Problem 3 - Algebra A Combined Function

Question

$$ \frac{4 m}{3 n}+\frac{5 m}{3 n} $$

Step-by-Step Solution

Answer

The simplified form is \(\frac{3m}{n}\).

Step 1: Identify the Common Denominator

Both fractions \( \frac{4m}{3n} \) and \( \frac{5m}{3n} \) have the same denominator, which is \( 3n \). This allows us to combine them directly.

Step 2: Add the Numerators

To add two fractions that have the same denominator, we simply add their numerators together while keeping the common denominator the same. The problem becomes: \[ \frac{4m + 5m}{3n}. \]

Step 3: Simplify the Expression

Now, add the terms in the numerator: \( 4m + 5m = 9m \). The fraction now looks like this: \[ \frac{9m}{3n}. \]

Step 4: Simplify the Fraction

Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor. Since both \(9m\) and \(3n\) are divisible by 3, we divide each by 3: \[ \frac{9m}{3n} = \frac{3m}{n}. \]

Key Concepts

Common Denominator

When working with fractions, a common denominator is essential for adding or subtracting them. A common denominator is a shared multiple of the denominators of two or more fractions. In simpler terms, it's the same bottom number across all fractions involved.
For the exercise provided, both \( \frac{4m}{3n} \) and \( \frac{5m}{3n} \) have the denominator \(3n\). This shared denominator, \(3n\), is what makes it straightforward to add these fractions together. When fractions have a common denominator, you can simply focus on adding or subtracting the numerators.
A good practice is to always look for the least common denominator if fractions have different denominators. However, in this case, since they are the same, it simplifies the process greatly.

Simplifying Fractions

Simplifying fractions is an important step to make them as simple as possible, ensuring you reach the most reduced form of a fraction. To simplify a fraction, you divide the numerator and the denominator by their greatest common divisor (GCD).
Once we reach the fraction \( \frac{9m}{3n} \) from our initial exercise, the next step is to simplify. The greatest common divisor here is 3, as it evenly divides both 9 and 3. Dividing both the numerator and the denominator by 3:

• \( 9m \div 3 = 3m \)
• \( 3n \div 3 = n \)

The simplified fraction then becomes \( \frac{3m}{n} \). Always aim to simplify fractions whenever possible to ensure accuracy and simplicity in your answers.

Adding Fractions

Adding fractions is straightforward when they have the same denominator. You keep the common denominator and add the numerators together. Let's break it down with the example from the exercise.
Given two fractions with the same denominator \( 3n \): \( \frac{4m}{3n} \) and \( \frac{5m}{3n} \), you add the numerators only:

• Numerator: \( 4m + 5m = 9m \)
• Denominator remains: \( 3n \)

The process simplifies to: \( \frac{9m}{3n} \). This result leads directly into the next step, which is simplifying the obtained fraction.

Greatest Common Divisor

The greatest common divisor (GCD) is a key concept in simplifying fractions. It is the largest number that can evenly divide both the top (numerator) and the bottom (denominator) of a fraction.
In our exercise, we have the fraction \( \frac{9m}{3n} \). Both 9 and 3 are divisible by 3. Therefore, the GCD of 9 and 3 is 3. This information allows us to reduce the fraction:

• Divide the numerator, 9m, by 3, resulting in 3m.
• Divide the denominator, 3n, by 3, resulting in n.

This simplification process yields the final fraction \( \frac{3m}{n} \). Using the GCD is efficient and ensures that the fraction is in its simplest terms.