Problem 34 - Fundamentals of Mathematics

Question

For the following 15 problems, convert each of the improper fractions to its corresponding mixed number. $$\frac{346}{15}$$

Step-by-Step Solution

Answer

\(23 \frac{1}{15}\) is the mixed number.

Step 1: Divide the Numerator by the Denominator

To convert the improper fraction \( \frac{346}{15} \) to a mixed number, start by dividing the numerator (346) by the denominator (15). Perform the division: 346 divided by 15 equals 23 with a remainder.

Step 2: Determine the Whole Number

The quotient from the division, 23, will be the whole number part of the mixed number.

Step 3: Find the Remainder

The remainder from the division is the part of the fraction that is left over. For 346 divided by 15, the remainder is 1.

Step 4: Write the Mixed Number

Combine the whole number and the remainder to express the mixed number as \( 23 \frac{1}{15} \). In this case, the remainder (1) is placed over the original denominator (15) to form the fractional part.

Key Concepts

Improper Fractions

An improper fraction is a type of fraction where the numerator (the number on top) is greater than or equal to the denominator (the number on the bottom). This is different from a proper fraction, where the numerator is smaller than the denominator. Improper fractions may look a bit unusual because the top number is larger, but they are quite common in math. They can be easily converted to mixed numbers, which is a more familiar way to express them. To convert an improper fraction into a mixed number, you'll need to understand the concept of division. Essentially, you are trying to see how many times the denominator fits into the numerator fully, which will give you the whole number part of a mixed number. The remainder, which is what's left over, becomes part of the fractional component.

Mixed Numbers

A mixed number is a way of expressing an improper fraction with a whole part and a fractional part. It's like mixing an integer with a fraction to better represent a number that is greater than a whole number.Here's how to convert an improper fraction to a mixed number:

• First, divide the numerator by the denominator. For example, with \( \frac{346}{15} \), divide 346 by 15.
• The result of this division without considering the remainder becomes the whole number. In our example, 346 divided by 15 equals 23.
• Next, the remainder from the division becomes the numerator of the fractional part. Here, the remainder is 1.
• The original denominator stays the same. Thus, our mixed number converts to \( 23 \frac{1}{15} \).

Using mixed numbers is beneficial as they make it simpler to comprehend quantities that are parts of a whole together with whole amounts.

Division

Division is a fundamental math operation used in various contexts, including converting improper fractions to mixed numbers. In the context of fractions, division helps us determine how many "whole parts" we have and what remains, which forms the fractional part. The division process involves:

• Identifying how many times the denominator fits fully into the numerator. This gives the whole number part of the mixed number.
• Calculating the remainder of this division, which becomes the new numerator of our fractional component.

For example, dividing 346 by 15 tells us that 15 goes into 346 a total of 23 times (that's our whole number), with 1 left over (that's our remainder). The entire process is central to simplifying and converting improper fractions into an easier-to-read format like mixed numbers. Understanding how to divide correctly ensures you can handle and interpret fractions comfortably in mathematical problems.