Problem 18 - Prealgebra

Question

Find each sum or difference. Write in simplest form. $$-\frac{1}{2}+\frac{3}{8}$$

Step-by-Step Solution

Answer

The sum is \(-\frac{1}{8}\).

Step 1: Identify a Common Denominator

The fractions have different denominators: 2 and 8. To add them, we need a common denominator. The least common multiple (LCM) of 2 and 8 is 8. So, we'll use 8 as the common denominator.

Step 2: Convert the First Fraction

Convert \(-\frac{1}{2}\) to an equivalent fraction with 8 as the denominator. Multiply both the numerator and the denominator of \(-\frac{1}{2}\) by 4 to get \(-\frac{4}{8}\).

Step 3: Add the Fractions

Now that both fractions have the same denominator, add the numerators: \(-\frac{4}{8} + \frac{3}{8} = \frac{-4 + 3}{8} = \frac{-1}{8}\).

Step 4: Write the Result in Simplest Form

The resulting fraction \(-\frac{1}{8}\) is already in its simplest form, as -1 and 8 have no common factors other than 1.

Key Concepts

Common Denominator

When working with fractions, one of the first hurdles we face is dealing with different denominators. A denominator is the bottom number in a fraction, showing the number of equal parts the whole is divided into. To add or subtract fractions, each needs to have the same bottom number, known as the 'common denominator'.

To find a suitable common denominator, we look for a number that can be divided by each of the denominators without leaving a remainder. Finding this number is crucial because it allows us to combine the fractions easily. A simple rule of thumb for small numbers is to use the least common multiple (LCM) of the denominators. Once the common denominator is found, we adjust the fractions accordingly by converting them so that their denominators are the same. This is an important step, as it sets the path for accurate addition or subtraction of fractions.

Least Common Multiple

The least common multiple (LCM) is essential when dealing with fractions having different denominators. It is the smallest non-zero number that is a multiple of two or more numbers. Finding the LCM helps us determine the smallest number that each denominator can divide into evenly.

To find the LCM of two numbers, you can start by listing multiples of each number until you find the smallest common one. For example, for the numbers 2 and 8:

• Multiples of 2 are 2, 4, 6, 8, 10, etc.
• Multiples of 8 are 8, 16, 24, 32, etc.

The smallest common multiple in these lists is 8, making it the LCM. Once the LCM is identified, it becomes the new denominator for the fractions, and you can then adjust the numerators to maintain the value of the fractions.

Simplifying Fractions

After adding or subtracting fractions, always check if the resulting fraction can be simplified. Simplifying a fraction means reducing it to its simplest form by ensuring the numerator and denominator have no common factors other than 1. This process makes fractions easier to understand and work with.

To simplify a fraction, divide both the numerator and the denominator by their greatest common divisor (GCD), which is the highest number that divides both numbers without leaving a remainder. For instance, if we have the fraction \(\frac{-1}{8}\), we check to see if there is a number greater than 1 that can divide both -1 and 8 evenly. Since 1 is the only number that can do this, \(\frac{-1}{8}\) is already in its simplest form. Simplifying fractions not only helps in developing better number sense but also in making computation faster and results clearer.