Problem 32 - Prealgebra

Question

Add and subtract the following mixed numbers as indicated. $$\begin{array}{r}18 \frac{7}{8} \\\\+19 \frac{1}{12} \\\\\hline\end{array}$$

Step-by-Step Solution

Answer

The result is \(37 \frac{23}{24}\).

Step 1: Convert Mixed Numbers to Improper Fractions

First, convert each mixed number to an improper fraction. For \(18 \frac{7}{8}\), multiply 18 by 8 and add 7, which gives \(\frac{151}{8}\). For \(19 \frac{1}{12}\), multiply 19 by 12 and add 1, yielding \(\frac{229}{12}\).

Step 2: Find a Common Denominator

Next, find a common denominator for the fractions. The denominators are 8 and 12, so the least common multiple is 24. Convert \(\frac{151}{8}\) to \(\frac{453}{24}\), and \(\frac{229}{12}\) to \(\frac{458}{24}\).

Step 3: Add the Fractions

Add the fractions by adding their numerators. Since both fractions now have the same denominator, add the numerators: \(453 + 458 = 911\). The result is \(\frac{911}{24}\).

Step 4: Simplify the Result

Simplify \(\frac{911}{24}\) by converting it back to a mixed number. Divide 911 by 24, which gives 37 with a remainder of 23. The mixed number is \(37 \frac{23}{24}\).

Key Concepts

Improper Fractions

When dealing with mixed numbers in arithmetic operations, converting them into improper fractions is a crucial step. An improper fraction is one where the numerator is greater than the denominator. This conversion makes it much easier to perform operations like addition or subtraction.
To convert a mixed number to an improper fraction:

• Multiply the whole number part by the denominator.
• Add the numerator to the result of this multiplication.
• Write this sum as the new numerator, keeping the original denominator.

For example, to convert the mixed number \(18 \frac{7}{8}\):

• Multiply 18 by 8 to get 144.
• Add 7 to 144, resulting in 151.
• Thus, \(18 \frac{7}{8} = \frac{151}{8}\).

Through this method, operations on mixed numbers become manageable as fractions can be added or subtracted directly.

Common Denominator

Finding a common denominator is essential when adding or subtracting fractions. This step ensures that both fractions share the same baseline, allowing their numerators to be combined directly. The common denominator is often the Least Common Multiple (LCM) of the original denominators.
To find the LCM:

• Write out the multiples of each denominator.
• Identify the smallest multiple that they share.

From the exercise, the denominators were 8 and 12. By listing their multiples, you find that 24 is their smallest shared multiple. Thus, it becomes the common denominator.
Once a common denominator is found, convert the fractions by adjusting their numerators:

• For \(\frac{151}{8}\), find an equivalent fraction with a denominator of 24, resulting in \(\frac{453}{24}\).
• For \(\frac{229}{12}\), convert it to \(\frac{458}{24}\).

This makes addition straightforward, as you can sum the numerators directly.

Mixed Number Conversion

After performing addition or subtraction operations with improper fractions, it's often necessary to convert the result back to a mixed number. This conversion helps to return your solution to a more readable and standard form.

• To convert an improper fraction like \(\frac{911}{24}\) to a mixed number, divide the numerator by the denominator.
• The integer part of the division result becomes the whole number part of the mixed number.
• The remainder becomes the new numerator, while the denominator remains the same.

For the example \(\frac{911}{24}\):

• Divide 911 by 24 to get 37 with a remainder of 23.
• This gives the mixed number \(37 \frac{23}{24}\).

Returning to mixed numbers in this way assures clarity in interpretation and presentation.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form, where the numerator and denominator have no common factors other than 1. Simplified fractions are easier to interpret and compare.

• First, determine if there are any common factors between numerator and denominator.
• Divide both by their greatest common factor (GCF).

In our provided solution, the final result, \(37 \frac{23}{24}\), is already in its simplest form, as 23 and 24 share no common factors other than 1. Though simplification wasn’t necessary here, this step is crucial when reducing complex fractions in other problems.
Always check your fractions to ensure clarity and precision in your final answer.