Problem 23 - Elementary Algebra

Question

For the following problems, perform each indicated operation. \(\frac{15}{23}-\frac{2}{23}\)

Step-by-Step Solution

Answer

Answer: The result is $\frac{13}{23}$.

Step 1: Identify the denominators

Both fractions in this problem have the same denominator, which is 23. Since the denominators are the same, we can perform the subtraction directly.

Step 2: Subtract the numerators

Subtract the numerators of the two fractions, which are the numbers in the top part of the fractions. In this case, we have: $$\frac{15-2}{23}$$

Step 3: Simplify the result

After subtracting the numerators, we get: $$\frac{13}{23}$$ There are no common factors between the numerator, 13, and the denominator, 23. Therefore, the fraction is already in its simplest form. So, the final result is: $$\frac{15}{23}-\frac{2}{23}=\frac{13}{23}$$

Key Concepts

Common Denominators

When dealing with fraction subtraction, a crucial first step is ensuring that the fractions share a common denominator. This makes the whole process much easier and more straightforward. The denominator is the bottom part of the fraction and indicates into how many parts something is divided. In the exercise given, both fractions have the same denominator of 23. This means they already have a common denominator, allowing us to move directly to the next step.

• Having a common denominator means you do not need to convert fractions.
• If the denominators were different, you would first need to find the least common multiple (LCM) to make them the same.
• This simplifies the operation since you're working with similar-size pieces.

When the denominators are the same, as in \( \frac{15}{23} - \frac{2}{23} \), you can simply focus on subtracting the numerators while keeping the denominator constant, as seen next.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form, which means making the numerator and denominator as small as possible while still retaining the value of the fraction. When a fraction is already in its lowest terms, it is easier to understand and compare to other fractions.

• To simplify a fraction, check if the numerator and denominator have any common factors.
• If common factors exist, divide both the numerator and the denominator by this greatest common factor (GCF).
• The fraction \( \frac{13}{23} \) from our problem cannot be further simplified because 13 and 23 have no common factors other than 1.

It's helpful to know that both 13 and 23 are prime numbers, meaning that their only divisors are 1 and themselves, underlining why this fraction is fully simplified.

Numerator Subtraction

Numerator subtraction is the step where we focus on the top numbers of the fractions, given they have common denominators. This involves a straightforward subtraction, taking the numerators of each fraction and performing the operation while keeping the denominator unchanged.

• The numerators are the top parts of the fractions, representing the number of counted parts.
• In our exercise, we subtract \(2\) from \(15\), resulting in \(13\).
• After the subtraction, the fraction becomes \( \frac{13}{23} \).

This step ensures that the fractional parts are correctly combined by subtracting only the parts counted (numerators) while the size of the parts (denominator) remains constant. This logic ensures accuracy and maintains the fraction's integrity.