Problem 42 - Fundamentals of Mathematics

Question

For the following 15 problems, convert each of the mixed numbers to its corresponding improper fraction. $$1 \frac{5}{12}$$

Step-by-Step Solution

Answer

The improper fraction is \(\frac{17}{12}\).

Step 1: Understand the Mixed Number Structure

A mixed number consists of a whole number and a proper fraction. In this case, the mixed number is \(1 \frac{5}{12}\), where '1' is the whole number and \(\frac{5}{12}\) is the proper fraction.

Step 2: Calculate the Improper Fraction Numerator

To convert a mixed number to an improper fraction, you first multiply the whole number by the denominator of the fraction: \(1 \times 12 = 12\). Then add the numerator of the proper fraction to this result: \(12 + 5 = 17\).

Step 3: Formulate the Improper Fraction

Use the sum from the previous step as the new numerator of the improper fraction, while the denominator remains the same. Therefore, the improper fraction is \(\frac{17}{12}\).

Key Concepts

Improper Fractions

An improper fraction is a type of fraction where the numerator is greater than or equal to the denominator. This means that the top number is bigger or equal to the bottom number, such as in \(\frac{17}{12}\).
Improper fractions are useful because they can represent quantities greater than one whole.
Improper fractions appear often when converting mixed numbers, like when you have one or more whole numbers along with a fraction. In calculations, it’s simpler to use improper fractions rather than mixed numbers.

• Numerator \(>\) Denominator
• Example: \(\frac{17}{12}\)
• Often used for conversion from mixed numbers

Numerator

The numerator is the number above the line in a fraction. It tells you how many parts you have out of the whole.
In the mixed number \(1 \frac{5}{12}\), when converting it to the improper fraction \(\frac{17}{12}\), the numerator 17 is found by calculating \((1 \times 12) + 5\).
This calculation combines the whole parts and the fractional parts together. Understanding the role of the numerator is crucial because it determines the size or "value" that a fraction represents compared to its whole.
Quick tips about numerators:

• It tells "how many."
• Found by combining whole numbers and fractions.
• Allows comparison to the denominator.

Denominator

The denominator is the bottom part of a fraction. It tells you how many total parts are in the whole. For example, in the fraction \(\frac{5}{12}\), 12 is the denominator.
When converting a mixed number like \(1 \frac{5}{12}\) to an improper fraction \(\frac{17}{12}\), the denominator 12 stays the same. This consistency is important in ensuring that the fractional value remains equivalent during conversion.
Important points about denominators:

• Defines the "whole."
• Does not change when converting mixed numbers to improper fractions.
• Helps maintain the fraction's equivalent value.

Proper Fraction

A proper fraction is a type of fraction where the numerator is less than the denominator.
This means the fraction represents a number less than one whole, such as \(\frac{5}{12}\).
In a mixed number, the fractional part is always a proper fraction. Proper fractions are important in math for comparing and understanding parts of a whole that are smaller than one.

• Numerator \(<\) Denominator
• Example: \(\frac{5}{12}\)
• Used in mixed numbers to represent less than a whole