Problem 32 - Fundamentals of Mathematics

Question

Convert each mixed number to an improper fraction. $$18 \frac{1}{3}$$

Step-by-Step Solution

Answer

The improper fraction is \(\frac{55}{3}\).

Step 1: Understand Mixed Numbers

A mixed number consists of a whole number and a fraction. In this case, the mixed number is \(18 \frac{1}{3}\), which is composed of the whole number 18 and the fraction \(\frac{1}{3}\).

Step 2: Multiply the Whole Number by the Denominator

Take the whole number part (18) and multiply it by the denominator of the fraction (3): \(18 \times 3 = 54\).

Step 3: Add the Numerator to the Product

Add the numerator of the fraction (1) to the product obtained in Step 2: \(54 + 1 = 55\).

Step 4: Write as an Improper Fraction

Use the sum obtained in Step 3 as the numerator of the improper fraction, with the original denominator (3) remaining the same. The improper fraction is \(\frac{55}{3}\).

Step 5: Verify the Conversion

Check your work by converting \(\frac{55}{3}\) back to a mixed number. Divide 55 by 3, which gives a quotient of 18 (whole number) and a remainder of 1, confirming that the improper fraction \(\frac{55}{3}\) is equivalent to the mixed number \(18 \frac{1}{3}\).

Key Concepts

Mixed Numbers

A mixed number combines a whole number and a fraction. It's an easy way to represent numbers that aren't whole.
For example,

• The mixed number \(18 \frac{1}{3}\) tells us we have 18 whole parts and an additional fraction, \(\frac{1}{3}\).

Understanding mixed numbers helps in various arithmetic operations, such as addition and subtraction, especially when dealing with non-whole quantities. They allow us to express a number that is greater than an integer but less than the next one.

Fractions

Fractions are a way to represent parts of a whole. They are composed of two main parts:

• The numerator: the top part that tells us how many parts we have.
• The denominator: the bottom part that tells us into how many parts the whole is divided.

For example, in the fraction \(\frac{1}{3}\), the numerator 1 indicates a single part, while the denominator 3 indicates the whole is divided into three equal parts. Fractions are useful in various calculations, providing a precise understanding of portion and division.

Numerator and Denominator

The numerator and denominator are essential components of fractions. The numerator signifies the parts we are focusing on, while the denominator indicates the total number of equal parts.
To understand better:

• Numerator is the number above the line in a fraction, showing "how many" parts are taken.
• Denominator is the number below the line, showing "of what" kind of total parts the portion is taken from.

For example, in \(\frac{4}{5}\), 4 (numerator) represents four parts taken from a total of five (denominator). This division allows us to precisely understand each part's size in relation to the whole.

Converting Fractions

Changing between improper fractions and mixed numbers is an important skill.
To convert a mixed number to an improper fraction:

• Multiply the whole number by the denominator – this gives you the total parts in whole numbers.
• Add this product to the numerator – this gives you the total number of parts.
• Place this sum over the original denominator to give the improper fraction.

For instance, converting \(18 \frac{1}{3}\) involves \(18 \times 3 = 54\), then adding 1 gives 55, forming \(\frac{55}{3}\).
The opposite process involves dividing the numerator by the denominator to get back to a mixed number, ensuring the flexibility of expressing numbers in both forms as needed.