Problem 98 - Elementary and Intermediate Algebra

Question

Perform the operations and, if possible, simplify. $$ 4-\frac{7}{3} $$

Step-by-Step Solution

Answer

The simplified result is \( \frac{5}{3} \).

Step 1: Identify the operation

The operation presented is a subtraction between the integer 4 and the fraction \( \frac{7}{3} \).

Step 2: Convert the integer into a fraction

Rewrite the integer 4 as a fraction with the same denominator as \( \frac{7}{3} \). This converts 4 into \( \frac{12}{3} \) since \( 4 = \frac{12}{3} \).

Step 3: Perform the subtraction

Subtract the fractions: \( \frac{12}{3} - \frac{7}{3} \). Since the fractions have a common denominator, the result is \( \frac{12 - 7}{3} = \frac{5}{3} \).

Step 4: Simplify the fraction

The fraction \( \frac{5}{3} \) is already in its simplest form because the numerator and the denominator have no common factors other than 1.

Key Concepts

Subtraction of Fractions

Subtracting fractions may seem intimidating at first, but it can become simple once you know the steps. When subtracting two fractions, it is essential to ensure that they have a common denominator. This makes the fractions easier to work with, as it aligns the fractional parts for a more straightforward calculation.
In our exercise, we have been given an integer to subtract from a fraction, specifically the operation is between 4 and \( \frac{7}{3} \). The critical step here is to first convert that integer into a fraction that shares the same denominator with \( \frac{7}{3} \). Once we achieve that, subtracting these fractions is merely a matter of subtracting the numerators while keeping the denominator unchanged.

• Ensure fractions have the same denominator.
• Subtract the numerators.
• The common denominator remains the same.

This leads us neatly to the next step in the operation, converting integers to fractions.

Converting Integers to Fractions

Converting integers into fractions is an important technique when dealing with fractional arithmetic. This process allows us to handle operations such as subtraction, addition, multiplication, and division seamlessly by bringing everything to the same type.
In our case, we needed to convert the integer 4 into a fraction with a denominator of 3, because our task involves the fraction \( \frac{7}{3} \).
Here's how you do it:

• Take the integer number, which is 4.
• Multiply this integer by the denominator of the fraction you're working with, which in this case is 3.
• As a result, 4 can be rewritten as \( \frac{12}{3} \), because \( 4 \times 3 = 12 \).

By converting integers into fractions with matching denominators, we pave the way for smooth subtraction or addition of fractions.
Now that the integer has been converted, we proceed with subtracting the fractions to get a result. After completing the subtraction, the next logical step is to see if further simplification is possible.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form, where the numerator and denominator have no common factors other than 1. This is a vital skill as it ensures results are presented in their most concise and recognizable states.
Once we performed the subtraction \( \frac{12}{3} - \frac{7}{3} = \frac{5}{3} \), we need to check if this fraction can be simplified further. To simplify a fraction, you look for the greatest common divisor (GCD) of the numerator and the denominator.
For \( \frac{5}{3} \), the number 5 and 3 have no common factors other than 1; hence, \( \frac{5}{3} \) is already in its simplest form. This step is significant as it confirms that our answer is as straightforward and clear as it can possibly be.
Simplification:

• Find the GCD of the numerator and denominator.
• If both can be divided by this GCD, do so to simplify.
• If the only common factor is 1, the fraction is already simplified.

Simplifying fractions often serves as the final step in confirming that our answer is correct and clean.