Problem 52 - Fundamentals of Mathematics

Question

For the following problems, find each value. $$5 \frac{1}{9} \div \frac{1}{18}$$

Step-by-Step Solution

Answer

The value of the expression is 92.

Step 1: Convert the Mixed Number to an Improper Fraction

First, we need to convert the mixed number \(5 \frac{1}{9}\) into an improper fraction. The whole number 5 can be expressed as \(\frac{45}{9}\) because when you multiply 5 by 9 (the denominator), you get 45. Then, add the numerator of the fraction (1) to this product: \(\frac{45}{9} + \frac{1}{9} = \frac{46}{9}\). So, \(5 \frac{1}{9}\) is equivalent to \(\frac{46}{9}\).

Step 2: Write the Division as Multiplication

Write the problem \(\frac{46}{9} \div \frac{1}{18}\) as \(\frac{46}{9} \times \frac{18}{1}\). Remember, when dividing by a fraction, multiply by the reciprocal of the fraction.

Step 3: Multiply the Fractions

Next, multiply the numerators together: \(46 \times 18 = 828\). Then multiply the denominators: \(9 \times 1 = 9\). Thus, the product of the fractions is \(\frac{828}{9}\).

Step 4: Simplify the Fraction

Finally, simplify \(\frac{828}{9}\) by dividing the numerator by the denominator. \(828 \div 9 = 92\). So, the simplified value of the initial expression is 92.

Key Concepts

Improper Fractions

Improper fractions are fractions where the numerator is greater than or equal to the denominator. This means the value is equal to or greater than 1. They often appear when converting mixed numbers in mathematical operations involving fractions. To understand how to deal with an improper fraction, let's see how to convert a mixed number to one. For example, given the mixed number \(5 \frac{1}{9}\), you can convert it to an improper fraction. Here's how:

• Multiply the whole number (5) by the denominator (9) to get 45.
• Add the result to the numerator (1) to get 46.
• Place this sum over the original denominator to get an improper fraction: \(\frac{46}{9}\).

This way of converting ensures that all parts of the mixed number are correctly represented in a single fraction. Understanding improper fractions helps in executing other operations such as division.

Mixed Numbers

Mixed numbers consist of a whole number and a fractional part. They are often used in everyday situations because they can make quantities easier to visualize and work with. While mixed numbers are friendly for simple calculations or measurements, they need to be converted to improper fractions for more complex arithmetic operations like multiplication or division involving fractions. Here is a short guide to quickly understand how to handle mixed numbers:

• Identify the whole number and the fraction part.
• Convert the mixed number to an improper fraction to simplify calculations.

When solving the division problem \(5 \frac{1}{9} \div \frac{1}{18}\), converting the mixed number into an improper fraction was the first essential step to streamlining later calculations.

Reciprocal of Fractions

The reciprocal of a fraction is what you multiply the original fraction by to get 1. It involves swapping the numerator and the denominator of a fraction. For instance, the reciprocal of \(\frac{1}{18}\) is \(\frac{18}{1}\). This concept is incredibly useful in division. Whenever you need to divide by a fraction, you can multiply by its reciprocal instead. This is because division by a number is equivalent to multiplying by its reciprocal. Here's a simple guide to working with reciprocals:

• Take the given fraction.
• Swap its numerator and denominator.
• Use this reciprocal for multiplying rather than dividing.

In our problem \(\frac{46}{9} \div \frac{1}{18}\), converting division into multiplication by using the reciprocal \(\frac{18}{1}\) simplified the process immensely.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest, or smallest, form where the numerator and the denominator have no common factors other than 1. This is an crucial step in solving fraction problems because it makes results more understandable and easier to perform further calculations on if needed. To simplify, divide both the numerator and the denominator by their greatest common divisor (GCD). In the exercise, after multiplying to find \(\frac{828}{9}\), we simplify by dividing both the numerator and the denominator by 9, because 9 is the GCD of 828 and 9. Here's a straightforward walkthrough on simplifying fractions:

• Identify the GCD of the numerator and denominator.
• Divide both the numerator and the denominator by this GCD.
• Write down the resulting simpler fraction.

The fraction \(\frac{828}{9}\) simplifies to 92, showing how crucial simplification is in finding the final answer in fraction problems.