Problem 59 - Fundamentals of Mathematics

Question

For the following 8 problems, use a calculator to convert each mixed number to its corresponding improper fraction. $$27 \frac{5}{61}$$

Step-by-Step Solution

Answer

The improper fraction is \( \frac{1652}{61} \).

Step 1: Understand the Problem

You are asked to convert the mixed number \( 27 \frac{5}{61} \) into an improper fraction. A mixed number consists of a whole number part and a fractional part.

Step 2: Multiply the Whole Number by the Denominator

Multiply the whole number part (27) by the denominator of the fractional part (61). \[ 27 \times 61 = 1647 \]

Step 3: Add the Numerator of the Fraction

Add the numerator of the fractional part (5) to the result from Step 2. This gives you the numerator of the improper fraction.\[ 1647 + 5 = 1652 \]

Step 4: Write the Result as an Improper Fraction

Combine the result from Step 3 with the denominator from the original fraction to form the improper fraction.\[ \frac{1652}{61} \]

Key Concepts

Mixed Numbers

Mixed numbers are a way of expressing numbers that have both a whole number part and a fractional part. For example, in the mixed number \(27 \frac{5}{61}\), 27 is the whole number part, and \(\frac{5}{61}\) is the fractional part.

They are often used in practical situations where you need to express amounts that are not exactly whole. Here are some important points about mixed numbers:

• They are useful for representing quantities that are greater than 1 but not whole.
• The fractional part of a mixed number always has a numerator that is smaller than the denominator.
• Mixed numbers are easier to understand and use in everyday life compared to improper fractions, their counterparts.

When performing mathematical operations, it is often useful to convert mixed numbers into improper fractions first, especially for multiplication and division. This helps simplify the calculations.

Conversion Methods

Converting mixed numbers to improper fractions is an important skill in fraction operations. The goal is to express the mixed number purely as a fraction, without a whole number part.

Here's how you can do this conversion step-by-step:

• Multiply the Whole Number by the Denominator: This converts the whole number part of the mixed number into a fraction. In our example, \(27 \times 61 = 1647\).
• Add the Numerator: Combine the fraction you found by multiplication with the numerator of the fractional part of the mixed number. Here, add 5 to 1647, getting 1652.
• Combine Into a Single Fraction: Place the result over the original denominator (61), forming the improper fraction \(\frac{1652}{61}\).

This method ensures a consistent approach to changing mixed numbers into improper fractions, making further calculations much easier.

Fraction Operations

When dealing with fractions, particularly improper fractions and mixed numbers, certain operations are common, like addition, subtraction, multiplication, and division. Understanding the right method to approach these operations can simplify complex math problems.

Let's consider these fraction operations:

• Addition and Subtraction: You need a common denominator to add or subtract fractions. Mixed numbers can be first converted into improper fractions for simplicity.
• Multiplication: This operation is straightforward with fractions, including improper ones. You multiply the numerators together and the denominators together, simplifying when possible.
• Division: To divide fractions, multiply by the reciprocal of the fraction you are dividing by.

Converting mixed numbers to improper fractions can make these operations more straightforward, focusing on managing numerators and denominators without dealing with whole numbers separately.