Problem 26 - Fundamentals of Mathematics

Question

For the following problems, find the reciprocal of each number. $$8 \frac{1}{4}$$

Step-by-Step Solution

Answer

The reciprocal of \(8 \frac{1}{4}\) is \(\frac{4}{33}\).

Step 1: Convert Mixed Number to Improper Fraction

The given number is a mixed number: \(8 \frac{1}{4}\). First, we will convert this into an improper fraction. To do this, multiply the whole number (8) by the denominator (4) and add the numerator (1). This gives us: \((8 \times 4) + 1 = 32 + 1 = 33\). Thus, \(8 \frac{1}{4}\) as an improper fraction is \(\frac{33}{4}\).

Step 2: Find the Reciprocal of the Improper Fraction

To find the reciprocal of a fraction, we swap the numerator and the denominator. The reciprocal of \(\frac{33}{4}\) is \(\frac{4}{33}\).

Key Concepts

Mixed Numbers

A mixed number is a combination of a whole number and a fraction, such as \(8 \frac{1}{4}\). This type of number is very useful when dealing with amounts that are more than a whole, but not complete enough to be considered another full unit. For instance, if a cake is cut into four equal parts and you eat one whole cake plus one piece, you would have eaten \(8 \frac{1}{4}\) cakes.Here’s how you might see mixed numbers:

• As part of recipes – "add \(1 \frac{1}{2}\) cups of sugar."
• In measurements – "The fence is \(3 \frac{1}{3}\) feet high."
• When dealing with time – "The race lasted for \(4 \frac{3}{4}\) hours."

Understanding mixed numbers helps you manage these situations comfortably. They provide a way to express values that are not perfectly whole, in a way that's easy to grasp at a glance.

Improper Fractions

Improper fractions occur when the numerator is larger than or equal to the denominator. For example, the improper fraction \(\frac{33}{4}\) came from the mixed number \(8 \frac{1}{4}\). This fraction suggests that we have more parts than needed to form a whole unit.Converting mixed numbers to improper fractions is simple:

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

For \(8 \frac{1}{4}\), you multiply \(8\) (the whole number) by \(4\) (the denominator), giving \(32\), and then add \(1\) (the numerator), resulting in \(33\). Therefore, this becomes the improper fraction \(\frac{33}{4}\).Improper fractions are essential in mathematics, especially when you need to perform operations like addition or multiplication, as they simplify calculations.

Numerator and Denominator

The terms numerator and denominator are foundational in understanding fractions, which are parts of a whole. In the fraction \(\frac{33}{4}\), "33" is the numerator, and "4" is the denominator.Here's a quick breakdown:

• Numerator: This is the top number of a fraction. It represents how many parts of the whole you have.
• Denominator: This is the bottom number and it indicates into how many parts the whole is divided.

When you find a reciprocal, the roles of the numerator and denominator switch. In the exercise, swapping them in \(\frac{33}{4}\) gives us \(\frac{4}{33}\). This reciprocal operation is critical in division with fractions.Understanding these parts of a fraction helps you know exactly what fraction of a whole is being described and how those fractions can be manipulated in mathematical operations.