Problem 23 - Prealgebra

Question

Find the following quotients. $$8 \frac{3}{5} \div 2$$

Step-by-Step Solution

Answer

The quotient is \(4 \frac{3}{10}\).

Step 1: Convert Mixed Number to Improper Fraction

First, convert the mixed number \(8 \frac{3}{5}\) to an improper fraction. To do this, multiply the whole number 8 by the denominator 5 and add the numerator 3. So, \(8 \times 5 + 3 = 40 + 3 = 43\). Therefore, \(8 \frac{3}{5}\) becomes \(\frac{43}{5}\).

Step 2: Reinterpret Division as Multiplication

Dividing by a fraction is the same as multiplying by its reciprocal. Since we are dividing by 2, we rewrite 2 as a fraction, \(\frac{2}{1}\), and find its reciprocal \(\frac{1}{2}\). So, \(8 \frac{3}{5} \div 2\) becomes \(\frac{43}{5} \times \frac{1}{2}\).

Step 3: Perform the Fraction Multiplication

Multiply the numerators and multiply the denominators of the fractions. So, \(\frac{43}{5} \times \frac{1}{2} = \frac{43 \times 1}{5 \times 2} = \frac{43}{10}\).

Step 4: Convert Improper Fraction to Mixed Number

Convert the improper fraction \(\frac{43}{10}\) back to a mixed number. Divide 43 by 10. The quotient is 4 with a remainder of 3. So, \(\frac{43}{10} = 4 \frac{3}{10}\).

Key Concepts

Mixed Numbers

A mixed number comprises a whole number and a fraction combined. It represents a quantity greater than a whole but less than the next whole number. For example, in the exercise where we begin with the mixed number \(8 \frac{3}{5}\), it means we have 8 whole parts and an additional \(\frac{3}{5}\) of another whole part.
To work with mixed numbers, we often need to convert them into improper fractions. This makes calculations involving fractions simpler, such as addition, subtraction, and division. In conversion, the whole number is multiplied by the denominator of the fractional part, and the resulting product is summed with the numerator of the fraction. Taking our example, \(8 \times 5 + 3 = 43\) gives us \(\frac{43}{5}\).

• **Advantages of Mixed Numbers:** Easy to understand and visualize.
• **Conversion Requirement:** Generally needed when performing arithmetic operations with fractions.

Improper Fractions

An improper fraction is a type of fraction where the numerator (top number) is greater than or equal to the denominator (bottom number). This means the fraction is equal to or greater than one. In our exercise, once we converted \(8 \frac{3}{5}\) into \(\frac{43}{5}\), this resulted in an improper fraction.
Improper fractions are convenient to work with for mathematical operations like multiplication and division. They simplify the process as there's no need to juggle between whole numbers and fractional parts. However, when interpreting the results, it might be more intuitive to convert back to a mixed number.

• **Benefits of Improper Fractions:** Simplifies calculations, particularly multiplication and division.
• **Converting Back:** After calculations, for ease of interpretation, convert back to mixed numbers.

Fraction Multiplication

Fraction multiplication involves multiplying the numerators to get the new numerator and the denominators to get the new denominator. It's a straightforward process, especially compared to fraction addition or subtraction.
In our exercise, once \(8 \frac{3}{5}\) was converted into \(\frac{43}{5}\) and division by 2 was reinterpreted as multiplication by \(\frac{1}{2}\), the multiplication became:
\[ \frac{43}{5} \times \frac{1}{2} = \frac{43 \times 1}{5 \times 2} = \frac{43}{10} \]
The outcome, \(\frac{43}{10}\), is the product of the multiplication. Make sure to always reduce the fraction to its simplest form if possible. If the result is an improper fraction, like in our case, it can be converted back to a mixed number for easier understanding. Therefore, \(\frac{43}{10}\) can be expressed as \(4 \frac{3}{10}\).

• **Steps for Multiplication:** Multiply across the numerators and denominators.
• **Reducing or Simplifying:** Always check if the result can be simplified.