Problem 77 - Algebra 1: Concepts and Skills

Question

Add. Write the answer as a mixed number in simplest form. $$ 2 \frac{3}{4}+\frac{19}{20} $$

Step-by-Step Solution

Answer

The solution to the addition \(2 \frac{3}{4}+\frac{19}{20}\) is \(3 \frac{7}{10}\) in simplest form.

Step 1: Convert the mixed number to an improper fraction

To convert the mixed number \(2 \frac{3}{4}\) to an improper fraction, multiply the whole number 2 by the denominator 4 and add the numerator 3 to get \(\frac{11}{4}\).

Step 2: Find common denominator

To add fractions, they must have the same denominator. The least common denominator (LCD) of 4 and 20 is 20. Multiply the numerator and the denominator of \(\frac{11}{4}\) by 5 to get \(\frac{55}{20}\). So, the fractions become \(\frac{55}{20} + \frac{19}{20}\).

Step 3: Perform Addition

Add the two fractions together by keeping the denominator the same and adding the numerators: \(\frac{55 + 19}{20} = \frac{74}{20}\).

Step 4: Convert back to mixed number and simplify

To convert \(\frac{74}{20}\) back to a mixed number, divide 74 by 20 to get 3 with a remainder of 14. So, the mixed number is \(3 \frac{14}{20}\). Simplifying this fraction by dividing the numerator and denominator by 2, we get the final answer \(3 \frac{7}{10}\).

Key Concepts

Improper Fractions

Understanding improper fractions is essential when dealing with mixed numbers and complex arithmetic operations. In basic terms, an improper fraction is one where the numerator (the top number) is greater than or equal to the denominator (the bottom number). Take for instance, converting a mixed number to an improper fraction is a preliminary step when adding mixed numbers.

As seen in our exercise, the mixed number is first converted to an improper fraction before addition can take place. For the mixed number \(2 \frac{3}{4}\), it was converted to \(\frac{11}{4}\) using the method of multiplying the whole number by the denominator and adding the numerator. This step is a bridge that connects mixed numbers and the streamlined process of fraction addition.

Common Denominator

When adding fractions, having a common denominator simplifies the task substantially. A common denominator is essentially the same bottom number for all fractions involved in the operation. To find the least common denominator (LCD), we look for the smallest multiple that the denominators of all fractions have in common.

In our exercise example, the denominators were 4 and 20. The LCD for these two numbers is 20 because 4 can be multiplied by 5 to reach 20 without changing the value of the fraction. Once both fractions have the same denominator, they can be added much like simple whole numbers, aligning the process and ensuring clarity and simplicity in the solution.

Simplifying Fractions

After performing addition and obtaining an improper fraction, it's often necessary to simplify the fraction, which means to reduce it to its simplest form. Simplifying makes fractions easier to understand, compare, and use in further calculations.

In the final part of our exercise, the improper fraction \(\frac{74}{20}\) needed to be simplified. This involves finding a common factor that both the numerator and the denominator share. In this instance, both 74 and 20 are divisible by 2. By dividing both by their greatest common divisor, the fraction is simplified to \(\frac{7}{10}\). This process ensures that the fraction is presented in its most basic and comprehensible form, aiding in the clear communication of mathematical results.