Problem 30 - Prealgebra

Question

Add and subtract the following mixed numbers as indicated. $$\begin{array}{r}12 \frac{7}{8} \\\\-3 \frac{5}{6} \\\\\hline\end{array}$$

Step-by-Step Solution

Answer

The result of the subtraction is \(9 \frac{1}{24}\).

Step 1: Convert to Improper Fractions

First, convert the mixed numbers into improper fractions. For \(12 \frac{7}{8}\), multiply the whole number 12 by the denominator 8: \(12 \times 8 = 96\). Add the numerator 7: \(96 + 7 = 103\). So, \(12 \frac{7}{8} = \frac{103}{8}\). Similarly for \(3 \frac{5}{6}\), multiply 3 by 6: \(3 \times 6 = 18\). Add the numerator 5: \(18 + 5 = 23\). So, \(3 \frac{5}{6} = \frac{23}{6}\).

Step 2: Find a Common Denominator

The denominators are 8 and 6. The least common multiple of 8 and 6 is 24. Convert the fractions to have a common denominator. For \(\frac{103}{8}\), multiply the numerator and denominator by 3 to get \(\frac{309}{24}\). For \(\frac{23}{6}\), multiply the numerator and denominator by 4 to get \(\frac{92}{24}\).

Step 3: Subtract the Improper Fractions

Now that the fractions have a common denominator, subtract \(\frac{92}{24}\) from \(\frac{309}{24}\). Perform the subtraction: \(309 - 92 = 217\). So, the result is \(\frac{217}{24}\).

Step 4: Convert Back to a Mixed Number

To convert \(\frac{217}{24}\) back to a mixed number, divide 217 by 24. 217 divided by 24 equals 9 with a remainder of 1. Thus, \(\frac{217}{24} = 9 \frac{1}{24}\).

Key Concepts

Improper Fractions

An improper fraction is a key concept when dealing with mixed numbers and can simplify calculations involving them. It is defined as a fraction where the numerator, the top part of the fraction, is larger than or equal to the denominator, the bottom part of the fraction. This form is particularly useful when adding or subtracting fractions.
An example from our problem involves converting mixed numbers to improper fractions. For instance, when converting the mixed number \(12 \frac{7}{8}\), multiply the whole number 12 by the denominator 8, and add the numerator 7:

• Calculate: \(12 \times 8 = 96\)
• Add: \(96 + 7 = 103\)
• Thus, \(12 \frac{7}{8} = \frac{103}{8}\)

This conversion simplifies the process of subtraction, allowing you to deal with straightforward numerical operations instead of juggling mixed numbers.

Least Common Multiple

The least common multiple (LCM) is the smallest number that is a multiple of two or more denominators. Finding the LCM is crucial when adding or subtracting fractions with different denominators, as it allows you to recast them under a common denominator.
In our exercise, the denominators of the fractions are 8 and 6. To find their least common multiple, we look for the smallest number that can be divided evenly by both using their multiples:

• Multiples of 8: 8, 16, 24, 32...
• Multiples of 6: 6, 12, 18, 24...
• LCM is 24, the smallest common number in the lists

By obtaining an LCM, you can convert fractions to equivalent forms with this shared denominator, making operations like addition and subtraction possible.

Denominators

Denominators play a central role in fractions, indicating how many parts each whole is divided into. When working with fractions, especially when adding or subtracting them, it is important that the denominators are the same or have been aligned by finding the least common multiple.
Handling denominators effectively involves navigating from unlike to like denominators. Using our problem as an illustration, converting denominators of 8 and 6 to their common form of 24 involved performing these operations:

• \(\frac{103}{8} \) converted to \(\frac{309}{24}\) by multiplying both numerator and denominator by 3
• \(\frac{23}{6} \) converted to \(\frac{92}{24}\) by multiplying both numerator and denominator by 4

By adjusting these fractions to a common denominator, subtraction becomes a straightforward operation.

Subtraction of Fractions

The subtraction of fractions is a process that involves several steps to ensure that you correctly handle fractions. After converting fractions to a common denominator, subtraction occurs in the same way you would subtract whole numbers, focusing on the numerators while keeping the common denominator unchanged.
Using the example, you have already converted the fractions to \(\frac{309}{24}\) and \(\frac{92}{24}\). Subtract the numerators to find the result:

• \(309 - 92 = 217\)
• The result is \(\frac{217}{24}\)

After getting this fraction, sometimes it is helpful to convert it back to a mixed number for easier interpretation, as shown by dividing 217 by 24 to yield \(9 \frac{1}{24}\). Subtracting fractions becomes more manageable when systematically working through these steps.