Problem 24 - Prealgebra

Question

Find sum or difference. Write in simplest form. \(\frac{17}{18}-\frac{5}{18}\)

Step-by-Step Solution

Answer

The simplest form of \( \frac{17}{18} - \frac{5}{18} \) is \( \frac{2}{3} \).

Step 1: Identify the denominators

In the fraction operation \( \frac{17}{18} - \frac{5}{18} \), both fractions have the same denominator, \( 18 \). This is necessary for direct subtraction.

Step 2: Subtract the numerators

Since the denominators are the same, subtract the numerators: \( 17 - 5 = 12 \). The resulting fraction is \( \frac{12}{18} \).

Step 3: Simplify the fraction

The fraction \( \frac{12}{18} \) can be simplified by dividing the numerator and the denominator by their greatest common divisor, which is \( 6 \). Thus, \( \frac{12 \div 6}{18 \div 6} = \frac{2}{3} \). The simplest form of the fraction is \( \frac{2}{3} \).

Key Concepts

Common Denominator

When you are working with fractions, especially when adding or subtracting them, the concept of a common denominator is vital. A denominator is the bottom part of a fraction, which tells us into how many equal parts the whole is divided. To add or subtract fractions, those parts must be the same size, which means the denominators of the fractions involved must be the same. This shared denominator is what we call a common denominator. In the problem we are considering, \( \frac{17}{18} - \frac{5}{18} \), both fractions already have the common denominator of 18. This allows us to subtract the fractions directly without any further adjustments. If the denominators were different, we would need to find their least common multiple to proceed with the subtraction.

• Common denominator makes fraction operations simpler by equalizing parts.
• Helps in performing subtraction and addition directly.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form, where the numerator and denominator have no common factors other than 1. This process makes the fraction easier to understand and use. In our example, after subtracting, we end up with the fraction \( \frac{12}{18} \). This fraction can be reduced by looking for the greatest common divisor.To simplify a fraction, follow these easy steps:

• Find the greatest common divisor (GCD) of the numerator and denominator.
• Divide both the numerator and the denominator by their GCD.

In our problem, simplifying \( \frac{12}{18} \) by dividing both parts by their GCD, which is 6, gives \( \frac{2}{3} \). A fraction in its simplest form is clearer and easier to compare or add with others.

Greatest Common Divisor

The greatest common divisor (GCD) is an essential concept when simplifying fractions. It refers to the largest number that can divide both the numerator and the denominator without leaving a remainder. In the context of the fraction \( \frac{12}{18} \), we determine that the GCD of 12 and 18 is 6. This is because:

• The divisors of 12 include: 1, 2, 3, 4, 6, 12.
• The divisors of 18 include: 1, 2, 3, 6, 9, 18.
• The largest common divisor among these is 6.

Using the GCD to simplify \( \frac{12}{18} \) gives \( \frac{2}{3} \), simplifying fraction operations and improving numerical understanding. Finding the GCD is a crucial step in reducing fractions to their simplest form.

Numerators and Denominators

Understanding the role of numerators and denominators is foundational for working with fractions. The numerator is the top part of a fraction and represents how many parts of the whole you have. Meanwhile, the denominator is the bottom part, which indicates how many parts the whole is divided into.In the subtraction problem \( \frac{17}{18} - \frac{5}{18} \), the numerators 17 and 5 tell us how many parts out of the 18 possible parts we have for each fraction before subtraction. Since the denominators are the same, 18, we can directly subtract these numerators to find the difference: \( 17 - 5 = 12 \).

• Numerators are the fractional parts we count or compare.
• Denominators represent total possible parts, setting the fraction size.

Having a solid grasp of these roles helps in easily maneuvering through addition, subtraction, and simplification of fractions. Remember, you can only combine or compare fractions with the same denominators by working through their numerators.