Problem 13 - Prealgebra

Question

Add and subtract the following mixed numbers as indicated. \(7 \frac{5}{12}-3 \frac{1}{3}\)

Step-by-Step Solution

Answer

The result of the subtraction is \(4 \frac{1}{12}\).

Step 1: Convert Mixed Numbers to Improper Fractions

First, convert each mixed number into an improper fraction. For 7 \(\frac{5}{12}\), multiply 7 by 12 and add 5: \(7 \times 12 + 5 = 84 + 5 = 89\). The improper fraction is \(\frac{89}{12}\). For 3 \(\frac{1}{3}\), multiply 3 by 3 and add 1: \(3 \times 3 + 1 = 9 + 1 = 10\). The improper fraction is \(\frac{10}{3}\).

Step 2: Find a Common Denominator

Determine the least common denominator (LCD) of the fractions \(\frac{89}{12}\) and \(\frac{10}{3}\). The LCD of 12 and 3 is 12. Convert \(\frac{10}{3}\) to an equivalent fraction with a denominator of 12: \(\frac{10}{3} \times \frac{4}{4} = \frac{40}{12}\).

Step 3: Subtract the Fractions

With a common denominator, subtract \(\frac{40}{12}\) from \(\frac{89}{12}\): \(\frac{89}{12} - \frac{40}{12} = \frac{49}{12}\).

Step 4: Convert Improper Fraction to Mixed Number

Convert the improper fraction \(\frac{49}{12}\) back to a mixed number. Divide 49 by 12: 49 divided by 12 is 4 with a remainder of 1. This results in the mixed number \(4 \frac{1}{12}\).

Key Concepts

Improper Fractions

To work easily with fractions, it's useful to understand improper fractions. Improper fractions have numerators that are larger than their denominators. This means the fraction represents a value greater than one. When dealing with operations involving mixed numbers, converting them to improper fractions can simplify the process.

• To convert a mixed number to an improper fraction, multiply the whole number by the denominator of the fractional part.
• Add this result to the numerator of the fractional part. The total becomes your new numerator over the original denominator.

For instance, to convert the mixed number 7 \(\frac{5}{12}\), we multiply 7 by 12, resulting in 84. Adding 5 to 84 gives us 89, making our improper fraction \(\frac{89}{12}\). This method allows us to easily handle arithmetic operations like addition or subtraction.

Least Common Denominator

Finding a common denominator is crucial when adding or subtracting fractions with different denominators. The Least Common Denominator (LCD) is the smallest number that both denominators can divide into evenly. Having a common denominator turns the fractions into like terms, making subtraction possible.

• Identify the denominators of both fractions you are working with.
• Find the smallest multiple they share, which becomes the LCD.

In our exercise, we looked at the denominators 12 and 3. The least common multiple of these two numbers is 12. We didn’t change \(\frac{89}{12}\) because it already had the denominator of 12. However, we converted \(\frac{10}{3}\) by multiplying both numerator and denominator by 4, giving us \(\frac{40}{12}\). Now, both fractions have the same denominator, so subtraction can occur.

Subtracting Fractions

Once fractions have a common denominator, they can be easily subtracted by working with the numerators alone. You keep the denominator the same and only focus on subtracting the numerators. This simplifies the process!

• Ensure both fractions have the same denominator.
• Subtract the second numerator from the first while keeping the common denominator unchanged.

In this exercise, we subtracted \(\frac{40}{12}\) from \(\frac{89}{12}\). The operation only affected the numerators: \(89 - 40 = 49\). Therefore, the resulting fraction is \(\frac{49}{12}\). Sometimes, the resulting fraction is improper, and you may want to convert it back to a mixed number for a clearer expression. This is what was done in Step 4 of the solution.