Problem 83 - Elementary Algebra

Question

For exercises 81-96, evaluate. $$ \frac{18}{5}-\frac{2}{15} $$

Step-by-Step Solution

Answer

\( \frac{52}{15} \)

Step 1: Find a common denominator

To subtract the fractions, the denominators must be the same. The least common multiple (LCM) of 5 and 15 is 15.

Step 2: Convert the fractions

Convert \(\frac{18}{5}\) to a fraction with a denominator of 15. \(\frac{18}{5} = \frac{18 \times 3}{5 \times 3} = \frac{54}{15}\). The fraction \(\frac{2}{15}\) already has the correct denominator.

Step 3: Subtract the fractions

Subtract the numerators of the fractions while keeping the denominator the same: \(\frac{54}{15} - \frac{2}{15} = \frac{54-2}{15} = \frac{52}{15}\).

Step 4: Simplify the fraction (if possible)

Check if \(\frac{52}{15}\) can be simplified further. Since 52 and 15 have no common factors other than 1, the fraction is already in its simplest form.

Key Concepts

Common Denominator

When subtracting fractions, it is crucial to have a common denominator. This allows you to combine the fractions into one. Think of the denominator as a type of 'base' that needs to be the same for both fractions.
For example, in the problem \(\frac{18}{5} - \frac{2}{15}\), the denominators are 5 and 15. These need to be made the same so you can easily subtract the fractions.
In this case, we find that the common denominator is 15.

Least Common Multiple (LCM)

The least common multiple (LCM) is the smallest number that is a multiple of two or more numbers. To subtract fractions with different denominators, you convert the fractions so they each have this least common multiple as the denominator.
For our example, we need the LCM of 5 and 15. The LCM of 5 and 15 is 15, because 15 is the smallest number that both 5 and 15 can divide into.
Once you find the LCM, you can adjust the fractions to have this new common denominator.

Simplifying Fractions

After subtracting the fractions, it's important to check if the resulting fraction can be simplified. Simplifying fractions means reducing them to their simplest form, where the numerator and the denominator are as small as possible.
This involves finding the greatest common divisor (GCD) of the numerator and denominator. If their GCD is 1, the fraction is already in its simplest form.
In our example, \(\frac{52}{15}\) cannot be simplified further because the GCD of 52 and 15 is 1.

Numerators

The numerator is the top part of a fraction and represents the number of parts we have. When subtracting fractions, you subtract the numerators and keep the common denominator.
In the original problem, after finding a common denominator of 15, the fractions become \(\frac{54}{15}\) and \(\frac{2}{15}\). You then subtract the numerators, so \(\frac{54 - 2}{15} = \frac{52}{15}\).
Always perform the subtraction on the numerators while keeping the denominator the same.