Problem 53 - Beginning and Intermediate Algebra

Question

Add or subtract as indicated. Write the answer in lowers ferms. See Example 7. $$ \frac{12}{5}-1 $$

Step-by-Step Solution

Answer

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

Step 1: Convert Whole Number to Fraction

Express the whole number 1 as a fraction with the same denominator as \( \frac{12}{5} \). This means rewriting 1 as \( \frac{5}{5} \).

Step 2: Subtraction of Fractions

Subtract the equivalent fraction from \( \frac{12}{5} \). So, compute \( \frac{12}{5} - \frac{5}{5} \).

Step 3: Calculation

Perform the subtraction: \( \frac{12 - 5}{5} = \frac{7}{5} \).

Step 4: Simplest Form

The fraction \( \frac{7}{5} \) is already in its simplest form, as 7 and 5 have no common factors other than 1.

Key Concepts

Subtracting Fractions

When you need to subtract one fraction from another, it is important that both fractions have the same denominator. The denominator is the bottom number of a fraction, indicating into how many equal parts the whole is divided. If fractions do not share the same denominator, you'll need to find a common denominator before you can subtract. However, in the example given, both \[ \frac{12}{5} \text{ and } \frac{5}{5} \] share the same denominator. This makes the subtraction straightforward: simply subtract the numerators (the top numbers of the fractions) and keep the common denominator unchanged. Thus, \[ \frac{12}{5} - \frac{5}{5} \] becomes \[ \frac{12 - 5}{5} = \frac{7}{5}. \] Remember: if dealing with fractions that have different denominators, you must adjust one or both fractions to have a common denominator before performing the subtraction.

Whole Number as Fraction

Fractions are versatile and can represent whole numbers by having a denominator and numerator that are the same. This concept helps in operations with fractions since a whole number can easily be integrated into a fraction equation. For example, transforming a whole number like 1 into a fraction involves writing it as a fraction whose numerator and denominator are equal. With respect to our problem, the whole number 1 is expressed as \[ \frac{5}{5}. \] This keeps the value of 1 unchanged but allows it to be easily combined with the fraction \[ \frac{12}{5} \text{ when performing subtraction.} \] This technique is extremely useful when performing operations on fractions, especially when needing matching denominators.

Simplest Form

Writing a fraction in its simplest form means expressing it in the smallest possible terms, with no common factors between the numerator and the denominator other than 1. This makes the fraction easier to read and understand. After performing an operation on fractions, always check if the result can be simplified. In the case of \[ \frac{7}{5}, \] 7 and 5 have no common factors other than 1. The fraction is already as simple as it gets. Simplifying a fraction involves dividing both the numerator and the denominator by their greatest common divisor (GCD). However, some fractions, like \[ \frac{7}{5}, \] are already in their simplest form as soon as the calculation is complete. Always double-check the final answer, as this ensures clarity and precision in your mathematical work.