, representing the total number of equal parts in the whole.

What is the Least Common Denominator (LCD)?

The least common denominator is the smallest positive number that is divisible by the denominators of all the fractions in question. Finding the LCD is crucial when we want to add, subtract, or compare fractions with different denominators. To find the LCD of two fractions, we’ll use the concept of the least common multiple (LCM) of their denominators.

The Least Common Multiple (LCM) of two or more numbers is the smallest number that is a multiple of all the numbers in question. It’s an important concept that’s particularly useful when solving problems involving fractions, ratios, or when finding equivalent fractions.

How To: Find the Least Common Multiple (LCM)

  1. List the Multiples: Start by listing a set of multiples for each number. Remember, multiples are what you get when you multiply the number by [latex]1, 2, 3,[/latex] and so on.
  2. Scan for Common Ground: Look for multiples that appear in both lists. These are the common multiples. If you don’t find any, continue listing more multiples for each number until you do.
  3. Identify the Least: Among the common multiples, pinpoint the smallest one. This is the LCM. It’s the “least common” because it’s the smallest number that all the original numbers can divide into without leaving a remainder.
  4. Confirmation: Ensure the number you’ve identified as the LCM is the smallest common multiple. There can be larger common multiples, but the LCM is always the smallest of these

Another method than can be used to find the least common multiple is the prime factorization method.

How To: Finding the Least Common Multiple Through Prime Factorization

  1. Find the prime factors of each denominator. You can use a factor tree or division method to break down each number into its prime factors.
  2. List down all the unique prime factors that appear in the prime factorization of each number.
  3. For each unique prime factor, identify the highest power to which it is raised in any of the given numbers.
  4. Multiply together the highest powers of all the unique prime factors. The result is the least common multiple (LCM) of the given numbers.

Now that we know how to find the least common multiple, we can apply this to find the least common denominator of fractions.

How To: Find the Least Common Denominator (LCD)

  1. Identify the denominators of the fractions you’re working with.
  2. Find the least common multiple (LCM) of these denominators using either the listing method or prime factorization method described above.
  3. The LCM you’ve found is the least common denominator (LCD) of the fractions.

Converting Fractions to Equivalent Fractions

Once we have the LCD, we can use it to convert our original fractions to equivalent fractions with the same denominator. Equivalent fractions are fractions that represent the same value or portion of a whole, even though they may look different. For example, [latex]\frac{1}{2}[/latex],[latex]\frac{2}{4}[/latex], and [latex]\frac{3}{6}[/latex] are all equivalent fractions because they all represent the same amount – half.

Converting fractions to equivalent forms with a common denominator allows us to:

  1. Add or subtract fractions easily
  2. Compare fractions accurately
  3. Simplify complex fraction expressions

The key principle in creating equivalent fractions is that we can multiply or divide both the numerator and denominator by the same non-zero number without changing the value of the fraction. This is because we’re essentially multiplying the fraction by 1 in a different form.

Equivalent Fraction Property: For [latex]b \ne 0[/latex] and [latex]c \ne 0[/latex], [latex]\dfrac{a}{b} = \dfrac{a \cdot c}{b \cdot c}[/latex]

How to: Convert Fractions to Equivalent Fractions with a Common Denominator

  1. Find the LCD of the fractions you’re working with.
  2. For each fraction:
    a. Calculate the “multiplier” by dividing the LCD by the fraction’s current denominator.
    b. Multiply both the numerator and denominator of the fraction by this multiplier.
    c. The denominator becomes the LCD.
  3. Verify that all fractions now have the LCD as their denominator.
Find the LCD for the fractions [latex]\dfrac{3}{4}[/latex] and [latex]\dfrac{5}{6}[/latex]. Then, convert both fraction to their equivalent fractions using the LCD.

LCD:

  • The denominators are [latex]4[/latex] and [latex]6[/latex].
  • The multiples of [latex]4[/latex] are: [latex]4, 8, 12, 16, 20, 24, ...[/latex]
  • The multiples of [latex]6[/latex] are: [latex]6, 12, 18, 24, 30, ...[/latex]
  • The least common multiple of [latex]4[/latex] and [latex]6[/latex] is [latex]12[/latex].

Thus, the LCD for the fractions [latex]\dfrac{3}{4}[/latex] and is [latex]12[/latex].

Convert each fraction: Adjust the numerator of each fraction so that the denominator equals the LCD [latex]=12[/latex].

  • [latex]\dfrac{3}{4} = \dfrac{3 \cdot 3}{4 \cdot 3} = \dfrac{9}{12}[/latex]
  • [latex]\dfrac{5}{6} = \dfrac{5 \cdot 2}{6 \cdot 2} = \dfrac{10}{12}[/latex]

Now that the fractions have the same denominator, you can easily add, subtract, or compare them.

Let’s consider a scenario where you need to find the least common denominator (LCD) for fractions that have denominators involving a single variable.

Find the LCD for the fractions [latex]\dfrac{5}{6x^3}[/latex] and [latex]\dfrac{7}{8x^5}[/latex]. Then, convert both fraction to their equivalent fractions using the LCD.

LCD: 

To find the LCD, we need to consider both the numeric coefficients and the variable parts:

  • Coefficient: The LCD of [latex]6[/latex] and [latex]8[/latex] can be found by determining their least common multiple. The factors of [latex]6[/latex] are [latex]1, 2, 3, 6[/latex] and the factors of [latex]8[/latex] are [latex]1, 2, 4, 8[/latex]. The smallest number that appears as a multiple of both is [latex]24[/latex].
  • Variable Factor: The variable [latex]x[/latex] appears as [latex]x^3[/latex] and [latex]x^5[/latex]. The highest power of [latex]x[/latex] is [latex]5[/latex], so [latex]x^5[/latex] will be part of the LCD.

Combining these findings, the LCD is [latex]24x^5[/latex].

Convert each fraction: Adjust the numerator of each fraction so that the denominator equals the LCD [latex]24x^5[/latex].

  • The first fraction’s denominator is [latex]6x^3[/latex]. To match [latex]24x^5[/latex], we need to multiply by [latex]\dfrac{24x^5}{6x^3} = 4x^2[/latex].

[latex]\dfrac{5}{6x^3} = \dfrac{5 \cdot 4x^2}{6x^3 \cdot 4x^2} = \dfrac{20x^2}{24x^5}[/latex]

  • The second fraction’s denominator is [latex]8x^5[/latex]. To match [latex]24x^5[/latex], we need to multiply by [latex]\dfrac{24x^5}{8x^5} = 3[/latex].

[latex]\dfrac{7}{8x^5} = \dfrac{7 \cdot 3}{8x^5 \cdot 3} = \dfrac{21}{24x^5}[/latex]