Problem 87 - Beginning and Intermediate Algebra

Question

In your own words, explain how to add two fractions with different denominators.

Step-by-Step Solution

Answer

To add fractions with different denominators, find a common denominator, convert to equivalent fractions, then add numerators.

Step 1: Identify the Fractions

First, look at the two fractions you want to add. For example, consider the fractions \( \frac{1}{3} \) and \( \frac{1}{4} \). Note that they have different denominators, which are 3 and 4.

Step 2: Find a Common Denominator

The next step is to find a common denominator. This is typically the least common multiple (LCM) of the two denominators. For 3 and 4, the LCM is 12. So, 12 is the common denominator we will use for our fractions.

Step 3: Convert Fractions to Equivalent Fractions

Convert each fraction to an equivalent fraction with the common denominator. To convert \( \frac{1}{3} \), multiply both the numerator and the denominator by 4 (since \( 3 \times 4 = 12 \)), resulting in \( \frac{4}{12} \).To convert \( \frac{1}{4} \), multiply both the numerator and the denominator by 3 (since \( 4 \times 3 = 12 \)), resulting in \( \frac{3}{12} \).

Step 4: Add the Equivalent Fractions

Now that both fractions have the same denominator, you can add the numerators directly. So add 4 and 3 to get 7. The fraction becomes \( \frac{7}{12} \).

Step 5: Simplify the Fraction

Finally, check if the resulting fraction can be simplified. In this case, \( \frac{7}{12} \) is already in its simplest form since 7 and 12 have no common factors other than 1.

Key Concepts

Common Denominator

When you want to add fractions, it's crucial to have a common denominator because it allows us to combine the values on a comparable scale. Denominators are like the parts of a whole, and common denominators mean each part is the same size.

• For example, consider trying to add thirds (\( \frac{1}{3} \)) to quarters (\( \frac{1}{4} \)). Their parts or divisions aren't the same, so we need a common denominator.
• By converting them into twelfths, both fractions can be related easily with a common reference point—the denominator 12. This is called the common denominator.

Finding a common denominator simplifies addition because it equalizes the size of each fractional part, which makes the numerators ready to be added directly. Moreover, each fraction is converted into an equivalent one, which exactly represents its original value using this common denominator.

Least Common Multiple

The least common multiple (LCM) is an essential concept when dealing with adding or comparing fractions with different denominators. The LCM is the smallest multiple that two numbers share.

• In the case of fractions, it ensures that we find a common denominator that is easiest to work with—often making further calculations simpler.
• For example, when adding \(\frac{1}{3}\) and \(\frac{1}{4}\), the LCM of 3 and 4 is 12. This is because 12 is the smallest number that both 3 and 4 can evenly divide into.

Thus, using the LCM to find a common denominator ensures that the values of the fractions are not altered incorrectly, preserving their ratios and making operations possible after conversion to equivalent fractions.

Equivalent Fractions

Equivalent fractions are different expressions of the same value. Converting a fraction to an equivalent one involves changing its numerator and denominator while keeping its overall value the same.

• For instance, \(\frac{1}{3}\) can become \(\frac{4}{12}\), and \(\frac{1}{4}\) becomes \(\frac{3}{12}\) by finding a common denominator of 12.
• This is achieved by multiplying the numerator and the denominator by the same factor, ensuring the fraction's value remains unchanged.

Using equivalent fractions allows us to operate on them (like addition or subtraction) easily, now that they share a common denominator. After the operation, the result can often be simplified to its simplest form, ensuring the fraction is expressed most efficiently.