Problem 76 - Algebra A Combined Function

Question

Simplify. Follow the circled steps in the order shown. $$ \left.\frac{\frac{9}{5}+\frac{6}{5}}{\frac{17}{6}+\frac{7}{6}}\right\\} \leftarrow \text { (1) Add. } $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{3}{4}\).

Step 1: Add the Numerator

We start by adding the fractions in the numerator: \(\frac{9}{5} + \frac{6}{5}\).- They have the same denominator, so we add the numerators:\[9 + 6 = 15\]- Therefore, the sum is:\[\frac{15}{5}\]- Simplifying this fraction:\[\frac{15}{5} = 3\]The simplified numerator is 3.

Step 2: Add the Denominator

Next, add the fractions in the denominator: \(\frac{17}{6} + \frac{7}{6}\).- They have the same denominator, so we add the numerators:\[17 + 7 = 24\]- Therefore, the sum is:\[\frac{24}{6}\]- Simplifying this fraction:\[\frac{24}{6} = 4\]The simplified denominator is 4.

Step 3: Divide the Results

Finally, divide the simplified results from the numerator and denominator.- With the simplified numerator being 3 and the denominator being 4, the division is:\[\frac{3}{4}\]This fraction is already in its simplest form.

Key Concepts

Adding Fractions

Adding fractions may seem tricky at first, but once you start, it's a straightforward process. The key is to ensure that both fractions involved have the same denominator. - **What is a common denominator?**
A common denominator is simply the same number on the bottom of each fraction. If your fractions already share this, you can directly add the numerators (the top numbers). When you have \( \frac{9}{5} + \frac{6}{5} \), you see that both fractions have 5 as their denominator. This allows you to directly add the numerators:
- 9 + 6, which equals 15. Thus, you end up with the fraction \( \frac{15}{5} \), which can often be simplified further. Remember, having the same denominator is essential for smooth addition!

Simplifying Fractions

Once you've added fractions, the result might not always be in the simplest form. When we say "simplifying," we mean making a fraction as simple as possible by reducing it. - **Why simplify?**
A simplified fraction is easier to understand and work with. To simplify \( \frac{15}{5} \), consider these steps:

• Check if the numerator (top number) and denominator (bottom number) have common factors apart from 1.
• If they do, divide them both by that common factor. For \( \frac{15}{5} \), both numbers can be divided by 5.
  
• This gives you the fraction \( 3 \) since \( \frac{15}{5} \) is simplified to just 3.
  

Always look for the greatest common factor to ensure the fraction is reduced properly and efficiently. Simplifying fractions makes calculations and comparisons easier!

Division of Fractions

After adding and simplifying fractions, you may need to divide them, as seen in our problem. Division of one fraction by another results in a new fraction.- **Performing division:**
Once you have fractions in place, use division by following this simple process:

• Take the simplified results from the numerator and denominator.
• You transform the division into a multiplication by its reciprocal. The reciprocal of a fraction just means flipping the numerator and the denominator.
• For instance, if you have \( \frac{3}{4} \), it is already a fraction, so you've completed the process when you reach this step in our example.

Understanding this helps solve more complex fraction problems and makes tackling tasks like converting mixed numbers or improper fractions seamless.