Problem 58 - Algebra A Combined Function

Question

Perform the indicated operations. See Examples 11 through \(23 .\) $$ \frac{13}{132}+\frac{35}{132} $$

Step-by-Step Solution

Answer

The answer is \(\frac{4}{11}\).

Step 1: Identify Common Denominator

Notice that both fractions \(\frac{13}{132}\) and \(\frac{35}{132}\) have the same denominator, 132.

Step 2: Add Numerators

Since both fractions have the same denominator, you can combine them by adding their numerators: \(13 + 35 = 48\).

Step 3: Form the Resulting Fraction

Place the sum from Step 2 over the common denominator to form the new fraction: \(\frac{48}{132}\).

Step 4: Simplify the Fraction

Determine the greatest common divisor (GCD) of 48 and 132, which is 12. Divide both the numerator and the denominator by their GCD: \(\frac{48}{12} = 4\) and \(\frac{132}{12} = 11\).

Step 5: Write the Simplified Fraction

The simplified fraction is \(\frac{4}{11}\).

Key Concepts

Common Denominator

A common denominator is the shared denominator between two or more fractions. In fraction addition, it is crucial to have a common denominator to combine the fractions effectively.
In our exercise, both fractions \(\frac{13}{132}\) and \(\frac{35}{132}\) already have a common denominator, which is 132. This makes the addition process straightforward.
Recognizing a common denominator simplifies the problem as you focus only on manipulating the numerators without altering the denominators.

Numerator Addition

Once fractions have a common denominator, the next step is to add their numerators, which are the top portions of fractions.
Here, with the numerators 13 and 35, we add them:

• 13 + 35 = 48

After this addition, the resulting number (48) becomes the new numerator of the fraction while the denominator remains the same. This step ensures the fractions are effectively summed up into a single fraction \(\frac{48}{132}\).
Adding numerators is a simple arithmetic process but pivotal in fraction addition.

Greatest Common Divisor (GCD)

The greatest common divisor (GCD) is used to find the largest number that divides two numbers without a remainder. When simplifying fractions, the GCD helps reduce them to their simplest form.
In our equation, the GCD of 48 and 132 is 12. You can find the GCD by listing the divisors of both numbers and selecting the largest one they share:

• Divisors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
• Divisors of 132: 1, 2, 3, 4, 6, 11, 12, 22, 33, 44, 66, 132
• Common Divisor: 12

Using the GCD ensures the fraction is expressed in the simplest and most understandable form.

Fraction Simplification

Fraction simplification makes fractions easier to interpret by reducing them to their simplest form. This is achieved by dividing both the numerator and the denominator by their GCD.
In our solved exercise, we simplify \(\frac{48}{132}\) by dividing both the numerator and denominator by their GCD, 12:

• Divide the numerator: \(\frac{48}{12} = 4\)
• Divide the denominator: \(\frac{132}{12} = 11\)

Thus, \(\frac{48}{132}\) simplifies to \(\frac{4}{11}\). Simplifying fractions not only makes them more digestible but also easier to use in further mathematical operations.