Problem 55 - Fundamentals of Mathematics

Question

For the following 15 problems, convert each of the mixed numbers to its corresponding improper fraction. $$19 \frac{7}{8}$$

Step-by-Step Solution

Answer

The improper fraction is \(\frac{159}{8}\).

Step 1: Identify the Mixed Number Components

The mixed number given is \(19 \frac{7}{8}\). It consists of a whole number part, \(19\), and a fractional part \(\frac{7}{8}\).

Step 2: Multiply the Whole Number by the Denominator

Calculate the product of the whole number and the denominator of the fraction. So, \(19 \times 8 = 152\).

Step 3: Add the Numerator

Add the product from Step 2 to the numerator of the fraction. Thus, \(152 + 7 = 159\).

Step 4: Form the Improper Fraction

The sum from Step 3 forms the numerator of the improper fraction. The denominator remains the same as that of the fraction part. The improper fraction is \(\frac{159}{8}\).

Key Concepts

mixed numbers

Mixed numbers are numbers that include both a whole number and a fractional part. They are often found in day-to-day situations where measurements and quantities cannot always be expressed as whole numbers. For example, when you have 19 cakes and additional 7/8 of another cake, you express this as the mixed number 19 7/8.
Mixed numbers are useful because they provide a simple way to express quantities larger than one but not a full number. They can also make understanding proportions and distributions in real-world scenarios more intuitive. To work with mixed numbers in calculations, though, it's often necessary to convert them to improper fractions, which allows for easier mathematical operations.

fraction conversion

Fraction conversion usually involves changing a mixed number into an improper fraction or vice versa. This is key in many mathematical operations because improper fractions are easier to use in arithmetic calculations such as addition, subtraction, multiplication, and division.
To convert a mixed number into an improper fraction, follow these steps:

• Multiply the whole number by the denominator of the fraction.
• Add the result to the numerator of the fraction.
• The sum becomes the new numerator, while the denominator remains the same.

For example, converting 19 7/8 involves multiplying 19 by 8, which gives 152, and then adding 7, resulting in the improper fraction 159/8.
Understanding these conversion techniques helps with solving complex problems that involve fractions and ensures accurate results.

numerator and denominator

Numerator and denominator are the key components of any fraction. The numerator is the number above the line in a fraction and represents parts of a whole. It tells us how many parts we have. Conversely, the denominator is the number below the line and signifies how many equal parts the whole is divided into.
In the fraction \(\frac{7}{8}\), 7 is the numerator, and 8 is the denominator. In the context of converting mixed numbers to improper fractions, you maintain the denominator unchanged as it represents the same size of each part. However, the numerator changes after the conversion process to reflect the total number of parts that includes both the whole and fractional parts.
Understanding numerator and denominator is fundamental when working with fractions, enabling accurate mathematical operations and results.

mathematical steps

Mathematical steps are the sequential processes followed to solve a problem accurately. Each step is essential for ensuring that your final answer is correct and easily understandable. When converting mixed numbers to improper fractions, it is important to follow these critical steps:

• Identify and separate the mixed number components: the whole and the fraction parts.
• Multiply the whole number by the denominator of the fraction part.
• Add the result to the numerator of the fraction part to find the improper fraction's numerator.
• Keep the same denominator for the improper fraction.

By adhering to these steps, you guarantee an accurate transformation from a mixed number to an improper fraction every time. Mastery over these steps not only aids in this specific conversion but also enhances problem-solving skills for broader mathematical challenges.