Problem 77 - Algebra 2

Question

Evaluate each expression. \(\frac{5}{6}-\frac{1}{5}\)

Step-by-Step Solution

Answer

The result of \(\frac{5}{6} - \frac{1}{5}\) is \(\frac{19}{30}\).

Step 1: Find a Common Denominator

To subtract fractions, we need a common denominator. The denominators are 6 and 5. The least common multiple of 6 and 5 is 30. This will be our common denominator.

Step 2: Convert Each Fraction

Convert each fraction to an equivalent fraction with the common denominator of 30. For \(\frac{5}{6}\):Multiply both numerator and denominator by 5 to get \(\frac{25}{30}\). For \(\frac{1}{5}\):Multiply both numerator and denominator by 6 to get \(\frac{6}{30}\).

Step 3: Subtract the Fractions

With the same denominator, subtract the numerators and place the result over the common denominator. Subtract \(\frac{6}{30}\) from \(\frac{25}{30}\). This gives:\(\frac{25 - 6}{30} = \frac{19}{30}\)

Step 4: Simplify the Result

Check if the resulting fraction \(\frac{19}{30}\) can be simplified. Since 19 is a prime number and does not divide evenly into 30, \(\frac{19}{30}\) is in its simplest form.

Key Concepts

Least Common Denominator

When subtracting fractions, a common denominator is needed. This shared denominator ensures we can combine or compare fractions directly. For any two denominators, we should find the least common multiple (LCM). In the case of the original exercise with the fractions \(\frac{5}{6}\) and \(\frac{1}{5}\), the denominators are 6 and 5. The least common multiple of these numbers helps us identify the least common denominator (LCD), which is 30 in this problem. This number brings both fractions to a common footing, making subtraction straightforward.

Equivalent Fractions

Once you have the least common denominator, it's time to convert your fractions to equivalent fractions with this common denominator. To create equivalent fractions:

• Multiply both the numerator and the denominator by the same number so that only the form changes, but they remain equal in value.
• For \(\frac{5}{6}\): Multiply by 5 to get \(\frac{25}{30}\).
• For \(\frac{1}{5}\): Multiply by 6 to get \(\frac{6}{30}\).

This ensures that both fractions now have denominators of 30 while keeping their original value intact. Equivalent fractions are truly the unsung heroes of fraction arithmetic, allowing for easy subtraction or addition with a uniform denominator.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form. This involves dividing both the numerator and the denominator by their greatest common divisor (GCD). In our exercise, the fraction \(\frac{19}{30}\) was the result. Here:

• Check if any number can evenly divide both the numerator (19) and the denominator (30).
• For \(\frac{19}{30}\), since 19 is a prime number and not a divisor of 30, it cannot be simplified further.

Therefore, \(\frac{19}{30}\) already represents its simplest form. Simplifying fractions not only makes them cleaner but helps in easier data handling in calculations.

Prime Numbers

Prime numbers are fundamental in mathematics, being numbers greater than 1 that have no divisors other than 1 and themselves. Identifying prime numbers is crucial, especially during simplification or factorization processes. The number 19, encountered in \(\frac{19}{30}\), is a prime number.

• Prime numbers help determine whether fractions can be simplified because they don't have divisors other than 1 and themselves.
• This aids in isolating the greatest common factors during fraction reduction, provided any exist.

If a numerator or denominator is prime and doesn't hint at further simplification possibilities, it signals the simplest form. Knowing prime numbers enhances your confidence in verifying the accuracy of your answer during fraction problems.