Problem 64 - Fundamentals of Mathematics

Question

For the following 8 problems, use a calculator to convert each mixed number to its corresponding improper fraction. $$708 \frac{42}{51}$$

Step-by-Step Solution

Answer

The improper fraction is \(\frac{36050}{51}\).

Step 1: Understanding Mixed Numbers

A mixed number is a combination of a whole number and a proper fraction. In this problem, the mixed number is given as \(708 \frac{42}{51}\), where 708 is the whole number and \(\frac{42}{51}\) is the fraction.

Step 2: Identify the Whole Number and Fraction

Identify the whole number, which is 708, and the fraction, which is \(\frac{42}{51}\). This separates the two parts for conversion.

Step 3: Convert the Whole Number to a Fraction with the Same Denominator

The whole number 708 can be expressed as a fraction with the denominator 51. This is done as follows: \(708 = \frac{708 \times 51}{51}\).

Step 4: Perform Multiplication

Multiply the whole number 708 by the denominator 51 to get the part of the improper fraction contributed by the whole number. Calculate: \(708 \times 51 = 36008\). Thus, \(708 = \frac{36008}{51}\).

Step 5: Add the Proper Fraction

Now, add the fraction \(\frac{42}{51}\) to the converted whole number fraction: \(\frac{36008}{51} + \frac{42}{51}\).

Step 6: Simplify and Add the Fractions

Since both fractions have the same denominator, add the numerators: \(36008 + 42 = 36050\). Thus, the improper fraction is \(\frac{36050}{51}\).

Step 7: Final Conversion Result

Now that the mixed number is converted into an improper fraction, verify the calculation: \(708 \frac{42}{51} = \frac{36050}{51}\).

Key Concepts

Understanding Mixed Numbers

Mixed numbers combine whole numbers with proper fractions. These expressions give you a more straightforward and visual way to understand portions over one whole. If you're wondering what makes a proper fraction, it's simply a fraction where the numerator (the top number) is smaller than the denominator (the bottom number).
For instance, in the mixed number \( 708 \frac{42}{51} \), the whole number is 708, and the proper fraction is \( \frac{42}{51} \). You typically encounter mixed numbers in everyday scenarios, like recipes or measurements.

• Whole number: Represents complete units.
• Proper fraction: Represents parts smaller than a whole.

Breaking a mixed number into these two parts helps make many mathematical processes easier, like converting into improper fractions.

Converting to Improper Fractions

Converting a mixed number to an improper fraction involves incorporating the entire value of that mixed number into a single fraction. This is practical for simplifying fraction operations, like addition and subtraction.
The process requires turning the whole number part into a fraction. To do this, multiply the whole number by the denominator of the fraction part. This makes both parts compatible with one another.
Here's the step-by-step breakdown:

• Identify the whole number and fraction.
• Convert the whole number to a fraction using the same denominator.
• Multiply the whole number by the denominator and combine with the fraction's numerator.

Using \( 708 \frac{42}{51} \) as an example, the conversion is: multiply 708 by 51 to get 36008, then add 42. The improper fraction becomes \( \frac{36050}{51} \).

Fraction Addition

Adding fractions can sometimes be a tricky task, but not when the denominators are already aligned. That's the beauty of converting a mixed number into an improper fraction first.
Once you have your improper fractions, adding them is straightforward because you're dealing with like fractions (same denominators). Simply add the numerators. This allows you to easily find the sum or solve equations, which might be part of more complex problems.

• Add numerators: Simply sum them up.
• Keep the denominator the same.

For instance, combining \( \frac{36008}{51} \) and \( \frac{42}{51} \), you add 36008 and 42, resulting in the improper fraction \( \frac{36050}{51} \).

Understanding Denominators

Denominators help determine the size of each piece in a fraction. In the world of fractions, the denominator tells us into how many parts the whole is divided.
It's critical in fraction operations like addition, as fractions must share the same denominator to be directly added or subtracted. For the mixed number \(708 \frac{42}{51}\), the denominator is 51, implying that each whole number is divided into 51 parts.
When you convert mixed numbers to improper fractions, the denominator stays the same. This consistency allows for seamless addition and subtraction.

• Common denominator: Necessary for fraction addition and subtraction.
• Denominator consistency: Ensures accurate calculations.

The consistent denominator facilitates the conversion and calculation processes, ensuring you accurately represent all parts of the number.