Problem 56 - Prealgebra

Question

The following problems all involve the concept of borrowing. Subtract in case. \(12 \frac{5}{6}-8 \frac{7}{8}\)

Step-by-Step Solution

Answer

The difference is \(\frac{95}{24}\).

Step 1: Convert to Improper Fractions

First, convert the mixed numbers into improper fractions. For the first number, \(12 \frac{5}{6}\), multiply 12 by 6 and add 5: \(12 \times 6 + 5 = 72 + 5 = 77\). Thus, \(12 \frac{5}{6} = \frac{77}{6}\). For the second number, \(8 \frac{7}{8}\), multiply 8 by 8 and add 7: \(8 \times 8 + 7 = 64 + 7 = 71\). Thus, \(8 \frac{7}{8} = \frac{71}{8}\).

Step 2: Find a Common Denominator

The denominators 6 and 8 need a common denominator. The least common multiple of 6 and 8 is 24. Convert each fraction to have a denominator of 24. \(\frac{77}{6} = \frac{77 \times 4}{6 \times 4} = \frac{308}{24}\). \(\frac{71}{8} = \frac{71 \times 3}{8 \times 3} = \frac{213}{24}\).

Step 3: Subtract the Improper Fractions

Now, subtract the fractions with a common denominator: \(\frac{308}{24} - \frac{213}{24}\). Perform the subtraction on the numerators: \(308 - 213 = 95\). This gives us \(\frac{95}{24}\).

Step 4: Simplify the Fraction

Finally, simplify \(\frac{95}{24}\) if possible. Since 95 and 24 have no common factors other than 1, \(\frac{95}{24}\) is already in its simplest form.

Key Concepts

Improper Fractions

Improper fractions appear when the numerator (the top part of the fraction) is larger than the denominator (the bottom part of the fraction). They're useful when dealing with mixed numbers—numbers that have both a whole number and a fraction. To convert a mixed number to an improper fraction, follow these steps:

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

For the example, converting the mixed number 12\(\frac{5}{6}\) to an improper fraction involves multiplying 12 by 6, then adding 5. Thus, we get \(\frac{77}{6}\). This conversion helps make subtraction (and other operations) between fractions much easier.

Least Common Denominator

When adding or subtracting fractions, it's important to have a common denominator. The least common denominator (LCD) is the smallest number that both denominators divide into without a remainder. Here's how to find it:

• List the multiples of each denominator.
• Identify the smallest number common to both lists.

For the fractions \(\frac{77}{6}\) and \(\frac{71}{8}\), list multiples of 6 and 8 to find the least common multiple, which is 24. Converting each fraction's denominator to the LCD allows you to easily subtract them. Multiply numerators and denominators by the same number to change each fraction's denominator to 24, resulting in \(\frac{308}{24}\) and \(\frac{213}{24}\). This process ensures the fractions can be subtracted directly.

Simplifying Fractions

Once you have performed operations with fractions, the next step is often to simplify the result. Simplifying a fraction means seeing if both numerator and denominator can be divided evenly by any common numbers except 1. This gives you the simplest form. Here’s what to do:

• Find the greatest common factor (GCF) of both the numerator and denominator.
• Divide both by this GCF.

In the case of \(\frac{95}{24}\), 95 and 24 do not share any common factors other than 1. Therefore, it remains in its simplest form. It is important to realize that some fractions are already as simple as they can get. Simplifying fractions helps keep answers neat and could make future calculations easier.