Problem 90 - Algebra 1

Question

Find the sum. $$\frac{2}{5}+\frac{1}{4}$$

Step-by-Step Solution

Answer

The sum of the fractions \(\frac{2}{5}\) and \(\frac{1}{4}\) is \(\frac{13}{20}\).

Step 1: Finding the LCD

The denominators here are 5 and 4. To find the least common denominator (LCD), find the least common multiple (LCM) of the two denominators. The LCM of 5 and 4 is 20.

Step 2: Convert the Fractions

With the LCD identified as 20, convert each of the fractions to have 20 as the denominator. This is done by multiplying the numerator and the denominator by the same value such that the denominator becomes 20. \[\frac{2}{5} = \frac{2 \times 4}{5 \times 4} = \frac{8}{20}\] and \[\frac{1}{4} = \frac{1 \times 5}{4 \times 5} = \frac{5}{20}\]

Step 3: Add the Fractions

Now that both fractions have the same denominators, we can add the numerators: \[\frac{8}{20} + \frac{5}{20} = \frac{8 + 5}{20}= \frac{13}{20}\]

Key Concepts

least common multiple (LCM)

The least common multiple, or LCM, is a crucial concept when dealing with the addition or subtraction of fractions. This is because you often need the denominators of the fractions to be the same in order to perform these operations. The LCM of two numbers is the smallest number that can be divided evenly by both numbers. In the context of the given exercise, we are looking for the LCM of the denominators 5 and 4. By listing the multiples of each number, you can find that 20 is the smallest multiple common to both 5 and 4. Knowing the LCM helps you convert different denominators to the same number, which makes addition or subtraction of fractions possible.

To determine the LCM, try the following steps:

• List the multiples of each denominator
• Find the smallest number that appears in both lists

Understanding how to find the LCM ensures that you maintain the balance needed to perform operations on fractions.

least common denominator (LCD)

When working with fractions, the least common denominator (LCD) is an essential tool. It refers to the smallest common multiple that two or more denominators have. In other words, it is the LCM of the denominators. This is used because both fractions need to have the same denominator to be added or subtracted.
In our example, fractions \(\frac{2}{5}\) and \(\frac{1}{4}\) need a common denominator. Using the LCM, we found that 20 is the least common denominator. This means both fractions need to be adjusted to have this denominator. To do that:

• Multiply the numerator and denominator of each fraction by the same number to get the new denominator of 20
• Ensure that the equivalent adjustments maintain the original value of the fractions

Starting with the LCD simplifies the process of addition or subtraction of fractions significantly, allowing you to focus purely on arithmetic without fractions' mismatched denominators complicating things.

equivalent fractions

Equivalent fractions might look different, but they represent the same value or portion of a whole. You create equivalent fractions by multiplying or dividing both the numerator and the denominator by the same number. This transformation comes into play when you need fractions to have an equal denominator, known as the LCD.
Consider \(\frac{2}{5}\) in the example. To change the denominator from 5 to 20, multiply both the numerator (2) and denominator (5) by 4, resulting in the equivalent fraction \(\frac{8}{20}\). Similarly, transform \(\frac{1}{4}\) by multiplying both by 5, yielding \(\frac{5}{20}\).

Here's how you can create equivalent fractions:

• Identify the needed common denominator
• Multiply both the numerator and denominator of each fraction accordingly
• Check whether the new fractions simplify the process of adding or subtracting

Understanding equivalent fractions is key in operations involving fractions, helping to transform and align them for easier computation.