Problem 4 - Fundamentals of Mathematics

Question

Convert each improper fraction to its corresponding mixed number. \(\frac{31}{13}\)

Step-by-Step Solution

Answer

\(2\frac{5}{13}\)

Step 1: Understand the Problem

An improper fraction is a fraction where the numerator is greater than or equal to the denominator. Our task is to convert the improper fraction \(\frac{31}{13}\) into a mixed number, which is a whole number and a proper fraction combined.

Step 2: Divide the Numerator by the Denominator

To convert \(\frac{31}{13}\) into a mixed number, first divide the numerator (31) by the denominator (13). Perform the division: \(31 \div 13 = 2\) with a remainder.

Step 3: Determine the Quotient, Remainder, and Denominator

The quotient obtained is the whole number part of the mixed number. The remainder becomes the numerator of the proper fraction, and the denominator remains the same. In this case, the division yields a quotient of 2 and a remainder of 5.

Step 4: Write the Mixed Number

Combine the whole number with the fraction form of the remainder over the original denominator. Therefore, \(\frac{31}{13}\) translates to the mixed number \(2\frac{5}{13}\).

Key Concepts

Understanding Mixed Numbers

Mixed numbers are composed of two distinct parts: a whole number and a fraction. This means that when you see a mixed number, like the example in our exercise, it will look something like this: \(2\frac{5}{13}\).

• The whole number represents complete units.
• The fraction shows a part of a unit.

Mixing these two elements gives a more concise way of writing partial and whole quantities together. Imagine having a pizza -- if you eat 2 whole pizzas and 5 slices out of the next 13-slice pizza, a mixed number accurately describes what you've eaten. Turning improper fractions into mixed numbers can make understanding or visualizing quantities much easier, especially when dealing with complex numbers.

Getting to Know the Numerator and Denominator

Fractions consist of two pivotal components: the numerator and the denominator. In the given improper fraction \(\frac{31}{13}\), the 31 is the numerator, and 13 is the denominator.
Here's how they function:

• The **numerator** tells us how many parts of the whole we have.
• The **denominator** tells us into how many equal parts the whole is divided.

In our conversion of \(\frac{31}{13}\) to a mixed number, understanding these roles is crucial. The numerator is larger, indicating that this is an improper fraction -- meaning it represents more than one whole. Hence, knowing what these components are helps us perform conversions such as division to find mixed numbers more effectively.

Division in Fractions Explained

When converting improper fractions to mixed numbers, division is a key step. This process may seem complicated, but it is all about breaking down the fraction into a whole number and the remainder that still stands for the leftover part of the fraction.
For \(\frac{31}{13}\), you divide 31 by 13. Let's break down the division:

• Perform the division: 31 divided by 13 equals 2 with a remainder.
• The **quotient** (result of division without remainder) gives us the whole number -> 2.
• The **remainder** becomes the new numerator of the fraction part and is 5.
• The denominator stays the same at 13.

The result is expressed as \(2\frac{5}{13}\). This demonstrates how division not only helps in computing operations but also in simplifying the expression of numerical quantities from improper fractions to mixed numbers.