Problem 50 - Fundamentals of Mathematics

Question

For the following problems, find each value. $$5 \frac{1}{6} \div \frac{31}{6}$$

Step-by-Step Solution

Answer

The value is 1.

Step 1: Convert Mixed Number to Improper Fraction

First, we need to convert the mixed number \(5 \frac{1}{6}\) into an improper fraction. A mixed number consists of a whole number and a fraction. To do this, multiply the whole number by the denominator of the fraction part, and then add the numerator. \(5 \times 6 + 1 = 30 + 1 = 31.\)So, \(5 \frac{1}{6} = \frac{31}{6}.\)

Step 2: Division of Fractions

The next step is to divide the improper fraction \(\frac{31}{6}\) by \(\frac{31}{6}\). In fraction division, you multiply by the reciprocal. Thus, \(\frac{31}{6} \div \frac{31}{6}\) becomes \(\frac{31}{6} \times \frac{6}{31}.\)

Step 3: Simplify the Fraction

When multiplying fractions, multiply the numerators and denominators together. Here, when you do this:\(\frac{31 \times 6}{6 \times 31} = \frac{186}{186}.\)The result can be simplified because the numerator and the denominator are the same, yielding \(1.\)

Step 4: Confirm Result

Since dividing any non-zero number by itself yields \(1\), this confirms the result:\[\frac{31}{6} \times \frac{6}{31} = 1\]

Key Concepts

Mixed Numbers

A mixed number is a combination of a whole number and a fraction. This representation is particularly useful for expressing numbers that are not integers but have a whole part. For instance, the mixed number \(5 \frac{1}{6}\) consists of the whole number 5 and the fraction \(\frac{1}{6}\).

To work with mixed numbers in calculations, you often need to convert them into improper fractions. This is because calculations like addition, subtraction, multiplication, and especially division of fractions, require fractions in an improper form. The conversion process is simple:

• Multiply the whole number by the denominator of the fraction part.
• Add the result to the numerator.

This gives you the improper fraction. Using our example, multiply 5 by 6 to get 30. Then, add 1 to get 31. Therefore, \(5 \frac{1}{6} = \frac{31}{6}\).

Understanding mixed numbers and how to convert them to improper fractions is essential for accurate calculations in arithmetic involving fractions.

Improper Fractions

Improper fractions are fractions where the numerator is larger than or equal to the denominator. These fractions are important because they allow us to easily perform arithmetic operations unlike mixed numbers.

Given the improper fraction \(\frac{31}{6}\), the numerator (31) is larger than the denominator (6). This means the fraction represents a number greater than or equal to 1, offering a straightforward way to handle division or multiplication.

Converting mixed numbers to improper fractions—like changing \(5 \frac{1}{6}\) to \(\frac{31}{6}\)—ensures consistency in mathematical operations. It simplifies processes like division of fractions, which involves multiplying by reciprocals. This uniformity makes improper fractions a preferred form when working with equations.

Practicing with improper fractions enriches understanding and flexibility in solving various math problems, especially in dividing one fraction by another.

Fraction Simplification

Simplifying fractions is the process of making the fraction as simple as possible. This means reducing it to its smallest numerator and denominator while still keeping the same value.

In the division problem \(\frac{31}{6} \div \frac{31}{6}\), after converting to multiplication, we obtain \(\frac{31}{6} \times \frac{6}{31}\). By multiplying, we get \(\frac{186}{186}\). Since the numerator equals the denominator, the fraction simplifies to 1, as any number divided by itself equals 1.

Here’s how you simplify fractions:

• Multiply numerators together and denominators together when multiplying fractions.
• Look for common factors between the numerator and the denominator.
• Divide both the numerator and the denominator by their greatest common factor (GCF).

Simplification helps to keep calculations neat and confirms that the solution is reduced to its simplest form, offering clarity and precision in mathematical solutions.