Problem 28 - Prealgebra

Question

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

Step-by-Step Solution

Answer

The quotient is \(\frac{64}{5}\).

Step 1: Convert Mixed Number to Improper Fraction

First, convert the mixed number \(1 \frac{1}{4}\) into an improper fraction. \(1 \frac{1}{4} = \frac{5}{4}\). This is done by multiplying the whole number (1) by the denominator (4) and then adding the numerator (1).

Step 2: Perform the Inner Division

Divide the improper fraction \(\frac{5}{4}\) by 2. To divide by a whole number, convert 2 into a fraction (\(\frac{2}{1}\)), and then multiply by the reciprocal. So, \(\frac{5}{4} \div \frac{2}{1} = \frac{5}{4} \times \frac{1}{2} = \frac{5}{8}\).

Step 3: Divide 8 by the Result

Now, divide 8 by \(\frac{5}{8}\). To do this, multiply 8 by the reciprocal of \(\frac{5}{8}\), which is \(\frac{8}{5}\). Thus, we get: \(8 \times \frac{8}{5} = \frac{64}{5}\).

Step 4: Simplify the Fraction (if needed)

The expression \(\frac{64}{5}\) represents a fraction and can also be expressed as a mixed number if desired, which is \(12 \frac{4}{5}\). However, it is often acceptable to leave it in improper fraction form.

Key Concepts

Mixed Number to Improper Fraction

A mixed number consists of a whole number and a fraction. To work with it in equations, we often convert it into an improper fraction. An improper fraction has a numerator larger than its denominator, allowing for easier calculations like division or multiplication.

To convert a mixed number, we use this simple method:

• Multiply the whole number by the denominator of the fractional part.
• Add the numerator to this result.
• This sum becomes the numerator of the improper fraction, while the denominator remains unchanged.

For example, converting the mixed number \(1 \frac{1}{4}\) involves multiplying 1 by 4 (gives 4), then adding the numerator 1 to get 5. Thus, \(1 \frac{1}{4}\) becomes \(\frac{5}{4}\).

Dividing Fractions

Dividing fractions is necessary to understand expressions involving fractions and division. When dividing by a fraction, you essentially multiply by its reciprocal. The reciprocal of a fraction simply switches its numerator and denominator.

Let's examine the process:

• Identify the reciprocal of the divisor. For example, the reciprocal of \(\frac{2}{1}\) (whole number 2) is \(\frac{1}{2}\).
• Multiply the dividend by this reciprocal instead of performing direct division.

Using our example, dividing \(\frac{5}{4}\) by 2 becomes \(\frac{5}{4} \times \frac{1}{2} = \frac{5}{8}\). This method simplifies division into something more manageable.

Fraction Multiplication

Multiplication of fractions requires a straightforward step-by-step approach. It involves multiplying the numerators together and the denominators together.

Here's how you do it:

• Multiply the numerators to find the new numerator.
• Multiply the denominators to get the new denominator.

For example, with \(\frac{5}{4} \times \frac{1}{2}\), we multiply the numerators 5 and 1 to get 5, and the denominators 4 and 2 to get 8. This gives us the product \(\frac{5}{8}\). Keeping steps separate helps avoid confusion and leads to the correct simplify operations if needed.

Simplifying Fractions

Simplifying fractions makes them easier to interpret and compare. A simplified fraction has a numerator and denominator with no common factors besides 1.

To simplify a fraction:

• Identify the greatest common divisor (GCD) of the numerator and denominator.
• Divide both the numerator and denominator by their GCD.

If a fraction is already in its simplest form, sometimes we can convert it into a mixed number for clarity. For instance, \(\frac{64}{5}\) doesn't simplify further, but you can express it as \(12 \frac{4}{5}\) to better understand the quantity involved. This mixed number shows how many whole parts and what portion of the whole is left over.