Problem 66 - Elementary Algebra

Question

For exercises \(23-74\), evaluate. $$ \frac{7}{15}-\frac{4}{15} $$

Step-by-Step Solution

Answer

\frac{1}{5}

Step 1: Identify Common Denominator

Both fractions \(\frac{7}{15}\) and \(\frac{4}{15}\) already have the same denominator, which is 15.

Step 2: Subtract the Numerators

Since the denominators are the same, subtract the numerators: \(7 - 4 = 3\)

Step 3: Form the Fraction

Place the result from the numerator subtraction over the common denominator: \(\frac{3}{15}\)

Step 4: Simplify the Fraction

To simplify \(\frac{3}{15}\), find the greatest common divisor (GCD) of 3 and 15, which is 3. Divide both the numerator and the denominator by the GCD: \(\frac{3 \div 3}{15 \div 3} = \frac{1}{5}\)

Key Concepts

Common Denominator

When subtracting fractions, we need to have the same denominator for both fractions. Denominator is the bottom part of a fraction. A common denominator makes the fraction parts comparable and easier to subtract.
For example, in the given problem \( \frac{7}{15} - \frac{4}{15} \), both fractions already have 15 as a common denominator. This saves an extra step. If the denominators were different, we would need to find a common multiple of both denominators.
In short, having the same denominator allows us to focus directly on subtracting the numerators.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form. This makes the fractions easier to understand and compare.

In our example, after subtracting the fractions, we got \( \frac{3}{15} \). To simplify it, we need to find the smallest form of this fraction by dividing the numerator and the denominator by their greatest common divisor (GCD).
Once we simplified \( \frac{3}{15} \), we obtained \( \frac{1}{5} \), which is much simpler to work with.

Greatest Common Divisor (GCD)

The Greatest Common Divisor (GCD) is the largest number that can evenly divide both the numerator and the denominator. It helps in simplifying the fraction to its simplest form.

To find the GCD of 3 and 15, list out the divisors:

• Divisors of 3: 1, 3
• Divisors of 15: 1, 3, 5, 15

The largest number in both lists is 3. Hence, 3 is the GCD of 3 and 15.
By dividing both the numerator and the denominator by 3, the fraction \( \frac{3}{15} \) is simplified to \( \frac{1}{5} \).

Numerator and Denominator

Understanding fractions involves knowing the roles of the numerator and the denominator. The numerator is the top part of the fraction and indicates how many parts we have. The denominator is the bottom part and shows the total number of equal parts the whole is divided into.

In the problem \( \frac{7}{15} - \frac{4}{15} \), both fractions have 15 as the denominator. When subtracting these fractions, we keep the denominator the same and subtract the numerators:

• Numerator: 7 - 4 = 3
• Denominator: 15

This gives us the initial fraction result \( \frac{3}{15} \), which we then simplify.
Always keep the denominator unchanged when the denominators are common during subtraction.