Problem 8 - Fundamentals of Mathematics

Question

Convert each mixed number to its corresponding improper fraction. $$5 \frac{3}{5}$$

Step-by-Step Solution

Answer

\( \frac{28}{5} \)

Step 1: Identify the Components

The mixed number is composed of a whole number and a fraction. In this case, the whole number is 5, and the fraction is \( \frac{3}{5} \).

Step 2: Convert the Whole Number

Multiply the whole number by the denominator of the fraction. Here, we multiply 5 (the whole number) by 5 (the denominator): \[ 5 \times 5 = 25 \]

Step 3: Add the Numerator

After converting the whole number, add the original numerator of the fraction to the result. Take the product from Step 2 and add 3 (the numerator): \[ 25 + 3 = 28 \]

Step 4: Write as an Improper Fraction

The sum from Step 3 becomes the numerator of the improper fraction, while the denominator remains the same as in the original fraction. Therefore, the improper fraction is: \( \frac{28}{5} \).

Key Concepts

Mixed Numbers

A mixed number is a combination of a whole number and a proper fraction. For example, in the mixed number \(5 \frac{3}{5}\), the "5" is the whole number and \(\frac{3}{5}\) is the fraction part. Mixed numbers are useful because they clearly show a quantity larger than a whole unit but less than the next whole unit, making them easier to visualize in certain contexts.

• They are often used in everyday situations, such as measurements and cooking.
• It's important to understand both parts of a mixed number to perform operations like conversion.

Understanding mixed numbers helps in performing calculations where whole units are involved along with fractional parts.

Numerator

The numerator is the top number in a fraction that indicates how many parts we are considering out of the whole. In the fraction \(\frac{3}{5}\), "3" is the numerator.

• It tells us that 3 parts of the total 5 parts are taken.
• In an improper fraction, the numerator is always greater than the denominator.

In the conversion process from mixed numbers to improper fractions, the numerator plays a crucial role as you add it to the product of the whole number and the denominator to form the new numerator.

Denominator

The denominator is the bottom part of a fraction and represents the total number of equal parts in the whole. In the case of the fraction \(\frac{3}{5}\), the "5" acts as the denominator.

• It is crucial for understanding the size of each part in the whole.
• The denominator remains constant when converting a mixed number to an improper fraction.

Having a grasp on what the denominator represents helps with comprehending fraction sizes and comparing them effectively.

Conversion Process

The conversion process involves changing a mixed number into an improper fraction. Here’s a step-by-step explanation:

• Multiply the whole number by the denominator of the fraction.
• Add the product to the original numerator.
• The result becomes the numerator of the improper fraction.
• Keep the original denominator as the new denominator.

For example, converting \(5 \frac{3}{5}\) into an improper fraction involves multiplying 5 (whole number) by 5 (denominator) to get 25. Then, add 3 (numerator) to get 28. Thus, the improper fraction is \(\frac{28}{5}\). This process allows us to express the same quantity in an alternate form that is often needed for more advanced calculations.