Problem 80 - Algebra 1: Concepts and Skills

Question

Add. Write the answer as a mixed number in simplest form. $$ 2 \frac{1}{2}+\frac{4}{3} $$

Step-by-Step Solution

Answer

The final answer is \(3 \frac{5}{6}\).

Step 1: Separate the Whole Number and Fraction

The first step is to separate the whole number and the fractional part. In this case, the given mixed number \(2 \frac{1}{2}\) can be separated into 2 and \(\frac{1}{2}\). The fraction stays as is.

Step 2: Convert Denominators to be the Same

To add the fractions \(\frac{1}{2}\) and \(\frac{4}{3}\), both fractions must first be converted to have the same denominator. The least common multiple of 2 and 3 is 6. Thus, multiply top and bottom of \(\frac{1}{2}\) by 3 to get \(\frac{3}{6}\) and multiply top and bottom of \(\frac{4}{3}\) by 2 to get \(\frac{8}{6}\).

Step 3: Add the Fractions and Simplify

Now, add the fractions \(\frac{3}{6}\) and \(\frac{8}{6}\) to get \(\frac{11}{6}\). This fraction can be simplified as \(1 \frac{5}{6}\) (as 6 goes into 11 one time with 5 remaining).

Step 4: Combine the Whole Numbers and Simplified Fraction

Finally, add the whole numbers 2 and 1 to get 3. The final answer is therefore \(3 \frac{5}{6}\).

Key Concepts

Least Common Multiple

When adding fractions, it's essential to have the denominators be the same so they can be easily combined. This is achieved by finding the Least Common Multiple (LCM) of the denominators of the fractions involved.
The LCM is the smallest number that is a multiple of each denominator.

• For example, with the fractions \( \frac{1}{2} \) and \( \frac{4}{3} \), the denominators are 2 and 3.


• The multiples of 2 are 2, 4, 6, 8, etc.


• The multiples of 3 are 3, 6, 9, 12, etc.


• The least value that appears in both lists is 6, making it the LCM.

Finding the LCM allows us to convert the fractions to a common denominator, enabling addition or subtraction.

Simplifying Fractions

After adding the fractions, your result might not be in the simplest form. To simplify a fraction, you need to divide the numerator (top number) and the denominator (bottom number) by their Greatest Common Divisor (GCD).

• In the context of the solution, \( \frac{11}{6} \) was obtained after adding the fractions.


• First, check if both numbers can be divided evenly by common factors other than 1.


• Since 11 and 6 have no common factors other than 1, the fraction is already simplified in its improper form.

Simplifying fractions is an important step as it provides the neatest and most reduced form of a fraction, which is often required in mathematical solutions.

Improper Fractions

An improper fraction is simply one where the numerator is greater than or equal to the denominator.

• For example, \( \frac{11}{6} \) is improper because 11 is greater than 6.


• To convert this to a mixed number, you divide the numerator by the denominator.


• In our case, 11 divided by 6 equals 1 with a remainder of 5.


• Thus, \( \frac{11}{6} \) converts to the mixed number \( 1 \frac{5}{6} \).

Understanding improper fractions is crucial as it often involves converting them to a more comprehensible form, especially in the context of word problems or real-life scenarios where mixed numbers make interpretation easier.