Problem 91 - Intermediate Algebra

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: Simplify Inside the Parentheses

First, focus on the expression inside the parentheses: \(\frac{3}{4} - \frac{5}{6}\).To subtract these fractions, we need a common denominator. The least common multiple of 4 and 6 is 12.Rewrite both fractions with the denominator of 12:\(\frac{3}{4} = \frac{9}{12}\) and \(\frac{5}{6} = \frac{10}{12}\).Perform the subtraction: \(\frac{9}{12} - \frac{10}{12} = -\frac{1}{12}\).

Step 2: Subtract the Result from the Outer Expression

Now, substitute the simplified expression back into the original expression:\(\frac{2}{3} - (-\frac{1}{12})\).Subtracting a negative is the same as adding, so rewrite it as:\(\frac{2}{3} + \frac{1}{12}\).

Step 3: Add the Fractions

Find a common denominator for \(\frac{2}{3}\) and \(\frac{1}{12}\). The least common multiple of 3 and 12 is 12.Rewrite \(\frac{2}{3} = \frac{8}{12}\).Now add the fractions:\(\frac{8}{12} + \frac{1}{12} = \frac{9}{12}\).

Step 4: Simplify the Final Fraction

Simplify \(\frac{9}{12}\) by finding the greatest common divisor (GCD) of 9 and 12, which is 3.Divide the numerator and denominator by 3:\(\frac{9}{12} = \frac{3}{4}\).

Key Concepts

Simplifying Fractions

Simplifying fractions is the process of making a fraction as simple as possible. A fraction is in its simplest form when the numerator and the denominator have no common factors other than 1. To simplify a fraction, you need to divide both the numerator and the denominator by their greatest common divisor (GCD).
For example, in the expression we are simplifying, we end up with \(\frac{9}{12}\). The GCD of 9 and 12 is 3, so to simplify, divide both 9 and 12 by their GCD:

• \(9 \div 3 = 3\)
• \(12 \div 3 = 4\)

Thus, \(\frac{9}{12}\) simplifies to \(\frac{3}{4}\).
Simplifying is important because it makes fractions easier to understand and often easier to work with in further calculations.

Least Common Denominator

The least common denominator (LCD) is a crucial concept when working with fractions, especially for adding and subtracting them. The LCD is the smallest number that both denominators can divide into evenly.
In more straightforward terms, it’s the least common multiple (LCM) of the denominators. For example, when subtracting \(\frac{3}{4}\) and \(\frac{5}{6}\), you need to find the LCD of 4 and 6. The smallest multiple that 4 and 6 share is 12, making it the LCD.

• Rewrite \(\frac{3}{4}\) as \(\frac{9}{12}\)
• Rewrite \(\frac{5}{6}\) as \(\frac{10}{12}\)

Using the LCD allows you to perform subtraction or addition of fractions smoothly, as shown in the initial simplification inside the parentheses.

Subtracting Fractions

Subtracting fractions might seem challenging, but it's straightforward if you follow the steps carefully. You must first ensure that both fractions share the same denominator. This often involves converting them using their least common denominator (LCD).
Take the example of subtracting \(\frac{3}{4}\) and \(\frac{5}{6}\). We rewrite both fractions with a common denominator of 12:

• \(\frac{3}{4} = \frac{9}{12}\)
• \(\frac{5}{6} = \frac{10}{12}\)

Once both fractions are over 12, you can easily perform the subtraction: \(\frac{9}{12} - \frac{10}{12} = -\frac{1}{12}\). Remember, you subtract the numerators and keep the common denominator.
Having a common denominator makes subtraction (or addition) much simpler, as the denominators align, allowing for direct arithmetic operations on the numerators.

Adding Fractions

Adding fractions follows similar steps to subtraction, where the key is having a common denominator. After simplifying the subtraction part of the exercise, the expression became \(\frac{2}{3} + \frac{1}{12}\).
To add these, first convert \(\frac{2}{3}\) to have the same denominator as \(\frac{1}{12}\), which is 12. Utilize the least common denominator:

• \(\frac{2}{3} = \frac{8}{12}\)

With the fractions \(\frac{8}{12}\) and \(\frac{1}{12}\), simply add the numerators, and the sum is over the common denominator:

• \(8 + 1 = 9\)

So \(\frac{8}{12} + \frac{1}{12} = \frac{9}{12}\).
Understanding the process of finding a common denominator and then adding the numerators allows for quick and accurate calculations, reinforcing the understanding of fraction addition.