Problem 67 - Algebra 1: Concepts and Skills

Question

Add. Write the answer as a fraction or as a mixed number in simplest form. $$ 9 \frac{12}{16}+15 \frac{13}{16} $$

Step-by-Step Solution

Answer

The simple form of the sum is \(25 \frac{9}{16}\)

Step 1: Convert mixed numbers to improper fractions

An improper fraction is a fraction in which the numerator is greater than the denominator. Each mixed number can be converted to an improper fraction in the following way: multiply the whole number by the denominator of the fraction, then add the numerator; this will be the numerator of the improper fraction. The denominator stays the same. So, \(9 \frac{12}{16}\) converts to \( \frac{(9*16)+12}{16} = \frac{156}{16}\) and \(15 \frac{13}{16}\) becomes \( \frac{(15*16)+13}{16} = \frac{253}{16} \)

Step 2: Add the improper fractions

Now add the two improper fractions \(\frac{156}{16} + \frac{253}{16}\), which is possible because they have the same denominator. The result is \(\frac{409}{16}\)

Step 3: Convert back to mixed number and simplify

Now, we convert \(\frac{409}{16}\) back to a mixed number by dividing 409 by 16. The quotient is the whole number part, and the remainder over 16 becomes the fraction part. So, \(\frac{409}{16} = 25 \frac{9}{16}\)

Key Concepts

Improper Fractions

Let's dive into the idea of improper fractions. An improper fraction occurs when the numerator, or the top number, is larger than the denominator, which is the bottom number. Understanding improper fractions makes it easier to work with fractions that derive from mixed numbers or large quantities. To convert a mixed number into an improper fraction:

• Multiply the whole number by the denominator.
• Add the result to the numerator.
• This sum becomes your new numerator, while the denominator remains unchanged.

For example, converting the mixed number \(9 \frac{12}{16}\) involves calculating \(9 \times 16 + 12\), resulting in the improper fraction \(\frac{156}{16}\). This approach is useful because it transforms the whole mixed number into a single fraction, simplifying addition.

Mixed Numbers

Mixed numbers are a combination of whole numbers and fractions. They often represent quantities that are greater than one, making them practical for various math problems and real-life situations. When you need to convert an improper fraction back into a mixed number:

• Divide the numerator by the denominator to find the whole number part.
• The remainder becomes the new numerator of the fraction part, with the denominator staying the same.

Consider the improper fraction \(\frac{409}{16}\): when you divide 409 by 16, the result is 25 with a remainder of 9. Thus, the mixed number is \(25 \frac{9}{16}\). This shows you how to switch between fractional forms and interpret results.

Simplifying Fractions

Simplifying fractions is about reducing them to their simplest form. This means the numerator and denominator have no common factors other than 1. Simplifying makes fractions straightforward and comparable.To simplify a fraction, follow these steps:

• Find the Greatest Common Divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and the denominator by this GCD.

For instance, if you needed to simplify \(\frac{12}{16}\), identify the GCD, which is 4.Thus, dividing both numbers by 4 gives \(\frac{12 ÷ 4}{16 ÷ 4} = \frac{3}{4}\).While not all fractions require simplifying in every problem, it's essential when presenting the final answer as neat and orderly as possible.