Problem 2 - Technical Mathematics with Calculus

Question

Combine and simplify. Don't use your calculator for these numerical problems. The practice you get working with common fractions will help you when doing algebraic fractions. $$\frac{1}{8}-\frac{3}{8}$$

Step-by-Step Solution

Answer

\(-\frac{1}{4}\)

Step 1: Identify Common Denominators

Observe that both fractions have the same denominator, which is 8.

Step 2: Subtract Numerators

Since the denominators are the same, subtract the numerators: 1 - 3.

Step 3: Simplify the Result

Calculate the result of the subtraction of the numerators which gives -2. The simplified fraction is then \(-\frac{2}{8}\) which can be further simplified.

Step 4: Reduce the Fraction

Divide both the numerator and the denominator by their greatest common divisor, which is 2. So, \(-\frac{2}{8} = -\frac{1}{4}\).

Key Concepts

Common Denominators

When it comes to adding or subtracting fractions, finding a common denominator is a critical step. A common denominator refers to a shared multiple of the denominators in two or more fractions. Having the same denominator allows fractions to be combined accurately. In the example of \(\frac{1}{8}-\frac{3}{8}\), the denominator is already common at 8.

In cases where the denominators are not the same, we would normally look for the least common multiple of the denominators to convert them into equivalent fractions with a common denominator. This makes it possible to directly compare and combine the fractions. However, when the denominators match, as in the given exercise, you can move directly to the next step without any additional work. This streamlines the process and is a vital foundation for operations with fractions.

Subtracting Numerators

Once a common denominator is established, the next step is subtracting the numerators. This is because when the denominators are the same, the pieces of the fractions are of equal size, and thus, we can simply subtract the number of pieces. Think of it like having slices of the same cake. If one person has 1 slice and another has 3 slices, if you take the 1 slice from the 3, you end up with 2 slices taken away, or in the case of our fractions, a negative value since we had fewer slices to begin with.

So in \(\frac{1}{8}-\frac{3}{8}\), we subtract the numerator 1 from 3, giving us \(1 - 3 = -2\). The result is then placed over the common denominator to form the new fraction \(\frac{-2}{8}\).

Reducing Fractions

The final step is reducing the fraction to its simplest form. Reducing, also known as simplifying, involves dividing the numerator and the denominator by their greatest common divisor (GCD). To further simplify \(\frac{-2}{8}\), we find that both 2 (or -2) and 8 are divisible by 2. When we divide them both by 2, we get the reduced fraction \(\frac{-1}{4}\).

It’s important to know that a fraction is in simplest form when the numerator and denominator are as small as possible, meaning they cannot be divided by the same number anymore. This makes the fraction easier to understand and work with, especially when dealing with more complex mathematical operations. Always remember that the goal of reducing fractions is to maintain the original value while using the smallest possible numerator and denominator.