Problem 65 - Algebra 1: Concepts and Skills

Question

Add. Write the answer as a fraction or as a mixed number in simplest form. $$ 5 \frac{3}{14}+8 \frac{9}{14} $$

Step-by-Step Solution

Answer

The sum of \(5 \frac{3}{14}\) and \(8 \frac{9}{14}\) is \(13 \frac{6}{7}\)

Step 1:Separate fractions and whole numbers

Separate the fraction and whole number part of both mixed numbers and add those parts separately. So, it looks like this: \(5 + 8\) and \(\frac{3}{14}+ \frac{9}{14}\)

Step 2:Add the whole numbers

Add the whole numbers, which are 5 and 8. \(5 + 8 = 13\)

Step 3:Add the fractions

Adjust the fractions so they have a common denominator (if they do not already), then add them. In this case the fractions already have the same denominator. Add the fraction parts from both, which will give \(\frac{3}{14}+ \frac{9}{14} = \frac{12}{14}\)

Step 4:Simplify the fraction

The fraction can be reduced by dividing the numerator and the denominator by their greatest common divisor. In this case, we can divide the numerator and the denominator by 2. So, the fraction \(\frac{12}{14}\) simplifies to \(\frac{6}{7}\)

Step 5:Combine the results

Combine the results of both parts. The resulting mixed number is \(13 \frac{6}{7}\)

Key Concepts

Adding Mixed Numbers

Understanding how to add mixed numbers is essential in mathematics. Mixed numbers consist of a whole number and a fraction, which can initially seem daunting to add. However, when we decompose these numbers into their components, the task becomes much simpler.

For example, let's look at the mixed numbers \(5 \frac{3}{14}\) and \(8 \frac{9}{14}\). The first step is to separate the whole numbers from the fractions: \(5 + 8\) and \(\frac{3}{14}+ \frac{9}{14}\). We then add the whole numbers and fractions separately. The whole numbers add to 13, and since the fractions already have a common denominator, they add up to \(\frac{12}{14}\).

Common Denominator

A 'common denominator' is required when adding or subtracting fractions to make the process possible. It refers to a shared denominator between two or more fractions. Having the same bottom number in a fraction ensures that we are comparing like with like.

In our example, \(\frac{3}{14}\) and \(\frac{9}{14}\) already have a common denominator of 14, which greatly simplifies the addition process. Without a common denominator, we would not be able to directly add the numerators (the top numbers) of the fractions. If they did not share a common denominator, we would need to find equivalent fractions that do have a common denominator before proceeding.

Simplest Form Fraction

A fraction is in its simplest form when no larger number than 1 evenly divides both the numerator and the denominator. Simplifying a fraction makes it easier to work with and understand.

After adding the fractions in our exercise, we get \(\frac{12}{14}\). To simplify, we need to find the greatest common divisor (GCD) of 12 and 14. The GCD in this case is 2. Dividing both the numerator and denominator by 2, we get the fraction \(\frac{6}{7}\), which is in its simplest form. In general, simplifying fractions involves dividing both parts of the fraction by their GCD until it cannot be reduced any further.