Problem 56 - Fundamentals of Mathematics

Question

(Section 4.3) Convert \(3 \frac{5}{12}\) to an improper fraction.

Step-by-Step Solution

Answer

The improper fraction is \(\frac{41}{12}\).

Step 1: Understand the Mixed Number

The given number is a mixed number, which consists of a whole number and a fractional part. In this case, it is given as \(3 \frac{5}{12}\), where 3 is the whole number and \(\frac{5}{12}\) is the fractional part.

Step 2: Convert the Whole Number to a Fraction

To convert the whole number into a fraction, multiply it by the denominator of the fractional part. Here, multiply 3 by 12: \(3 \times 12 = 36\). This gives the fraction \(\frac{36}{12}\).

Step 3: Add the Fractional Part

Add the fractional part to the fraction you derived from the whole number. The fractional part is \(\frac{5}{12}\). Add this to \(\frac{36}{12}\): \(\frac{36}{12} + \frac{5}{12} = \frac{41}{12}\).

Step 4: State the Improper Fraction

The resulting improper fraction from the addition is \(\frac{41}{12}\). This fraction is improper because the numerator (41) is larger than the denominator (12).

Key Concepts

Mixed Numbers

Mixed numbers are a combination of a whole number and a fraction. They are typically written in the form of, say, \( a \frac{b}{c} \), where \( a \) is the whole number and \( \frac{b}{c} \) is the fractional part. Mixed numbers are useful in representing numbers that are not whole but more intuitive to read than improper fractions. For instance, \( 3 \frac{5}{12} \) clearly indicates a quantity larger than 3 but not as concise when it comes time to perform calculations.

• Whole Number: The integer part of the mixed number.
• Fractional Part: Proper fraction, meaning numerator < denominator.

Understanding mixed numbers is crucial for performing operations like addition, subtraction, etc., with ease. When calculations are involved, it's often efficient to convert them to improper fractions.

Fraction Conversion

Fraction conversion is the process of changing a mixed number into an improper fraction, or vice versa. This is fundamental for operations on fractions like addition or subtraction. Conversion to an improper fraction involves a straightforward process:

• Multiply the whole number by the denominator of the fractional part.
• Add the result to the numerator of the fractional part.
• Write this sum over the original denominator.

For example, to convert \( 3 \frac{5}{12} \), multiply 3 by 12, getting 36. Add 5 to this result (36 + 5 = 41), placing it over the denominator, resulting in \( \frac{41}{12} \). This method ensures that we accurately transfer the "value" expressed by the mixed number.

Addition of Fractions

Adding fractions involves a need for common denominators. If the fractions share the same denominator, this process is simple:

• Add the numerators directly.
• Maintain the denominator as it is.

In the case of adding \( \frac{36}{12} \) and \( \frac{5}{12} \), since both fractions have 12 as their denominator, we simply add the numerators: 36 + 5, to get 41. Thus, the resulting fraction is \( \frac{41}{12} \).For fractions with different denominators, however, you must first find a common denominator. This usually involves multiplying the denominators and adjusting the numerators accordingly. This way, the fractions can be combined easily, leading to correct results. Mastery in adding fractions ensures proper handling of various mathematical scenarios.