Problem 45 - Algebra 1: Concepts and Skills

Question

Add. Write the answer as a fraction or a mixed number in simplest form. $$ \frac{3}{4}+\frac{9}{4} $$

Step-by-Step Solution

Answer

The answer is 3. When adding fractions, remember to only add the numerators while keeping the denominator the same. Then simplify the fraction if possible.

Step 1: Identify the denominators

Identify the denominators of the fractions, which in this case are both 4. When the denominators are the same, you can easily add or subtract the fractions.

Step 2: Add the numerators

Now you add the numerators of the fractions. That is, 3 + 9 = 12, while keeping the denominator the same. This results in the fraction \(\frac{12}{4}\) .

Step 3: Simplify the fraction

The fraction \(\frac{12}{4}\) can be simplified by dividing both the numerator and the denominator by the same number. In this case, divide both by 4 to get the fraction down to its simplest form, which is \(\frac{3}{1}\).

Key Concepts

Simplest Form

When working with fractions, the term 'simplest form' refers to the version of a fraction where the numerator and the denominator are as small as possible. To achieve this, you must find the greatest common divisor (GCD) that both the numerator and the denominator share and divide them by it.

In the example exercise, we start with the fraction \(\frac{12}{4}\). If we list the factors of the numerator, 12 (1, 2, 3, 4, 6, 12) and the factors of the denominator, 4 (1, 2, 4), we see that the GCD is 4. Dividing both the numerator and the denominator by 4, we get the fraction \(\frac{3}{1}\), which is in its simplest form. Importantly, when a fraction has a denominator of 1, it can also be written as just the numerator, which in this case is simply 3.

Common Denominators

When adding fractions, having common denominators is crucial. The denominator tells us the total number of equal parts into which the whole is divided, and for fractions to be added directly, they need to share this value.

In our example, the fractions \(\frac{3}{4}\) and \(\frac{9}{4}\) already have common denominators of 4. This means we can just add the numerators together. If the denominators had been different, we would need to find the least common denominator (LCD) which is the smallest number that both denominators can divide into without leaving a remainder, and then adjust the numerators accordingly before adding them.

Simplifying Fractions

Simplifying fractions, also known as reducing fractions, involves expressing a fraction in its simplest form, as mentioned earlier. It's not just about dividing the numerator and denominator by their GCD though; understanding how to factor numbers is an essential skill here.

Factoring Numbers

• Identify the prime numbers that multiply together to give you the numerator and denominator.
• Cancel out the common prime factors.

After simplifying, you might end up with an improper fraction (where the numerator is larger than the denominator). When this happens, you can convert it to a mixed number for ease of understanding. However, in the given example, \(\frac{12}{4}\) simplifies directly to \(\frac{3}{1}\) or simply 3, which is an integer.