Problem 63 - Fundamentals of Mathematics

Question

For the following 8 problems, use a calculator to convert each mixed number to its corresponding improper fraction. $$816 \frac{19}{25}$$

Step-by-Step Solution

Answer

The improper fraction is \(\frac{20419}{25}\).

Step 1: Understand Mixed Numbers

A mixed number consists of a whole number and a fractional part. Here, 816 is the whole number and \(\frac{19}{25}\) is the fractional part.

Step 2: Convert Whole Number to Fraction Form

Express the whole number as a fraction with the same denominator as the fractional part. In this case, convert 816 to \(\frac{816 \times 25}{25}\), which is \(\frac{20400}{25}\).

Step 3: Add Fractions Together

Add the fractions \(\frac{20400}{25}\) and \(\frac{19}{25}\). Adding these gives \(\frac{20400 + 19}{25} = \frac{20419}{25}\).

Step 4: Write the Final Answer

The improper fraction corresponding to the mixed number 816 \(\frac{19}{25}\) is \(\frac{20419}{25}\).

Key Concepts

Improper Fractions

Improper fractions are an interesting type of fraction where the numerator is greater than or equal to the denominator. This means the fraction's value is equal to or greater than 1. An improper fraction might look a bit overwhelming at first, but it simply tells you how many times the whole is contained within the fraction. For example, if you have \(\frac{7}{3}\), it means you have two wholes with a remainder of \(\frac{1}{3}\). Converting a mixed number, like 816 \(\frac{19}{25}\), into an improper fraction is a way of expressing the entire quantity as just one fraction instead of a whole number and a fraction. This can make mathematical calculations like addition or subtraction more manageable.

Fraction Addition

Adding fractions might seem complicated, but it's all about having a common denominator. When two fractions have the same denominator, you can simply add their numerators. Imagine you have \(\frac{3}{5}\) and \(\frac{2}{5}\). Both of these fractions have 5 as the denominator, so you just add the numerators: \(3 + 2 = 5\), giving you \(\frac{5}{5}\), which is equal to 1.
In the case of the problem, after expressing 816 as \(\frac{20400}{25}\), and with \(\frac{19}{25}\) already having the denominator 25, you just add the numerators. Calculating \(20400 + 19 = 20419\), the sum is expressed as \(\frac{20419}{25}\). This shows how the addition of fractions with the same denominators simplifies the computation.

Whole Numbers as Fractions

Whole numbers can be expressed as fractions with any specified denominator. This is a fundamental concept when working with mixed numbers. By converting a whole number into a fraction, you align it with the fractional component to perform operations such as addition. Take the whole number 816. To express it as a fraction with a denominator of 25, you multiply 816 by 25 to get the numerator and keep 25 as the denominator, resulting in \(\frac{816 \times 25}{25} = \frac{20400}{25}\). This way, you're expanding the whole number into a fraction that can easily engage in further mathematical operations.