Problem 84 - Fundamentals of Mathematics

Question

For the following problems, find the products. Be sure to reduce. $$6 \frac{1}{4} \cdot 2 \frac{4}{15}$$

Step-by-Step Solution

Answer

The product is \(14 \frac{1}{6}\).

Step 1: Convert Mixed Numbers to Improper Fractions

Convert the mixed number \(6 \frac{1}{4}\) to an improper fraction. Multiply the whole number by the denominator and add the numerator: \(6 \times 4 + 1 = 25\). So, \(6 \frac{1}{4} = \frac{25}{4}\).Convert \(2 \frac{4}{15}\) similarly: \(2 \times 15 + 4 = 34\). Thus, \(2 \frac{4}{15} = \frac{34}{15}\).

Step 2: Multiply the Improper Fractions

Multiply the fractions \(\frac{25}{4}\) and \(\frac{34}{15}\) by multiplying the numerators and denominators: \[\frac{25 \cdot 34}{4 \cdot 15} = \frac{850}{60}\]

Step 3: Simplify the Resulting Fraction

Simplify \(\frac{850}{60}\) by finding the greatest common divisor (GCD). The GCD of 850 and 60 is 10.Divide both the numerator and the denominator by 10:\[\frac{850 \div 10}{60 \div 10} = \frac{85}{6}\].This is the simplified form of the fraction.

Step 4: Convert the Improper Fraction to a Mixed Number

Convert the improper fraction \(\frac{85}{6}\) back into a mixed number by dividing 85 by 6. This gives a quotient of 14 and a remainder of 1. Therefore, \(\frac{85}{6}\) as a mixed number is \(14 \frac{1}{6}\).

Key Concepts

Mixed Numbers

A mixed number is a combination of a whole number and a proper fraction. Think of it as an easy way to express something that is more than one whole but less than two wholes. For example, in the mixed number \(6 \frac{1}{4}\), \(6\) is the whole number part and \(\frac{1}{4}\) is the fractional part.
Mixed numbers are practical because they give a clear view of how much there is beyond the whole, making them more intuitive for some real-life applications. However, to perform operations like multiplication, they must first be converted into improper fractions, which can directly participate in arithmetic calculations.

Improper Fractions

Improper fractions are fractions where the numerator (top number) is larger than the denominator (bottom number). This means the fraction is equivalent to more than one whole.
To convert a mixed number to an improper fraction, follow these simple steps:

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

Using our example, \(6 \frac{1}{4}\) is converted into \(\frac{25}{4}\) because \((6 \times 4) + 1 = 25\). Improper fractions are useful in simplifying multiplication and division operations, as they can be easily multiplied or divided without requiring separate operations for the whole and fractional parts.

Simplification

Simplification is the process of reducing a fraction to its most basic form, where the numerator and denominator have no common factors other than 1.
This process can make calculations easier and results clearer by offering the simplest equivalent fraction. After multiplying fractions like \(\frac{25}{4} \cdot \frac{34}{15} = \frac{850}{60}\), simplification is crucial.
To simplify, find the Greatest Common Divisor (GCD) of the numerator and denominator, and divide both by this number.
The fraction \(\frac{850}{60}\) simplifies to \(\frac{85}{6}\) because the GCD of 850 and 60 is 10.

Greatest Common Divisor (GCD)

The Greatest Common Divisor, or GCD, is the largest number that can exactly divide two numbers without leaving a remainder.
In fraction simplification, the GCD helps minimize both the numerator and the denominator as far as possible.

• List the factors of each number.
• Find the largest factor they have in common.

For example, the GCD of 850 and 60 is 10. We can find this by listing the factors or using the Euclidean algorithm.
By dividing both the numerator and the denominator of \(\frac{850}{60}\) by their GCD, 10, we get \(\frac{85}{6}\), the simplest form of the fraction. Knowing how to find the GCD efficiently is vital to simplifying fractions correctly.