Problem 8 - Prealgebra

Question

Change each mixed number to an improper fraction. $$17 \frac{3}{4}$$

Step-by-Step Solution

Answer

The improper fraction is \(\frac{71}{4}\).

Step 1: Understand the components

Identify the components of the mixed number. The whole number part is 17, and the fractional part is \(\frac{3}{4}\).

Step 2: Multiply whole number by the fraction's denominator

Multiply the whole number 17 by the denominator of the fraction part, which is 4. Compute \(17 \times 4 = 68\).

Step 3: Add the numerator to the result

Take the result from Step 2 and add the numerator of the fractional part. So, compute \(68 + 3 = 71\).

Step 4: Form the improper fraction

Place the result as the new numerator over the original denominator. Thus, the improper fraction is \(\frac{71}{4}\).

Key Concepts

Improper Fractions

Improper fractions are fractions where the numerator (the top number) is greater than or equal to the denominator (the bottom number). This is the opposite of a proper fraction, where the numerator is smaller. Improper fractions are useful in mathematics as they simplify various calculations, especially during arithmetic operations.An example to illustrate converting a mixed number to an improper fraction is the mixed number \(17 \frac{3}{4}\). Here's how you do it:

• First, take the whole number part of the mixed number, which is 17 in this case.
• Multiply the whole number by the denominator of the fraction part: \(17 \times 4 = 68\).
• Add the result to the numerator of the fraction part: \(68 + 3 = 71\).
• The improper fraction is formed by placing the result as the numerator above the original denominator, \(\frac{71}{4}\).

Improper fractions are advantageous for certain calculations because they eliminate the two-part nature of mixed numbers, making arithmetic operations, such as addition or multiplication, more straightforward.

Fractions

Fractions represent parts of a whole and consist of a numerator and a denominator. The numerator is the number above the line, and the denominator is the number below. They are used to express quantities that are not whole numbers.Fractions can be classified into three primary types:

• Proper Fractions: Here, the numerator is less than the denominator, like \(\frac{3}{4}\). This represents less than one whole.
• Improper Fractions: Already discussed in the previous section, these have numerators larger than or equal to the denominators, like \(\frac{71}{4}\).
• Mixed Numbers: These are combinations of whole numbers and proper fractions, such as \(17 \frac{3}{4}\).

Fractions form a significant part of mathematics and everyday life. They are omnipresent in recipes, financial calculations, and measurements, among others. Understanding fractions is fundamental because they allow us to divide whole numbers into parts, to perform arithmetic operations on them, and to convert them as needed for different applications.

Whole Numbers

Whole numbers are the set of numbers that include all the positive integers and zero. They do not contain fractions, decimals, or negative numbers. Whole numbers are the building blocks of arithmetic and are fundamental in both academic and everyday contexts.Some characteristics of whole numbers include:

• They are countable and begin from zero, progressing to infinity like 0, 1, 2, 3, etc.
• Mixed numbers, like \(17 \frac{3}{4}\), include a whole number part, which in this case is 17.
• Whole numbers can be easily added, subtracted, multiplied, and divided (except by zero).

Whole numbers are essential for basic counting and for laying the groundwork for more complex number systems. Understanding their role is critical when dealing with fractions or mixed numbers, as you often need to operate on both whole and fractional components in calculations.