Problem 22 - Algebra 1: Concepts and Skills

Question

Write the sum in simplest form. $$ \frac{3}{2 z}+\frac{1}{z} $$

Step-by-Step Solution

Answer

\(\frac{5}{2 z}\)

Step 1: Identify the common denominator

The fractions \(\frac{3}{2 z}\) and \(\frac{1}{z}\) have a common denominator, \(z\). However, \(\frac{3}{2 z}\) has an additional factor of \(2\) in the denominator that needs to be accounted for. To make the fractions have exactly the same denominator, multiply the numerator and denominator of \(\frac{1}{z}\) by \(2\), which gives us \(\frac{2}{2 z}\).

Step 2: Perform the addition

Now, the problem is \(\frac{3}{2 z} + \(\frac{2}{2 z}\). Since the denominators are the same, we can add the numerators together, which gives us \(\frac{3+2}{2 z}\).

Step 3: Simplify the result

When we add the numerators, we get \(\frac{5}{2 z}\). This is the simplest form of the sum.

Key Concepts

Common Denominator

When adding fractions, finding a common denominator is crucial. The denominator is the lower part of a fraction and shows into how many parts the whole is divided. To add fractions seamlessly, both fractions should share this same denominator. In the exercise, the fractions were \(\frac{3}{2z}\) and \(\frac{1}{z}\).

At first glance, you might think that \(z\) is the common denominator. However, because \(\frac{3}{2z}\) has an extra factor of 2, we actually need \(2z\) as the common denominator. To fix this difference, we multiply the numerator and denominator of \(\frac{1}{z}\) by 2, converting it into \(\frac{2}{2z}\).

This step ensures that both fractions now have the same base in the denominators, making addition straightforward. Remember, dealing with a common denominator simplifies the process of handling fractions.

Simplifying Fractions

Simplifying fractions means reducing them to their smallest form, where the numerator and denominator share no common factors other than 1. In our example, after finding the common denominator and adding the fractions, we reached \(\frac{5}{2z}\).

Before calling a fraction fully simplified, ensure there are no common factors other than 1 between the numerator and denominator. Here, in \(\frac{5}{2z}\), 5 and 2 are already as simple as they can be, since they don’t share any other factors.

Always check if both numbers can be divided by the same number. If not, you have completed your fraction addition and reached the simplest form.

Numerators

Numerators are the top numbers in fractions and indicate how many parts of the whole are being considered. In our example, after adjusting for a common denominator, we had \(\frac{3}{2z}\) and \(\frac{2}{2z}\).

With fractions sharing the common denominator \(2z\), adding them becomes a matter of simply adding the numerators. Here, you sum the numerators, which are 3 and 2, together to get 5. This results in \(\frac{5}{2z}\).

It's crucial to remember that while adding fractions, only numerators are added directly. Denominators remain unchanged once a common one is established. This fundamental rule simplifies the entire addition process and ensures accuracy.