Problem 27 - Intermediate Algebra

Question

Perform the operations. See Example 2 . $$ \text { Subtract }-\frac{3}{5} \text { from } \frac{1}{2} $$

Step-by-Step Solution

Answer

\( \frac{1}{2} - (-\frac{3}{5}) = \frac{11}{10} \) or \( 1\frac{1}{10} \).

Step 1: Understand the Expression

The expression to solve is \( \frac{1}{2} - (-\frac{3}{5}) \). This means we need to subtract \( -\frac{3}{5} \) from \( \frac{1}{2} \). When subtracting a negative number, it is equivalent to adding its positive counterpart.

Step 2: Convert Subtraction to Addition

Subtracting a negative is the same as adding. Thus, the expression \( \frac{1}{2} - (-\frac{3}{5}) \) can be rewritten as \( \frac{1}{2} + \frac{3}{5} \).

Step 3: Find a Common Denominator

To add the fractions \( \frac{1}{2} \) and \( \frac{3}{5} \), we need a common denominator. The least common denominator of 2 and 5 is 10.

Step 4: Convert Fractions to Common Denominator

Convert \( \frac{1}{2} \) to have a denominator of 10: \( \frac{1}{2} = \frac{5}{10} \).Convert \( \frac{3}{5} \) to have a denominator of 10: \( \frac{3}{5} = \frac{6}{10} \).

Step 5: Add the Fractions

Now that both fractions have a common denominator, add them: \( \frac{5}{10} + \frac{6}{10} = \frac{11}{10} \).

Step 6: Simplify if Necessary

\( \frac{11}{10} \) is an improper fraction. As a mixed number, it is \( 1\frac{1}{10} \). However, \( \frac{11}{10} \) is simplest in this context.

Key Concepts

Subtracting Negative Numbers

Handling negative numbers in subtraction can be tricky at first. But there's a simple rule to remember: subtracting a negative number is equivalent to adding its positive counterpart. This concept is central to various mathematical operations.
Imagine owing a friend \(3. If they forgive the debt (subtracting the negative), you effectively gain \)3 – as if someone just handed you $3!
Let's apply it to fractions. If you subtract \(-\frac{3}{5}\) from \(\frac{1}{2}\), you'll actually add \(+\frac{3}{5}\) instead. This transition helps simplify many math problems.

Common Denominator

Adding or subtracting fractions requires a common denominator. This means each fraction must have the same number in the bottom part (the denominator) for easy calculation.
For example, to add \(\frac{1}{2}\) and \(\frac{3}{5}\), find the least common denominator, which in this case is 10.
With this common number:

• Convert \(\frac{1}{2}\) to \(\frac{5}{10}\)
• Convert \(\frac{3}{5}\) to \(\frac{6}{10}\)

Now, both fractions can easily be added or subtracted.

Improper Fractions

An improper fraction is when the top number (numerator) is bigger than the bottom number (denominator). It looks a bit strange compared to typical fractions, but it has its uses.
For instance, when you add \(\frac{5}{10}\) and \(\frac{6}{10}\) to get \(\frac{11}{10}\), the result is an improper fraction because 11 is greater than 10.
These fractions can also be expressed as mixed numbers but often stay in improper form for clarity in certain operations like this exercise. Understanding both forms will enhance your overall number comprehension.

Mixed Numbers

A mixed number combines a whole number with a fraction. It provides an easily understandable way to express improper fractions.
For instance, turning \(\frac{11}{10}\) into a mixed number gives you \(1\frac{1}{10}\), which conveys that there's one whole and a bit more.
Knowing how to switch between improper fractions and mixed numbers is beneficial, especially in real-life situations like measuring. It clarifies the quantity you have, beyond the simple fraction.