Problem 60 - Algebra A Combined Function

Question

Perform the indicated operations. See Examples 11 through \(23 .\) $$ \frac{18}{35}-\frac{11}{35} $$

Step-by-Step Solution

Answer

The result is \(\frac{1}{5}\).

Step 1: Identify the Operation

The exercise asks us to subtract two fractions, \(\frac{18}{35}\) and \(\frac{11}{35}\). Since the fractions have the same denominator, we can directly perform the subtraction of their numerators.

Step 2: Subtract the Numerators

Subtract the numerators of the fractions while keeping the denominator the same. This means subtracting \(18\) and \(11\). The operation is: \(18 - 11 = 7\). Thus, the result is \(\frac{7}{35}\).

Step 3: Simplify the Fraction

We need to check if \(\frac{7}{35}\) can be simplified. The greatest common divisor (GCD) of \(7\) and \(35\) is \(7\), so divide both the numerator and the denominator by \(7\). \(\frac{7}{35} \div \frac{7}{7} = \frac{1}{5}\).

Step 4: Conclusion

The result of performing the operation \(\frac{18}{35} - \frac{11}{35}\) and simplifying, is \(\frac{1}{5}\).

Key Concepts

Fraction Simplification

Fraction simplification is all about making a fraction as simple as possible. This involves reducing both the numerator and denominator by dividing them by their greatest common factor, or GCF. When you simplify a fraction, you're not changing its overall value, but making it easier to understand or work with.

• For instance, the fraction \( \frac{7}{35} \) can be simplified.
• We look for the greatest number that divides both numerals cleanly, which is 7 in this case.
• When we divide both the numerator (7) and the denominator (35) by 7, we get \( \frac{1}{5} \).

This process of simplification is crucial especially when preparing fractions for further mathematical operations, as it makes calculations neater and reduces the chance of errors later on.

Numerator and Denominator

Fractions are made up of two key parts: the numerator and the denominator. The numerator is the top number, and the denominator is the bottom number.

• The numerator tells you how many parts of a whole you are considering. For example, in \( \frac{18}{35} \), the numerator is 18.
• The denominator shows into how many parts the whole is divided. So in the same fraction, 35 is the denominator.

Understanding these parts helps us perform operations like addition and subtraction on fractions. If the denominators are the same, as seen with \( \frac{18}{35} \) and \( \frac{11}{35} \), subtraction becomes straightforward: simply subtract the numerators while keeping the denominator constant.

Greatest Common Divisor (GCD)

The greatest common divisor, or GCD, is the largest number that can evenly divide two numbers. It plays an important role when simplifying fractions by reducing them to their simplest forms.

• To find the GCD of two numbers, list the factors of each number.
• For example, factors of 7 are just {1, 7}, and 35 has factors {1, 5, 7, 35}.
• The largest common factor, in this case, is 7.
• Thus, the GCD of 7 and 35 is 7.

Using the GCD, we can simplify the fraction \( \frac{7}{35} \) by dividing both the numerator and the denominator by 7. This simplifies the fraction to \( \frac{1}{5} \), which is much easier to work with in subsequent calculations.