Problem 27 - Prealgebra

Question

Find the following quotients. $$\left(8 \div 1 \frac{1}{4}\right) \div 2$$

Step-by-Step Solution

Answer

The quotient is \(\frac{16}{5}\).

Step 1: Convert Mixed Number to Improper Fraction

The first step is to convert the mixed number \(1 \frac{1}{4}\) into an improper fraction. A mixed number is a combination of a whole number and a fraction. To convert it: \(1 \frac{1}{4} = \frac{5}{4}\).

Step 2: Rewrite the division as multiplication

Division by a fraction is equivalent to multiplication by its reciprocal. For \(8 \div \frac{5}{4}\), we rewrite it as: \(8 \times \frac{4}{5}\).

Step 3: Perform the multiplication

Now multiply: \(8 \times \frac{4}{5} = \frac{32}{5}\). This fraction is the result of the first operation.

Step 4: Divide by 2

The problem now becomes dividing \(\frac{32}{5}\) by 2. Divide the numerator of the fraction by 2: \(\frac{32}{5} \div 2 = \frac{32}{5} \times \frac{1}{2} = \frac{32}{10}\).

Step 5: Simplify the fraction

Finally, simplify the fraction \(\frac{32}{10}\). Both numerator and denominator can be divided by 2: \(\frac{32}{10} = \frac{16}{5}\).

Key Concepts

Improper fractions

An improper fraction is a type of fraction where the numerator, which is the top number, is larger than or equal to the denominator, the bottom number. This means the fraction represents a value greater than or equal to one whole. For example, \( \frac{5}{4} \) is an improper fraction because 5 is larger than 4.

• To convert a mixed number into an improper fraction, multiply the whole number by the denominator and add the numerator.
• For \(1 \frac{1}{4}\), multiply 1 by 4 (denominator) which gives us 4, and then add 1 (numerator) to get 5. This makes \( \frac{5}{4} \).
• Thus, knowing how to switch between these forms is useful in division and multiplication problems involving fractions.

Doing this conversion ensures calculations are always with fractions if that was the original task context, keeping consistency in mathematical operations.

Mixed numbers

A mixed number combines a whole number and a proper fraction together. It indicates parts of a whole plus whole numbers. For instance, the number \(1 \frac{1}{4}\) consists of the whole number 1 and the fraction \(\frac{1}{4}\). To easily read and understand it:

• It's useful to convert them into improper fractions when carrying out operations such as division or multiplication.
• This makes computations smoother, especially when a problem involves multiple fraction operations.
• Always convert mixed numbers like \(1 \frac{1}{4}\) into improper fractions \(\frac{5}{4}\) to simplify computation steps directly.

This step simplifies the problem by maintaining a single fraction format.

Reciprocal

A reciprocal of a fraction is what you multiply that fraction by to get the result as 1. In simpler terms, it's flipping a fraction upside down. This is a powerful tool

• The reciprocal of \( \frac{5}{4} \) is \( \frac{4}{5} \). It is used to transform division into easier multiplication problems.
• To find the reciprocal, just switch the numerator and the denominator of the fraction.
• Using reciprocals in operation allows us to handle fractions systematically in multi-step arithmetic operations.

For example, in our problem \(8 \div \frac{5}{4}\), we'd instead do \(8 \times \frac{4}{5}\). Using reciprocals can simplify calculations, enabling easier solving of complex fraction problems.