Problem 19 - Fundamentals of Mathematics

Question

Perform the following multiplications. Convert improper fractions to mixed numbers. $$7 \frac{1}{8} \cdot 12$$

Step-by-Step Solution

Answer

The result is \( 85 \frac{1}{2} \).

Step 1: Convert Mixed Number to Improper Fraction

First, convert the mixed number \( 7 \frac{1}{8} \) into an improper fraction. To do this, multiply the whole number 7 by the denominator 8 and add the numerator 1. This gives:\[7 \times 8 + 1 = 56 + 1 = 57\]So, \( 7 \frac{1}{8} = \frac{57}{8} \).

Step 2: Multiply Fractions

Now, multiply the improper fraction \( \frac{57}{8} \) by the integer 12. Convert 12 to a fraction by writing it as \( \frac{12}{1} \), and then multiply:\[\frac{57}{8} \cdot \frac{12}{1} = \frac{57 \times 12}{8 \times 1} = \frac{684}{8}\]

Step 3: Simplify the Fraction

Simplify the fraction \( \frac{684}{8} \). Divide both the numerator and the denominator by 8:\[684 \div 8 = 85.5\]This means \( \frac{684}{8} = 85.5 \).

Step 4: Convert Decimal to Mixed Number

Since it's often preferable to have answers in fractional form, we convert the decimal 85.5 back to a mixed number. Split 85.5 into 85 and \( \frac{1}{2} \) since 0.5 is equivalent to \( \frac{1}{2} \). Thus, \( 85.5 = 85 \frac{1}{2} \).

Key Concepts

Improper Fractions

When working with improper fractions, the numerator (top number) is larger than the denominator (bottom number). An example of this is when you have a fraction like \( \frac{57}{8} \). This means you have 57 parts, each of size \( \frac{1}{8} \), which results in a whole and additional parts. Improper fractions are often used in calculations because they simplify the multiplication and division processes.

To convert a mixed number, like \( 7 \frac{1}{8} \), into an improper fraction, follow these steps:

• Multiply the whole number by the denominator: \( 7 \times 8 = 56 \).
• Add the numerator to that product: \( 56 + 1 = 57 \).
• Put the result over the original denominator: \( \frac{57}{8} \).

This transformation allows you to do operations without worry about the combination of whole numbers and fractions separately.

Mixed Numbers

Mixed numbers combine whole numbers and fractions. They look like this: \( 7 \frac{1}{8} \). Sometimes, it is necessary to convert mixed numbers into improper fractions, especially when multiplying or dividing, because it simplifies mathematical operations.

To understand mixed numbers better:

• The whole number represents complete units.
• The fraction shows the additional parts that are less than a whole.

When you need to return to a mixed number format (like from a decimal), identify the whole part and the fractional part by splitting the decimal. For example, \( 85.5 \) as a mixed number would be \( 85 \frac{1}{2} \), since 0.5 converts to \( \frac{1}{2} \). This approach helps ensure your answer is clear and understandable.

Fraction Simplification

Simplifying fractions is the process of making them as simple as possible, with the smallest whole number values for the numerator and the denominator while maintaining the same value. This can be crucial in mathematics to make calculations easier and results to understand at a glance.

For example, \( \frac{684}{8} \) can be simplified by dividing both the numerator and the denominator by their greatest common divisor. Here's how you can do it:

• Divide 684 by 8 to find the simplest form: \( 684 \div 8 = 85.5 \).

Since it ends up as a decimal (not a simplified fraction), returning it to a mixed number is often clearer. The process of simplification not only helps in reducing the complexity of a number but also provides clarity when interpreting numerical results, especially in practical applications.