Problem 65 - Fundamentals of Mathematics

Question

For the following 8 problems, use a calculator to convert each mixed number to its corresponding improper fraction. $$ 6,012 \frac{4,216}{8,117} $$

Step-by-Step Solution

Answer

The mixed number converts to the improper fraction \(\frac{48,788,620}{8,117}\).

Step 1: Understanding the Mixed Number

In this problem, you need to convert the mixed number \(6,012 \frac{4,216}{8,117}\) into an improper fraction. A mixed number consists of a whole number and a fraction. Here, the whole number is 6,012, and the fraction is \(\frac{4,216}{8,117}\).

Step 2: Converting to Improper Fraction

To convert the mixed number to an improper fraction, we use the formula:\[ \text{Improper Fraction} = (\text{Whole Number} \times \text{Denominator}) + \text{Numerator} \]Substitute the given values into the formula:\[ (6,012 \times 8,117) + 4,216 \]

Step 3: Calculating the Numerator

First, calculate \(6,012 \times 8,117\) using a calculator.\[ 6,012 \times 8,117 = 48,784,404 \]Add the numerator of the fraction:\[ 48,784,404 + 4,216 = 48,788,620 \]

Step 4: Forming the Improper Fraction

The numerator of the improper fraction is the result from the previous calculation, and the denominator stays the same.So, the improper fraction is:\[ \frac{48,788,620}{8,117} \]

Step 5: Simplifying the Fraction (if possible)

Check if the fraction can be simplified by finding the greatest common divisor (GCD) of the numerator and denominator, \(48,788,620\) and \(8,117\), respectively, using a calculator. In this case, no simplification is possible.

Key Concepts

Improper Fractions

An improper fraction is a type of fraction where the numerator (the top number) is greater than or equal to the denominator (the bottom number). This means that the value of the fraction is greater than or equal to one. Improper fractions are useful in various mathematical calculations, especially when dealing with mixed numbers, because they can simplify arithmetic operations like addition, subtraction, multiplication, and division.

• Example: An improper fraction like \( \frac{9}{4} \) represents a value greater than one.
• Purpose: Improper fractions make calculations easier by avoiding the complications that arise when dealing with mixed numbers.

To convert a mixed number to an improper fraction, we typically multiply the whole number part by the denominator and then add the numerator. This process allows us to work with a single numerical value in the fraction form, streamlining many mathematical processes.

Fraction Conversion

Fraction conversion is the process of changing one form of a fraction to another, such as converting a mixed number into an improper fraction or vice versa. This skill is key in mathematics, as it allows us to express numbers in different forms based on what is most useful for a given calculation or context.
When converting a mixed number like \(6,012 \frac{4,216}{8,117}\) to an improper fraction, follow these steps:

• Multiply the whole number (6,012) by the denominator (8,117).
• Add the numerator (4,216) to the result of the multiplication.
• Form the improper fraction by placing the sum over the original denominator.

This conversion provides a consistent method to handle larger values expressed in a combined whole-fraction format. Being able to switch between mixed numbers and improper fractions allows for flexibility in problem-solving.

Mathematical Calculations

Mathematical calculations involving mixed numbers and improper fractions require precision and attention to detail. Using a calculator for large numbers ensures accuracy, especially in complex computations.
For the problem \(6,012 \times 8,117 + 4,216\), follow these calculation steps:

• Calculate the product of 6,012 and 8,117.
• Add the resulting product to the numerator of the fraction, 4,216.
• The result gives you the new numerator for the improper fraction.

Ensure to keep the denominator the same when converting from a mixed number to an improper fraction. By using these steps, you can tackle any problem involving mixed numbers and ensure that you're working with precise figures throughout mathematical procedures.