Problem 96 - Elementary Algebra

Question

For exercises \(95-98\), evaluate. $$ \frac{5}{7}-\frac{2}{9} $$

Step-by-Step Solution

Answer

\(\frac{5}{7}-\frac{2}{9}=\frac{31}{63}\)

Step 1: Find a common denominator

To subtract fractions, find a common denominator. The denominators are 7 and 9. The least common multiple (LCM) of 7 and 9 is 63.

Step 2: Convert fractions to the common denominator

Convert \(\frac{5}{7}\) and \(\frac{2}{9}\) to fractions with the common denominator of 63. \(\frac{5}{7}=\frac{5 \cdot 9}{7 \cdot 9}=\frac{45}{63}\) and \(\frac{2}{9}=\frac{2 \cdot 7}{9 \cdot 7}=\frac{14}{63}\).

Step 3: Subtract the numerators

Subtract the numerators of the fractions with the common denominator: \(\frac{45}{63}-\frac{14}{63}=\frac{45-14}{63}=\frac{31}{63}\).

Step 4: Simplify the fraction

Simplify \(\frac{31}{63}\) if possible. In this case, 31 and 63 have no common factors other than 1, so the fraction is in its simplest form.

Key Concepts

least common multiple //the value of concept will be the headline later on so DO NOT restate the headline in text

To subtract fractions with different denominators, you need a common ground. This is where the least common multiple (LCM) comes in. The LCM of two numbers is the smallest number that is a multiple of both. For example, to find the LCM of 7 and 9 in the exercise, you list the multiples:
* Multiples of 7: 7, 14, 21, 28, 35, 42, 49, 56, 63 ...
* Multiples of 9: 9, 18, 27, 36, 45, 54, 63 ...
The smallest common multiple is 63.
Finding the LCM ensures that you can combine the fractions easily. This is a key step for fraction subtraction because it enables the use of a common denominator.

common denominator //the value of concept will be the headline later on so DO NOT restate the headline in text

A common denominator is a shared multiple of the denominators of two or more fractions. It allows fractions to be added or subtracted. In our exercise, 63 is the common denominator for \(\frac{5}{7}\) and \(\frac{2}{9}\).

By converting each fraction to have the common denominator 63, the fractions become:
* \(\frac{5}{7} = \frac{45}{63}\)
* \(\frac{2}{9} = \frac{14}{63}\)

Once they share the same denominator, you can simply subtract the numerators while maintaining the common denominator. This step is crucial in fraction arithmetic.

simplifying fractions //the value of concept will be the headline later on so DO NOT restate the headline in text

Simplifying fractions makes them easier to understand and work with. To simplify a fraction, you divide both the numerator and the denominator by their greatest common divisor (GCD). In our solution \(\frac{31}{63}\), 31 and 63 have no common factors other than 1, so it's already in its simplest form.

Simplifying fractions helps in:
* Making calculations easier
* Providing a clear, final answer
Understanding how to simplify fractions is essential to math proficiency and helps in various applications.