Question

Add the following fractions and mixed numbers. Reduce to lowest terms. \(\frac{15}{47}+\frac{15}{47}=\) ______

Short answer

The sum is \( \frac{30}{47} \).

Step-by-step solution

Identify Like Fractions

The fractions are \( \frac{15}{47} \) and \( \frac{15}{47} \). Since both fractions have the same denominator, they are like fractions.

Add the Numerators

Since the denominators are the same, we only add the numerators. So, we have \( 15 + 15 = 30 \).

Write the Resulting Fraction

Place the sum of the numerators over the common denominator to find \( \frac{30}{47} \).

Simplify the Fraction

Check if \( \frac{30}{47} \) can be simplified. Since 47 is a prime number and does not divide 30 evenly, \( \frac{30}{47} \) is already in its lowest terms.

Key concepts

Like Fractions

When dealing with fractions, a common task is to determine if fractions are "like fractions". Like fractions are simply fractions that have the same denominator. In the exercise, both fractions are \( \frac{15}{47} \), sharing the denominator 47, thus they are like fractions.Recognizing like fractions is important because it simplifies the addition or subtraction process:

• You can add or subtract their numerators directly without worrying about finding a common denominator.

Identifying like fractions makes the operations with fractions straightforward by reducing extra steps, allowing a focus on the numerators.

Numerators

The numerator in a fraction represents the top part of the fraction. It shows how many parts of the whole are being considered. In like fractions, you only need to add these numerators, as the denominators are the same.In our example, \( \frac{15}{47} + \frac{15}{47} \):

• Numerators: 15 and 15

Adding the numerators is straightforward:

• \( 15 + 15 = 30 \)

Then, you simply put the sum over the common denominator 47, resulting in our new fraction:

• \( \frac{30}{47} \)

This step shows the ease of working with like fractions, where the task is made simple by managing just the numerators.

Simplifying Fractions

Once you have a fraction after addition, the next step is usually to simplify it. Simplifying a fraction means reducing it to its simplest form, where the numerator and denominator have no common factors other than 1.For example, to simplify \( \frac{30}{47} \):

• Check if the numerator and denominator share any common factors.
• If no common factors exist, other than 1, the fraction is already simplified.

In this case:

• 47 is a prime number and does not divide 30 evenly.

So, \( \frac{30}{47} \) remains unsimplified because it is already in its simplest form.

Lowest Terms

The term "lowest terms" refers to a fraction being expressed in its simplest form. There are no common factors between the numerator and the denominator, except for 1.To check if a fraction is in its lowest terms:

• Identify any common factors for the numerator and the denominator.
• If none other than 1 appear, the fraction is already in lowest terms.

In our exercise, \( \frac{30}{47} \) is already in its lowest terms since:

• 47 is a prime number and has no factors in common with 30.

This means that \( \frac{30}{47} \) is fully simplified, representing the fraction as compactly as possible.