Problem 70 - Beginning and Intermediate Algebra

Question

Find each difference. $$ -\frac{5}{6}-\frac{1}{2} $$

Step-by-Step Solution

Answer

\( -\frac{4}{3} \)

Step 1: Find a Common Denominator

To subtract fractions, the denominators must be the same. The denominators here are 6 and 2. The least common denominator (LCD) of 6 and 2 is 6.

Step 2: Convert Fractions to Have the Same Denominator

Convert \(-\frac{1}{2}\) so that its denominator is 6. \[ -\frac{1}{2} = -\frac{1 \times 3}{2 \times 3} = -\frac{3}{6} \] Now the fractions are \(-\frac{5}{6}\) and \(-\frac{3}{6}\).

Step 3: Subtract the Fractions

Subtract the numerators since the denominators are now the same: \[ -\frac{5}{6} - \frac{3}{6} = \frac{-5 - 3}{6} = \frac{-8}{6} \]

Step 4: Simplify the Fraction

Simplify \(-\frac{8}{6}\) by dividing the numerator and the denominator by their greatest common divisor (GCD), which is 2: \[ \frac{-8 \div 2}{6 \div 2} = \frac{-4}{3} \]

Key Concepts

Least Common Denominator

When subtracting fractions, you need to have the same denominator for both fractions. This shared number is called the Least Common Denominator (LCD). The LCD is the smallest number that both denominators can divide into without leaving a remainder. In the exercise, we had the denominators 6 and 2. The LCD for 6 and 2 is 6 because 6 is the smallest number that both 6 and 2 can evenly divide into.
To find the LCD, you can list the multiples of each denominator until you find the smallest one that appears in both lists:

• Multiples of 6: 6, 12, 18, ...
• Multiples of 2: 2, 4, 6, 8, ...

The first common number is 6. So, 6 is our LCD.
Finding the LCD is the first step in preparing fractions for subtraction.

Converting Fractions

After identifying the LCD, you will need to convert each fraction to have this common denominator. In this case, \(-\frac{1}{2}\) needs to be converted to have a denominator of 6.
You do this by multiplying both the numerator and the denominator by a number that will change the denominator to the LCD.
For \-\frac{1}{2}\, the denominator 2 needs to be multiplied by 3 to become 6. So you also multiply the numerator (which is -1) by 3: \[ \-\frac{1}{2} = -\frac{1 \times 3}{2 \times 3} = -\frac{3}{6} \] Now, \(-\frac{1}{2}\) is \-\frac{3}{6}\. Both fractions in the problem now have the same denominator, making it possible to subtract them.

Simplifying Fractions

The final step in subtracting fractions is simplifying the result. You simplify a fraction by dividing its numerator and denominator by their greatest common divisor (GCD).
In the exercise, after subtracting the fractions, we get -\frac{8}{6}\. Now, we need to find the GCD of 8 and 6. The GCD is the largest number that can evenly divide both numbers. For 8 and 6, the GCD is 2.
To simplify \-\frac{8}{6}\, you divide both the numerator and the denominator by 2: \[ \-\frac{8}{6} = -\frac{8 \div 2}{6 \div 2} = -\frac{4}{3} \] So, the final simplified answer is \-\frac{4}{3}\.
Simplifying makes the fraction easier to understand and work with in future calculations.