Problem 52 - Prealgebra

Question

Find each sum or difference. Write in simplest form. $$\frac{5}{9}+\left(-\frac{1}{12}\right)$$

Step-by-Step Solution

Answer

The sum is \(\frac{17}{36}\).

Step 1: Find the Least Common Denominator (LCD)

First, identify the denominators: 9 and 12. The LCD is the smallest number that both denominators can divide without a remainder. To find this, list the multiples of each: - 9: 9, 18, 27, 36, 45, ... - 12: 12, 24, 36, 48, ... The smallest common multiple is 36, so the LCD is 36.

Step 2: Convert Fractions to Have the Same Denominator

Rewrite each fraction with the denominator of 36:- For \(\frac{5}{9}\), multiply both the numerator and the denominator by 4 to get \(\frac{20}{36}\).- For \(-\frac{1}{12}\), multiply both the numerator and the denominator by 3 to get \(-\frac{3}{36}\).

Step 3: Perform the Addition

Now with the same denominator, add the two fractions:\[\frac{20}{36} + \left(-\frac{3}{36}\right) = \frac{20 - 3}{36} = \frac{17}{36}.\]

Step 4: Simplify the Result (if needed)

Check if the fraction can be simplified. Since 17 is a prime number and does not divide 36, \(\frac{17}{36}\) is already in its simplest form.

Key Concepts

Least Common Denominator

When adding or subtracting fractions, having a common denominator makes things much simpler. This common denominator is known as the "Least Common Denominator" (LCD).
To find the LCD, you first need to identify the denominators of the fractions involved. In our example, the denominators are 9 and 12.

• List the multiples of each denominator:
• Multiples of 9: 9, 18, 27, 36, 45...
• Multiples of 12: 12, 24, 36, 48...

The smallest common multiple from both lists is 36 which becomes the LCD. Using the LCD allows you to easily perform operations on fractions by aligning them with a common base.

Simplifying Fractions

After finding a common denominator, the next step is to make sure the fractions are in a form that can be worked with. Sometimes, fractions have larger numerators and denominators than necessary. Simplifying means reducing the fraction to its smallest form while maintaining its original value.

• Find a number that divides evenly into both the numerator and the denominator.
• Divide both the top and bottom by that number.

In the exercise, the fraction \( \frac{17}{36} \) is already in its simplest form because 17 is a prime number and doesn’t divide 36. Thus, knowing how to recognize when a fraction is already simplest can save you extra steps.

Adding and Subtracting Fractions

Once fractions have a common denominator, you can easily add or subtract them by focusing just on the numerators. Remember that the common denominator remains unchanged.

• With the same denominator, subtract the numerators numerically.
• Keep the common denominator.
• For instance, with the fractions \( \frac{20}{36} \) and \( -\frac{3}{36} \), you combine numerators directly: \( 20 - 3 = 17 \).

The result is \( \frac{17}{36} \), which is then checked to see if it can be simplified. It’s a straightforward process that becomes intuitive with practice. After performing these simple steps, fractions can easily be added or subtracted to solve even more complex problems.