Problem 87 - Algebra for College Students

Question

Simplify each numerical expression. $$\frac{2}{3}-\left(\frac{3}{4}-\frac{5}{6}\right)$$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{3}{4} \).

Step 1: Understand the Expression

We need to simplify the expression \( \frac{2}{3} - \left( \frac{3}{4} - \frac{5}{6} \right) \). It involves both subtraction within the parentheses and subtraction outside the parentheses.

Step 2: Simplify Inside the Parentheses

First, we focus on the inner expression: \( \frac{3}{4} - \frac{5}{6} \). To subtract these fractions, we need a common denominator. The least common multiple of 4 and 6 is 12.

Step 3: Convert to Common Denominator

Convert \( \frac{3}{4} \) to \( \frac{9}{12} \) by multiplying both numerator and denominator by 3. Convert \( \frac{5}{6} \) to \( \frac{10}{12} \) by multiplying both numerator and denominator by 2.

Step 4: Perform the Subtraction

Subtract \( \frac{10}{12} \) from \( \frac{9}{12} \), which gives \( \frac{9}{12} - \frac{10}{12} = -\frac{1}{12} \).

Step 5: Substitute Back into the Original Expression

Now substitute the result from Step 4 into the original expression: \( \frac{2}{3} - \left(-\frac{1}{12}\right) \).

Step 6: Simplify Further

Subtracting a negative is the same as adding. Thus, the expression becomes \( \frac{2}{3} + \frac{1}{12} \).

Step 7: Find a Common Denominator

The common denominator for \( \frac{2}{3} \) and \( \frac{1}{12} \) is 12. Convert \( \frac{2}{3} \) to \( \frac{8}{12} \).

Step 8: Perform the Addition

Add \( \frac{8}{12} \) and \( \frac{1}{12} \), which gives \( \frac{9}{12} \).

Step 9: Simplify the Result

Simplify \( \frac{9}{12} \) by dividing both the numerator and the denominator by 3 to get \( \frac{3}{4} \).

Key Concepts

Common Denominator

To work with fractions effectively, especially when adding or subtracting, one must have a shared basis for comparison known as a "common denominator." A common denominator allows fractions to speak the same language so to speak, making arithmetic operations straightforward.
For example, when subtracting \( \frac{3}{4} \) from \( \frac{5}{6} \), the denominators (4 and 6) differ, making it tricky to directly carry out the operation. The least common multiple (LCM) of 4 and 6 is 12, which becomes our common denominator. By converting both fractions to equivalent fractions with this denominator, you turn \( \frac{3}{4} \) into \( \frac{9}{12} \) and \( \frac{5}{6} \) into \( \frac{10}{12} \). With this shared platform, subtraction becomes simple.

Subtracting Fractions

When subtracting fractions, the primary goal is to subtract the numerators while maintaining a common denominator. Let's explore this step-by-step

• Align the fractions using a common denominator—this ensures that you're comparing like quantities.
• With denominators aligned, subtract the numerators.
• Simplify the resulting fraction if necessary (more on that later).

For our expression \( \frac{3}{4} - \frac{5}{6} \), after finding a common denominator of 12, we subtract \( \frac{10}{12} \) from \( \frac{9}{12} \), giving a difference of \(-\frac{1}{12}\).

Adding Fractions

Adding fractions follows a similar pattern to subtraction but requires adding rather than subtracting the numerators. Using a common denominator is essential once again.
For the expression \( \frac{2}{3} + \left(-\frac{1}{12}\right) \), after resolving the subtraction as an addition, apply the following steps:

• Identify the common denominator—in this case, it's 12.
• Convert \( \frac{2}{3} \) into \( \frac{8}{12} \).
• Add the fractions: \( \frac{8}{12} + \frac{1}{12} = \frac{9}{12} \).
• Simplify if needed.

Our result, \( \frac{9}{12} \), can be simplified further.

Simplifying Fractions

Simplifying a fraction involves reducing it to its simplest form, where the numerator and denominator share no common factors other than 1. This makes the fraction easier to understand and use in future calculations.
First, check if both the numerator and the denominator are divisible by the same number. In our example, \( \frac{9}{12} \), both 9 and 12 can be divided by 3. Dividing gives us \( \frac{3}{4} \), which is the simplest form of the fraction.

• Check for common factors in the numerator and denominator.
• Divide both by the greatest common factor (GCF).
• Write down the simplified fraction.

Simplifying fractions keeps the solution neat and helps in recognizing equivalent fractions more efficiently.