Problem 20 - Fundamentals of Mathematics

Question

For the following 15 problems, identify each expression as a proper fraction, an improper fraction, or a mixed number. $$31 \frac{6}{7}$$

Step-by-Step Solution

Answer

The expression is a mixed number.

Step 1: Convert Mixed Number

First, let's identify the type of number we have. This is a mixed number because it consists of a whole number (31) and a fraction \(\frac{6}{7}\). Mixed numbers combine both an integer and a proper fraction.

Step 2: Verify Fraction Component

Check if the fractional part of the mixed number is a proper fraction. A proper fraction has a numerator that is smaller than the denominator. Here, \(\frac{6}{7}\) is indeed a proper fraction because 6 is less than 7.

Step 3: Final Classification

Since we started with a whole number and a proper fraction, the entire expression \(31 \frac{6}{7}\) is a mixed number, defined as a combination of a whole number and a proper fraction.

Key Concepts

Proper Fraction

A proper fraction is a type of fraction where the numerator is less than the denominator. In simpler terms, the top number in a proper fraction is always smaller than the bottom number.
This means that the value of a proper fraction is less than one. For example, in the fraction \( \frac{6}{7} \), the numerator is 6, and the denominator is 7. Since 6 is less than 7, \( \frac{6}{7} \) is a proper fraction.
Proper fractions are very useful in everyday life when dividing things into smaller parts, like splitting a pizza into slices. Understanding that the numerator is smaller helps grasp that these fractions represent less than a whole.

Improper Fraction

An improper fraction stands out because its numerator is larger than or equal to its denominator. This means the fraction's value is greater than or equal to one.
For instance, if you have the fraction \( \frac{9}{5} \), notice that 9, the numerator, is larger than 5, the denominator. Therefore, \( \frac{9}{5} \) is classified as an improper fraction.
Improper fractions are important because they can be converted into mixed numbers, which often make more sense in practical contexts. For example, \( \frac{9}{5} \) can be expressed as \(1 \frac{4}{5}\), combining a whole number with a proper fraction, showing a part that exceeds one whole.

Fraction Conversion

Converting between different forms of fractions, such as from a mixed number to an improper fraction, is a crucial skill. It helps in making calculations easier and comparing fractions more straightforward.
To convert a mixed number like \(31 \frac{6}{7}\) into an improper fraction, multiply the whole number by the denominator and add the numerator. This means \(31 \times 7 + 6 = 217 \), giving us the improper fraction \( \frac{217}{7} \).
Conversion also goes the other way. For instance, turning an improper fraction back into a mixed number involves dividing the numerator by the denominator. The quotient is the whole part, and the remainder is the new numerator of the proper fraction. This process makes it simple to understand and interpret values greater than one.