Problem 6 - Prealgebra

Question

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

Step-by-Step Solution

Answer

The result of the subtraction is 5 \(\frac{1}{3}\).

Step 1: Convert Mixed Numbers to Improper Fractions

The problem involves subtracting mixed numbers. First, we need to convert them into improper fractions. For the first mixed number, 12 \(\frac{5}{12}\), multiply 12 by 12 and add 5 to get \(\frac{149}{12}\). For the second mixed number, 7 \(\frac{1}{12}\), multiply 7 by 12 and add 1 to get \(\frac{85}{12}\).

Step 2: Subtract the Improper Fractions

Subtract the improper fractions obtained from Step 1: \(\frac{149}{12} - \frac{85}{12}\). Since they have the same denominator, subtract the numerators: 149 - 85 = 64. The result is \(\frac{64}{12}\).

Step 3: Simplify the Fraction

Simplify \(\frac{64}{12}\) by finding the greatest common divisor (GCD) of 64 and 12, which is 4. Divide the numerator and the denominator by 4: \(\frac{64}{4} = 16\) and \(\frac{12}{4} = 3\). So, \(\frac{64}{12}\) simplifies to \(\frac{16}{3}\).

Step 4: Convert the Improper Fraction Back to a Mixed Number

Convert \(\frac{16}{3}\) back to a mixed number by dividing 16 by 3. The quotient is 5 and the remainder is 1, thus the mixed number is 5 \(\frac{1}{3}\).

Key Concepts

Improper Fractions

When dealing with mixed numbers, converting them into improper fractions is often the first step. A mixed number such as 12 \( \frac{5}{12} \) is composed of a whole number and a fraction. To convert it, you multiply the whole number by the fraction's denominator and add the numerator. In this example:

• Multiply the whole number 12 by the denominator 12: \( 12 \times 12 = 144 \)
• Add the numerator 5: \( 144 + 5 = 149 \)

So, 12 \( \frac{5}{12} \) = \( \frac{149}{12} \). Repeat this process with the second mixed number, 7 \( \frac{1}{12} \), to get \( \frac{85}{12} \). This conversion helps simplify operations like addition or subtraction, because it involves only the numerators if they have a common denominator.

Greatest Common Divisor

The greatest common divisor (GCD) is vital for simplifying fractions. It is the largest integer that divides both the numerator and the denominator without leaving a remainder. For simplifying the fraction \( \frac{64}{12} \):

• List the divisors for 64 (1, 2, 4, 8, 16, 32, 64) and 12 (1, 2, 3, 4, 6, 12).
• Identify the largest common divisor in both lists, which is 4.

Using the GCD of 4, divide both the numerator and the denominator:

• \( \frac{64 \div 4}{12 \div 4} = \frac{16}{3} \)

This technique reduces the fraction to its simplest form, making further calculations more manageable.

Fraction Simplification

Simplifying fractions is a crucial step in dealing with improper fractions. Once you've identified the GCD, apply it for simplification. Simplification is dividing both parts of the fraction by their GCD. In our exercise:

• The improper fraction \( \frac{64}{12} \) can be simplified using the GCD 4.
• Divide the numerator 64 by 4, giving 16.
• Divide the denominator 12 by 4, giving 3.

Thus, \( \frac{64}{12} \) simplifies neatly to \( \frac{16}{3} \). Simplification helps in clearer understanding and easier practical usage of fractions in further operations.

Subtraction of Fractions

Subtracting fractions is straightforward when they share the same denominator, as there's no need to find a common one. You simply subtract the numerators and keep the denominator unchanged. Using the derived improper fractions \( \frac{149}{12} \) and \( \frac{85}{12} \):

• Subtract the numerators: \( 149 - 85 = 64 \)
• Keep the denominator 12 the same.

This gives us the resulting fraction \( \frac{64}{12} \). With this resultant fraction, next comes simplification, as simplified fractions are more intuitive and useful in applications.