Problem 13 - Elementary and Intermediate Algebra

Question

Perform the operations. Simplify, if possible. $$ \frac{7 a}{8}+\frac{4 a}{5} $$

Step-by-Step Solution

Answer

The result is \(\frac{67a}{40}\).

Step 1: Find a Common Denominator

To add fractions, we first need to find a common denominator. The denominators here are 8 and 5. The least common multiple of 8 and 5 is 40.

Step 2: Convert Fractions to Common Denominator

Convert each fraction to an equivalent fraction with the common denominator of 40. For \(\frac{7a}{8}\), multiply both the numerator and the denominator by 5: \(\frac{7a \times 5}{8 \times 5} = \frac{35a}{40}\).For \(\frac{4a}{5}\), multiply both the numerator and the denominator by 8: \(\frac{4a \times 8}{5 \times 8} = \frac{32a}{40}\).

Step 3: Add the Fractions

Now, add the fractions with the same denominator:\(\frac{35a}{40} + \frac{32a}{40} = \frac{35a + 32a}{40} = \frac{67a}{40}\).

Step 4: Simplify the Result

Check to see if the fraction \(\frac{67a}{40}\) can be simplified. Since 67 and 40 have no common factors (67 is a prime number), the fraction is in its simplest form.

Key Concepts

Common Denominator

Adding algebraic fractions involves finding a common denominator, which is essential for combining fractions. When fractions have different denominators, like \( \frac{7a}{8} \) and \( \frac{4a}{5} \), they must be converted to have the same denominator before they can be added.
The common denominator is often the least common multiple (LCM) of the original denominators. This ensures you have the smallest denominator possible, making calculations and simplification easier.
Remember:

• Identify the denominators in your fractions (e.g., 8 and 5).
• Find the LCM of these denominators (for 8 and 5, it's 40).
• Convert each fraction to have this common denominator.

With a common denominator, addition of the fractions becomes straightforward. You simply add the numerators while keeping the denominator the same.

Least Common Multiple

The least common multiple (LCM) is the smallest number that is a multiple of two or more numbers. It's vital in finding the common denominator for fractions. When the denominators of fractions are relatively prime (they have no common factors other than 1), the LCM is their product.
To find the LCM of 8 and 5 in our example:

• List the multiples of 8: 8, 16, 24, 32, 40...
• List the multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40...
• Identify the smallest common multiple: This is 40.

Using the LCM as a common denominator simplifies the task of adding fractions. This step ensures that each fraction is expressed in terms of the same 'whole,' facilitating easy addition.

Simplifying Fractions

Once fractions have been added together, the next step is often simplification. Simplifying fractions involves reducing them to their simplest form. This occurs when the numerator and denominator have no common factors other than 1.
In our exercise, after adding the fractions \( \frac{35a}{40} + \frac{32a}{40} = \frac{67a}{40} \), we need to check if this fraction can be simplified. Look for common factors:

• The numerator (67) is a prime number, meaning it only has factors of 1 and itself.
• The denominator (40) factors into 1, 2, 4, 5, 8, 10, 20, 40.
• Since 67 and 40 share no common factors other than 1, the fraction is already in its simplest form.

Simplifying fractions is crucial because it makes the results clearer and easier to work with. In this case, \( \frac{67a}{40} \) is already fully simplified.