Problem 20 - Prealgebra

Question

Add and subtract the following mixed numbers as indicated. \(8 \frac{5}{6}+9 \frac{5}{6}\)

Step-by-Step Solution

Answer

The sum is \(18 \frac{2}{3}\).

Step 1: Write the Mixed Numbers as Improper Fractions

First, convert each mixed number into an improper fraction. For the first mixed number, \(8 \frac{5}{6}\), multiply the whole number 8 by the denominator 6, then add the numerator 5: \(8 \times 6 + 5 = 48 + 5 = 53\). So, \(8 \frac{5}{6} = \frac{53}{6}\). For the second mixed number, \(9 \frac{5}{6}\), multiply the whole number 9 by the denominator 6, then add the numerator 5: \(9 \times 6 + 5 = 54 + 5 = 59\). So, \(9 \frac{5}{6} = \frac{59}{6}\).

Step 2: Add the Improper Fractions

Add the two improper fractions: \(\frac{53}{6} + \frac{59}{6}\). Since the denominators are the same, you can add the numerators directly: \(53 + 59 = 112\). Therefore, \(\frac{53}{6} + \frac{59}{6} = \frac{112}{6}\).

Step 3: Simplify the Resulting Improper Fraction

Simplify the fraction \(\frac{112}{6}\) by finding the greatest common divisor (GCD) of 112 and 6, which is 2. Divide both the numerator and denominator by 2: \(\frac{112}{6} = \frac{56}{3}\).

Step 4: Convert the Improper Fraction to a Mixed Number

Convert \(\frac{56}{3}\) into a mixed number by dividing 56 by 3. The quotient is 18 and the remainder is 2, so \(\frac{56}{3} = 18 \frac{2}{3}\).

Key Concepts

Converting Mixed Numbers to Improper Fractions

Mixed numbers are numbers that consist of an integer and a proper fraction. To make calculations easier, it's often beneficial to convert these mixed numbers into improper fractions. An improper fraction is where the numerator is larger than or equal to the denominator.

To convert, follow these three simple steps:

• Multiply the whole number by the denominator of the fraction.
• Add the numerator to the result from the first step.
• Write this result as the numerator over the original denominator.

For example, with the mixed number \(8 \frac{5}{6}\), you multiply 8 by 6 to get 48, add 5 to get 53, resulting in \(\frac{53}{6}\). This method ensures accuracy, and the resulting improper fraction can be used for further arithmetic operations like addition or subtraction.

Simplifying Fractions

Once you've performed addition or subtraction and obtained your result as a fraction, it's essential to simplify it. Simplifying makes a fraction easier to understand and work with by reducing it to its simplest form. This involves dividing both the numerator and the denominator by their greatest common divisor (GCD).

Finding the GCD means identifying the largest number that divides both the numerator and the denominator without leaving a remainder. For the fraction \(\frac{112}{6}\), the GCD is 2. By dividing the numerator (112) and the denominator (6) each by 2, you simplify the fraction to \(\frac{56}{3}\). Simplification keeps numbers manageable and is an essential skill in solving fraction problems efficiently.

Converting Improper Fractions to Mixed Numbers

Sometimes, after performing operations on fractions, you end with an improper fraction. Converting it back to a mixed number can make it more understandable. To switch an improper fraction into a mixed number, you proceed with the following steps:

• Divide the numerator by the denominator to get an integer result called the quotient.
• The remainder from this division becomes the new numerator of the proper fraction part.
• The denominator remains the same.

For instance, with \(\frac{56}{3}\), dividing 56 by 3 gives 18 with a remainder of 2. Thus, \(\frac{56}{3}\) becomes \(18 \frac{2}{3}\). This conversion provides a clear representation of the amount and is often more intuitive in real-life contexts.