Problem 61 - Prealgebra

Question

The following problems all involve the concept of borrowing. Subtract in case. \(13 \frac{1}{6}-12 \frac{5}{8}\)

Step-by-Step Solution

Answer

The result of the subtraction is \(\frac{13}{24}\).

Step 1: Convert Mixed Numbers to Improper Fractions

First, convert the mixed numbers into improper fractions. For \(13 \frac{1}{6}\), multiply the whole number \(13\) by \(6\) and add \(1\) to get \(79\). Thus, \(13 \frac{1}{6} = \frac{79}{6}\). Similarly, for \(12 \frac{5}{8}\), multiply \(12\) by \(8\) and add \(5\) to get \(101\). So, \(12 \frac{5}{8} = \frac{101}{8}\).

Step 2: Find a Common Denominator

To subtract the fractions \(\frac{79}{6}\) and \(\frac{101}{8}\), find a common denominator. The least common multiple of \(6\) and \(8\) is \(24\). Convert \(\frac{79}{6}\) to \(\frac{316}{24}\), and \(\frac{101}{8}\) to \(\frac{303}{24}\).

Step 3: Subtract the Improper Fractions

Subtract the fractions: \(\frac{316}{24} - \frac{303}{24} = \frac{13}{24}\). Both fractions have the same denominator, so subtract the numerators directly.

Step 4: Simplify if Possible

Check if the resulting fraction \(\frac{13}{24}\) can be simplified. Since \(13\) is a prime number and does not divide \(24\), the fraction is already in its simplest form.

Key Concepts

Improper Fractions

Improper fractions can sometimes seem tricky at first, but they're just a different way to express a number. Let's clarify what they mean. An improper fraction has a numerator that is larger than its denominator. This is different from a proper fraction where the numerator is smaller. For example, in the fraction \( \frac{79}{6} \), 79 is the numerator and 6 is the denominator, making it an improper fraction.

Why use improper fractions? They arise naturally when dealing with mixed numbers. A mixed number like \( 13 \frac{1}{6} \) can be daunting to subtract if left in that form. By converting it to an improper fraction \( \frac{79}{6} \), it simplifies arithmetic operations like addition and subtraction. Here is how you do it:

• Multiply the whole number by the denominator.
• Add the numerator to the result.
• Write the sum over the original denominator.

This process makes calculations much more manageable and is especially useful for homework and exams.

Common Denominator

Finding a common denominator is essential when adding or subtracting fractions. To perform these operations, the fractions need to "speak the same language," so to speak. This means they should have the same denominator.

In the exercise, we needed a common denominator to subtract \(\frac{79}{6}\) and \(\frac{101}{8}\). The easiest way is to find the least common multiple (LCM) of the two denominators. For 6 and 8, the LCM is 24. Here's how you find a common denominator:

• List the multiples of each denominator.
• Identify the smallest multiple they have in common.

Once you find this common denominator, rewrite each fraction with it. For example:

• Convert \(\frac{79}{6}\) to \(\frac{316}{24}\).
• Convert \(\frac{101}{8}\) to \(\frac{303}{24}\).

Now these fractions can be easily subtracted from each other.

Simplifying Fractions

Simplifying fractions is about making them as simple as possible while still representing the same value. This makes them easier to understand and work with.

For simplification, you'll want to look for the greatest common factor (GCF) of the numerator and the denominator, and divide them both by it. This process does not change the fraction's value, just its appearance. For example, consider the subtraction result \(\frac{13}{24}\) from the exercise. Although 13 is a prime number and doesn't divide 24, which means the fraction is already simplified, it's good to know the steps:

• List factors of the numerator and denominator.
• Find the largest factor they have in common.
• Divide both by this factor.

By applying these steps, you ensure your final answer is in its simplest form, making it neat and easier to understand.