Problem 88 - Algebra 1: Concepts and Skills

Question

Write the answer as a fraction or as a mixed number in lowest terms. (Skills Review p. 764) $$\frac{19}{24}+\frac{11}{12}$$

Step-by-Step Solution

Answer

The sum of the fractions is \(1 \frac{17}{24}\).

Step 1: Align Denominators

The first step is to ensure that the fractions have the same denominator. Notice that the denominator 12 in the fraction \(\frac{11}{12}\) is a divisor of the denominator 24 in the fraction \(\frac{19}{24}\). So, align denominators by multiplying the numerator and denominator of \(\frac{11}{12}\) by 2, which gives \(\frac{22}{24}\). So, the operation becomes \(\frac{19}{24} + \frac{22}{24}\).

Step 2: Add the Fractions

Since the two fractions now have the same denominators, add the numerators and keep the denominator. This gives: \(\frac{19 + 22}{24} = \frac{41}{24}\).

Step 3: Simplify Fraction to Lowest Terms

\(\frac{41}{24}\) cannot be simplified further as a fraction but it represents a number greater than 1. This must be represented as a mixed fraction. Divide 41 by 24 (the denominator). The quotient is the whole number and the remainder over the denominator 24 constitutes the fractional part. So, it simplifies to \(1 \frac{17}{24}\).

Key Concepts

Aligning Denominators

Before adding fractions, ensure that they have the same denominator. This process is called aligning denominators. It allows us to add the fractions directly by focusing only on the numerators.

For example, if we have the fractions \( \frac{19}{24} \) and \( \frac{11}{12} \), we notice that their denominators are different. To align them, find a common multiple of the denominators, which in this case is 24.

Here’s how you align them step by step:

• If one denominator is a multiple of the other (like 24 is of 12), convert the fraction with the smaller denominator.
• Turn \( \frac{11}{12} \) into an equivalent fraction with denominator 24 by multiplying both its numerator and denominator by 2: \( \frac{11 \times 2}{12 \times 2} = \frac{22}{24} \).
• Now both fractions are \( \frac{19}{24} \) and \( \frac{22}{24} \).

Fraction Addition

Once you have aligned the denominators, adding fractions becomes straightforward. Just add the numerators while keeping the denominator the same. This process consolidates the two fractions into one.

Here is how you add them:

• Given \( \frac{19}{24} \) and \( \frac{22}{24} \), add their numerators: \( 19 + 22 = 41 \).
• Keep the denominator 24 the same, resulting in \( \frac{41}{24} \).

Adding fractions is that simple once the denominators are aligned, making the entire process less complex. Don't forget to ensure numerators add together as whole numbers, showing your work can prevent errors.

Mixed Numbers

When a fraction's numerator is greater than its denominator, the fraction is an improper fraction. In these cases, it's often beneficial to express it as a mixed number, combining whole numbers with fractions.

Here are the steps to convert \( \frac{41}{24} \) into a mixed number:

• Divide the numerator by the denominator: \( 41 \div 24 = 1 \) with a remainder of \( 17 \).
• The whole number part of the mixed number is 1, as 24 fits into 41 once.
• The remainder becomes the new numerator over the original denominator: \( \frac{17}{24} \).
• Combine the whole number and the fraction to get the mixed number: \( 1 \frac{17}{24} \).

Mixed numbers offer a clearer way to represent improper fractions, especially when interpreting real-world values.