Problem 46 - Prealgebra

Question

Add or subtract as indicated. $$\frac{x}{8}-\frac{5}{6}$$

Step-by-Step Solution

Answer

The result is \( \frac{3x - 20}{24} \).

Step 1: Find the Least Common Denominator (LCD)

To add or subtract fractions, they must have a common denominator. The denominators here are 8 and 6. The least common denominator (LCD) is the least common multiple of these two numbers. The LCM of 8 and 6 is 24.

Step 2: Convert Each Fraction

We need to adjust each fraction to have the common denominator of 24. - For \( \frac{x}{8} \), multiply both the numerator and the denominator by 3 to get \( \frac{3x}{24} \).- For \( \frac{5}{6} \), multiply both the numerator and the denominator by 4 to get \( \frac{20}{24} \).

Step 3: Subtract the Fractions

Now that the fractions have the same denominator, subtract the second fraction from the first: \[ \frac{3x}{24} - \frac{20}{24} = \frac{3x - 20}{24} \].

Step 4: Simplify the Result (if possible)

Examine \( \frac{3x - 20}{24} \) to see if it can be simplified further. In this case, because \(3x\) and 20 do not have common factors, the expression cannot be simplified further.

Key Concepts

Least Common Denominator

When adding or subtracting fractions, the key is to have a shared bottom number, which is known as the denominator. To find a common ground between the denominators, we calculate the Least Common Denominator (LCD). The LCD is simply the Least Common Multiple (LCM) of the individual denominators.
To find the LCM of two numbers, list their multiples and identify the smallest one they share. For example, with the denominators 8 and 6:

• Multiples of 8: 8, 16, 24, 32, 40...
• Multiples of 6: 6, 12, 18, 24, 30...

The smallest common multiple is 24, which becomes our LCD. Now, both fractions can be transformed to have 24 as their denominator, simplifying the process of addition or subtraction.

Least Common Multiple

The Least Common Multiple (LCM) refers to the smallest number that is a multiple of two or more numbers. Finding the LCM is essential when working with fractions, as it helps determine the Least Common Denominator needed for calculations.
Here's how to find the LCM for the numbers 8 and 6:

• Break down each number into its prime factors: 8 is 2 × 2 × 2 and 6 is 2 × 3.
• Identify the highest power of each unique factor: For 2, use 2 × 2 × 2 = 8, and for 3, use 3 (as 3¹).

Multiply these together to get the LCM: 8 × 3 = 24. This number acts as a universal denominator, allowing fractions to be easily added or subtracted.

Simplifying Fractions

After combining fractions using their common denominator, the result should be simplified if possible. Simplifying means reducing a fraction to its simplest form, where the numerator and denominator have no common factors other than 1.
To simplify:

• Find the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both by the GCD to reduce them to the simplest form.

In our example, the fraction resulting from subtraction was \(\frac{3x - 20}{24}\). Since 3x and 20 do not share any common factors, this fraction is at its simplest form already. Thus, knowing how to identify and simplify fractions ensures accuracy and neatness in your work.