Problem 4 - Prealgebra

Question

Find the quotient in each case by replacing the divisor by its reciprocal and multiplying. $$-\frac{5}{8} \div \frac{1}{4}$$

Step-by-Step Solution

Answer

The quotient is \(-\frac{5}{2}\).

Step 1: Understand the Problem

We need to find the quotient of dividing a fraction by another fraction, specifically \(-\frac{5}{8} \div \frac{1}{4}\). According to division of fractions rules, we replace the divisor by its reciprocal and multiply.

Step 2: Determine the Reciprocal

The reciprocal of a fraction is obtained by flipping the numerator and the denominator. For the divisor \(\frac{1}{4}\), the reciprocal is \(4\), which can be considered as \(\frac{4}{1}\).

Step 3: Rewrite the Division Problem as Multiplication

Replace the division operation with multiplication by the reciprocal of the divisor. Thus, we rewrite the problem as \(-\frac{5}{8} \times \frac{4}{1}\).

Step 4: Perform the Multiplication

Multiply the numerators together and the denominators together. This gives: \(-\frac{5}{8} \times \frac{4}{1} = \frac{-5 \times 4}{8 \times 1} = \frac{-20}{8}\).

Step 5: Simplify the Fraction

Simplify \(\frac{-20}{8}\) by finding the greatest common divisor. Both 20 and 8 are divisible by 4. So, divide both numerator and denominator by 4: \(\frac{-20 \div 4}{8 \div 4} = \frac{-5}{2}\).

Key Concepts

Reciprocals

When dividing fractions, the concept of a reciprocal is key. A reciprocal of a fraction is simply the inverse of that fraction. For instance, if you have a fraction \(\frac{a}{b}\), its reciprocal would be \(\frac{b}{a}\).
This means you switch the numerator (top number) and the denominator (bottom number).

• The reciprocal of \(\frac{1}{4}\) is \(\frac{4}{1}\).
• This allows us to transform a division problem into a multiplication problem.

This is extremely helpful when dealing with division of fractions, because multiplying by the reciprocal is often simpler.

Multiplication of Fractions

Once you have the reciprocal, the next step is to replace the division operation with multiplication. This is the magic that allows for easy operation!

• In our problem, \(-\frac{5}{8} \div \frac{1}{4}\) becomes \(-\frac{5}{8} \times \frac{4}{1}\).
• You then multiply straight across calculating directly with the numerators and denominators.

So, you multiply the numerators: -5 times 4 gives -20. Then you multiply the denominators: 8 times 1 gives 8.
This results in the fraction \(-\frac{20}{8}\).

Simplifying Fractions

Simplifying a fraction means reducing it to its smallest form. A fraction is in its simplest form when the numerator and denominator are as small as possible but still have the same ratio.

• In our example, \(-\frac{20}{8}\) can be simplified.
• We first look for the greatest common divisor (GCD) of 20 and 8, which is 4.

Divide both the numerator and the denominator by their GCD:
\(\frac{-20 \div 4}{8 \div 4} = \frac{-5}{2}\)
By simplifying, you ensure the fraction is in its most efficient and easily understandable form.