Problem 30 - Fundamentals of Mathematics

Question

For the following 15 problems, convert each of the improper fractions to its corresponding mixed number. $$\frac{71}{8}$$

Step-by-Step Solution

Answer

\(\frac{71}{8}\) as a mixed number is \(8\frac{7}{8}\).

Step 1: Understand the Improper Fraction

Identify the numerator and the denominator in the improper fraction \( \frac{71}{8} \). Here, 71 is the numerator (the number on top), and 8 is the denominator (the number at the bottom).

Step 2: Divide the Numerator by the Denominator

Perform the division \( 71 \div 8 \) to find out how many whole times 8 can go into 71. The quotient represents the whole number of the mixed number. Calculate: 71 divided by 8 equals 8 with a remainder.

Step 3: Determine the Whole Number

The quotient from the division is 8, which is the whole number part of the mixed number.

Step 4: Find the Remainder

Calculate the remainder from the division. Multiply the whole number (8) by the denominator (8), getting 64, then subtract from the numerator (71). The remainder is \(71 - 64 = 7\).

Step 5: Assemble the Mixed Number

Combine the whole number and the remainder into a mixed number. The remainder becomes the new numerator and the original denominator stays the same. Therefore, \( \frac{71}{8} = 8 \frac{7}{8} \).

Key Concepts

Mixed Numbers

Mixed numbers are an interesting way to express numbers that are greater than a whole but less than another whole number. They consist of two parts: a whole number and a proper fraction. Consider them as a way of simplifying the view of improper fractions, which can sometimes seem awkward or harder to interpret.

• The whole number represents how many full sets of the denominator fit into the numerator.
• The fraction part indicates the leftover part of the overall number.

For instance, the improper fraction \( \frac{71}{8} \) translates into the mixed number \( 8 \frac{7}{8} \), where 8 is the whole number and \( \frac{7}{8} \) represents the remaining portion. This mix allows for a clearer understanding of the size of the fraction relative to whole numbers.

Numerator and Denominator

In any fraction, you'll encounter two essential players: the numerator and the denominator. Understanding their roles can significantly simplify working with fractions.

• The numerator is the number on top. It shows how many parts of the whole we are considering.
• The denominator, on the bottom, tells us how many parts the whole is divided into.

In the fraction \( \frac{71}{8} \), 71 is the numerator, and 8 is the denominator. This structure explains that we have "71 parts out of 8 possible equal parts." It sets the stage for converting improper fractions into more digestible mixed numbers.

Division in Fractions

Converting improper fractions to mixed numbers primarily involves performing division. This process helps us determine how many full groups of the denominator exist in the numerator.
Take \( 71 \div 8 \) as an example:

• Divide the numerator (71) by the denominator (8).
• The result, or quotient, represents the whole number in the mixed number.

Here, dividing 71 by 8 gives us 8, with some left over. The quotients of such divisions are vital as they help us uncover the whole number part of a mixed number.

Remainder in Division

When dividing fractions to convert an improper fraction to a mixed number, the divisor does not always go into the dividend perfectly. This results in a remainder—the part left after the whole number part is determined.

• The remainder is the leftover part after subtracting the total "whole" multiples of the denominator from the numerator.
• In our example, after multiplying \( 8 \times 8 = 64 \), subtracting \( 64 \) from \( 71 \) leaves us with a remainder of 7.

This remainder becomes the new numerator in the fractional part of the mixed number. So, in \( 8 \frac{7}{8} \), the remainder 7 tells us how much more than 8 whole sets of the denominator we have.