Problem 50 - Prealgebra and Introductory Algebra

Question

Evaluate the expression \(x+y\) for the given values of \(x\) and \(y .\) $$x=-\frac{3}{8}, y=\frac{2}{9}$$

Step-by-Step Solution

Answer

The solution is \(-\frac{11}{72}\).

Step 1: Identify the Values of x and y

From the exercise, we know that \(x=-\frac{3}{8}\) and \(y=\frac{2}{9}\).

Step 2: Find Common Denominator

Since the denominators of the given fractions are not the same, we need to find a common denominator before adding them. The least common denominator for 8 and 9 is 72.

Step 3: Convert x and y to Equivalent Fractions

To express the fraction \(-\frac{3}{8}\) with the denominator 72, multiply the numerator and the denominator by 9. This gives the equivalent fraction \(-\frac{27}{72}\). Similarly, to express the fraction \(\frac{2}{9}\) with the denominator 72, multiply the numerator and the denominator by 8. This gives the equivalent fraction \(\frac{16}{72}\).

Step 4: Add the Fractions

Now we can add the two fractions: \(-\frac{27}{72} + \frac{16}{72} = -\frac{11}{72}\).

Key Concepts

Common Denominator

When adding fractions, it's crucial for the fractions to share the same denominator. This common base allows us to directly add the numerators while the denominators remain unchanged, simplifying the process. To find a common denominator:

• Identify the denominators of the given fractions. In this exercise, they are 8 and 9.
• Calculate the least common denominator (LCD), which is the smallest number that both denominators can divide into evenly. For 8 and 9, the LCD is 72.
• A common denominator enables the conversion of different fractional parts into a unified system, allowing addition or subtraction to occur effortlessly.

Identifying the common denominator is often the first and most vital step in operations involving fractions.

Equivalent Fractions

Once you have a common denominator, the next step is to convert the original fractions to equivalent fractions with the new denominator. Equivalent fractions represent the same value or proportion, even though their numerators and denominators might differ.To create equivalent fractions:

• Multiply both the numerator and the denominator of each fraction by the same number. This number is chosen such that the denominator becomes the common denominator found in the previous step.
• For example, to rewrite \(-\frac{3}{8}\) with a denominator of 72, multiply by 9: \(-\frac{3}{8} \times \frac{9}{9} = -\frac{27}{72}\).
• Similarly, convert \(\frac{2}{9}\) by multiplying by 8: \(\frac{2}{9} \times \frac{8}{8} = \frac{16}{72}\).

These steps ensure that both fractions now have the same denominator, enabling easier addition or subtraction.

Fraction Addition

With both fractions now having a common denominator, you can proceed to add the numerators while keeping the denominator the same.To perform fraction addition:

• Take the equivalent fractions. In this case, they are \(-\frac{27}{72}\) and \(\frac{16}{72}\).
• Add their numerators: \(-27 + 16 = -11\).
• Keep the common denominator the same, which is 72. Thus, \(-\frac{27}{72} + \frac{16}{72} = -\frac{11}{72}\).

Adding fractions becomes straightforward once they share a base (common denominator), completing the process by simply adjusting the numerators while maintaining the established denominator. This systematic approach ensures accuracy and simplifies fractional operations.