Problem 77 - Prealgebra

Question

The following problems review addition and subtraction with fractions and mixed numbers. $$\frac{3}{8}+\frac{1}{4}$$

Step-by-Step Solution

Answer

The sum is \(\frac{5}{8}\).

Step 1: Find a Common Denominator

To add the fractions, we need to find a common denominator. The denominators are 8 and 4. The smallest common multiple of both is 8.

Step 2: Convert Fractions to Like Denominators

Convert the fraction \(\frac{1}{4}\) to have the same denominator as \(\frac{3}{8}\). Since \(\frac{1}{4} = \frac{2}{8}\), write \(\frac{1}{4}\) as \(\frac{2}{8}\).

Step 3: Add the Fractions

Now that both fractions have the same denominator, add them: \(\frac{3}{8} + \frac{2}{8} = \frac{5}{8}\).

Step 4: Simplify the Result

Check if the resulting fraction can be simplified. \(\frac{5}{8}\) is already in its simplest form.

Key Concepts

Common Denominator

When adding fractions, it is important to first find a common denominator. The common denominator is a shared multiple of the denominators in the fractions you are working with. This is crucial because fractions can only be directly added or subtracted when they have like denominators. A common denominator helps to line up the fractions just like aligning columns in a spreadsheet.

To find a common denominator, you can use the following steps:

• Identify the denominators in your fractions. In the exercise, they are 8 and 4.
• Determine the least common multiple (LCM) of these denominators. For 8 and 4, the LCM is 8, since 8 is the smallest number that both 4 and 8 can divide into without leaving a remainder.

Having a common denominator will allow you to rewrite the fractions so they can be added easily, making the entire process more straightforward.

Fractions with Like Denominators

After finding a common denominator, it is often necessary to convert fractions to give them the same denominator. This is known as creating fractions with like denominators. Once achieved, adding fractions becomes as simple as adding their numerators.

In our example, we needed to convert the fraction \(\frac{1}{4}\) to have the same denominator as \(\frac{3}{8}\). Here's how to do it:

• Multiply the numerator and the denominator of \(\frac{1}{4}\) by 2, which gives \(\frac{2}{8}\). This ensures both fractions now share the same denominator, 8.
• Leaving the first fraction as it is, \(\frac{3}{8}\), ensures that no further conversion is needed.

Now, with like denominators, fractions are ready to be added or subtracted effortlessly, just as in single-column arithmetic.

Simplifying Fractions

Once the fractions have been added, the next step is to simplify the result if possible. Simplifying a fraction means reducing it to its smallest form, where the numerator and the denominator have no common factors other than 1. This makes the fraction easier to understand and often more intuitive to work with in further calculations.

Let's say the resulting fraction from our exercise is \(\frac{5}{8}\). Here are the steps to check if it can be simplified:

• Identify any common factors of the numerator and the denominator other than 1.
• In this example, 5 and 8 have no common divisors other than 1, meaning \(\frac{5}{8}\) is already in its simplest form.

When a fraction cannot be simplified further, you have reached the simplest expression of your solution. Simplifying is an important step as it provides a clean and concise representation of the answer.