Problem 73 - Algebra A Combined Function

Question

Perform the indicated operations. See Examples 11 through \(23 .\) $$ \frac{12}{5}-1 $$

Step-by-Step Solution

Answer

The result is \(\frac{7}{5}\).

Step 1: Convert the Subtraction into Addition

First, consider the expression \( \frac{12}{5} - 1 \). Notice that subtracting 1 can be thought of as adding the negative of 1. Essentially, we are dealing with \( \frac{12}{5} + (-1) \).

Step 2: Express 1 as a Fraction

To perform the addition (or subtraction) of fractions, we need to express 1 as a fraction with the same denominator as \( \frac{12}{5} \). Hence, rewrite 1 as \( \frac{5}{5} \).

Step 3: Subtract the Fractions

Now perform the subtraction: \( \frac{12}{5} - \frac{5}{5} = \frac{12 - 5}{5} \). Simplifying the numerator, we have \( \frac{7}{5} \).

Key Concepts

subtracting fractions

Subtracting fractions is an essential skill often used in math problems. It involves the removal of one fraction from another. The process may seem daunting at first, but it can be simplified with some steps. To subtract fractions effectively, it’s crucial to understand how to deal with different denominators and simplify the expressions. In essence, subtracting fractions requires making the denominators identical before performing the operation. This is because only with a common denominator can fractions be directly subtracted just like regular numbers. We'll touch on these steps more deeply in the following sections, ensuring you grasp not just the how, but the why, behind each move.

common denominators

When subtracting fractions, having a common denominator is a critical step. What exactly is a common denominator? Simply put, it’s a shared multiple of the denominators of the fractions you are working with.
To subtract fractions like \( \frac{12}{5} - 1 \), you first need to express the number 1 as a fraction with the same denominator as \( \frac{12}{5} \).
Here's how:

• Convert the whole number 1 to a fraction with denominator 5. Thus, 1 becomes \( \frac{5}{5} \).
• This step allows both fractions to be compared directly, since they now share the same baseline - the denominator.

With a common denominator established, the pathway to subtraction becomes clearer and easier.

fraction simplification

After subtracting fractions, it’s often necessary to simplify the result. Fraction simplification means reducing the fraction to its simplest form, where the numerator and denominator have no common factors other than 1.
In the example \( \frac{12}{5} - \frac{5}{5} = \frac{7}{5} \), we need to check if the fraction can be simplified.

• Check for common factors in the numerator and the denominator.
• If there are none, the fraction is already in its simplest form.
• In this case, 7 and 5 have no common factors besides 1, so \( \frac{7}{5} \) is already simplified.

Simplifying fractions makes them cleaner and often easier to interpret, enabling better understanding and quicker calculations in future operations.

improper fractions

Improper fractions are fractions where the numerator is larger than the denominator. In many exercises, such as the one with \( \frac{12}{5} - 1 \), you might end up with an improper fraction like \( \frac{7}{5} \).
Understanding how to handle improper fractions is crucial.
Here are a few pointers:

• An improper fraction can be converted into a mixed number by dividing the numerator by the denominator.
• In our example, divide 7 by 5. The quotient is 1, with a remainder of 2.
• This means \( \frac{7}{5} \) can be expressed as the mixed number 1\( \frac{2}{5} \).

Converting improper fractions isn't always necessary, but it can provide a clearer insight into the value of the fraction, especially in practical contexts.