Problem 58 - Fundamentals of Mathematics

Question

For the following problems, find each value. $$\frac{16}{3} \div 6 \frac{2}{5}$$

Step-by-Step Solution

Answer

The value is \(\frac{5}{6}\).

Step 1: Convert the Mixed Number to an Improper Fraction

The problem involves dividing by a mixed number. First, convert the mixed number \(6\frac{2}{5}\) into an improper fraction. Multiply the whole number by the denominator and add the numerator: \(6 \times 5 + 2 = 32\). So, \(6\frac{2}{5} = \frac{32}{5}\).

Step 2: Rewrite the Division as Multiplication

To divide by a fraction, multiply by its reciprocal. Rewrite \(\frac{16}{3} \div \frac{32}{5}\) as \(\frac{16}{3} \times \frac{5}{32}\).

Step 3: Simplify the Expression

Multiply the numerators and the denominators: \(\frac{16 \times 5}{3 \times 32}\) which equals \(\frac{80}{96}\).

Step 4: Reduce the Fraction to its Simplest Form

Find the greatest common divisor (GCD) of 80 and 96, which is 16. Divide both the numerator and the denominator by 16: \(\frac{80}{96} = \frac{5}{6}\). This final fraction \(\frac{5}{6}\) is in its simplest form.

Key Concepts

Mixed Numbers

A mixed number is a combination of a whole number and a fraction. For example, in the exercise, we have the mixed number \(6\frac{2}{5}\). Mixed numbers are common when dealing with quantities that exceed whole units by a little extra. They are used to describe amounts that are more than a whole number but not quite another whole number.
When working with fractions, it's often necessary to convert mixed numbers into improper fractions, especially in operations like division or multiplication.

• The whole number is multiplied by the fraction's denominator.
• The numerator is added to the result.

For instance, converting \(6\frac{2}{5}\) involves multiplying 6 by 5, then adding 2, resulting in the improper fraction \(\frac{32}{5}\). Understanding this conversion is crucial for performing arithmetic operations involving mixed numbers.

Improper Fractions

Improper fractions have a numerator that is larger than or equal to the denominator. This might initially seem confusing, as we’re used to seeing numerators smaller than denominators in proper fractions. Improper fractions are important when performing arithmetic operations such as division or multiplication.
In our example, the fractions \(\frac{16}{3}\) and \(\frac{32}{5}\) are both improper. Often, improper fractions are easier to work with in calculations compared to mixed numbers.

• They allow for consistent application of multiplication and division rules.
• You can easily change improper fractions back into mixed numbers by dividing the numerator by the denominator.

For instance, with \(\frac{32}{5}\), dividing 32 by 5 gives the mixed number \(6\frac{2}{5}\) again.

Simplifying Fractions

Simplifying fractions involves reducing them to their smallest possible numerator and denominator while keeping the value the same. This is often the final step in problems involving fraction operations.
When you have a fraction like \(\frac{80}{96}\), simplifying it means finding the greatest common divisor (GCD) of the numerator and the denominator and dividing both by that number.

• For \(\frac{80}{96}\), the GCD is 16.
• Dividing the numerator 80 by 16 gives 5.
• Dividing the denominator 96 by 16 gives 6.

Thus, the simplified fraction is \(\frac{5}{6}\). Simplification helps to make fractions easier to understand and work with, ensuring results are in their simplest form.

Reciprocal

The reciprocal of a fraction is basically flipping the numerator and denominator. It’s a crucial concept when dealing with fraction division because dividing by a fraction is the same as multiplying by its reciprocal.
In the example from the exercise, to divide \(\frac{16}{3}\) by \(\frac{32}{5}\), you multiply by the reciprocal of \(\frac{32}{5}\), which is \(\frac{5}{32}\).

• The division problem \(\frac{16}{3} \div \frac{32}{5}\) becomes the multiplication problem \(\frac{16}{3} \times \frac{5}{32}\).
• This switch allows for straightforward multiplication of fractions, adhering to the general rules for multiplying numerators and denominators.

Understanding reciprocals is not only essential for division but also makes many fraction operations simpler and more intuitive.