Problem 60 - Prealgebra

Question

Find quotient. Write in simplest form. \(-\frac{5}{8} \div \frac{1}{3}\)

Step-by-Step Solution

Answer

The quotient is \(-\frac{15}{8}\).

Step 1: Identify the Problem Type

The exercise asks us to find the quotient of two fractions: \(-\frac{5}{8}\) divided by \(\frac{1}{3}\).

Step 2: Recall the Division Rule for Fractions

To divide fractions, multiply the first fraction by the reciprocal of the second fraction. The reciprocal of a fraction is found by swapping its numerator and denominator.

Step 3: Find the Reciprocal of the Second Fraction

The reciprocal of \(\frac{1}{3}\) is \(\frac{3}{1}\).

Step 4: Multiply the Fractions

Multiply \(-\frac{5}{8}\) by \(\frac{3}{1}\). This gives us:\[-\frac{5}{8} \times \frac{3}{1} = -\frac{5 \times 3}{8 \times 1} = -\frac{15}{8}\]

Step 5: Simplify if Possible

Check if \(-\frac{15}{8}\) can be simplified further. Since 15 and 8 have no common factors other than 1, \(-\frac{15}{8}\) is already in its simplest form.

Key Concepts

Reciprocal of a Fraction

When you're working with fractions, sometimes you'll need to find what's known as the "reciprocal." This simply means flipping the fraction. You turn the numerator into the denominator and the denominator into the numerator.
For example, if you have a fraction like \( \frac{1}{3} \), its reciprocal would be \( \frac{3}{1} \). Essentially, you're inverting the fraction.
This is particularly useful when dividing fractions. Instead of directly performing division, you multiply by the reciprocal of the second fraction. So, in the exercise we're looking at, we took the reciprocal of \( \frac{1}{3} \) to easily solve the problem by multiplying instead.

Simplifying Fractions

Simplifying a fraction means making it as simple as possible, where the numerator and the denominator have no common factors other than 1. This makes the fraction easier to understand and work with.
Imagine you have a fraction like \( \frac{15}{8} \). You might want to check if the fraction can be simplified by finding a common factor of both the numerator and the denominator. If it's possible to divide both by the same number (other than 1) to make them smaller, then go ahead and do it.
However, sometimes, like in our exercise with \( \frac{15}{8} \), the only common factor is 1. So, \( \frac{15}{8} \) is already as simple as it can be. This means \( \frac{15}{8} \) is its simplest form.

Numerators and Denominators

In a fraction, the top number is called the numerator, and the bottom number is the denominator. These two numbers show us exactly what part of a whole we're working with.
For instance, in \( \frac{5}{8} \), 5 is the numerator and 8 is the denominator. This tells us we have 5 parts out of 8 total parts.
The numerator tells us "how many," while the denominator tells us "of what." Understanding these roles helps when performing operations with fractions, as you'll know what to change and when, especially when finding reciprocals or simplifying.
So in a division problem like \( -\frac{5}{8} \div \frac{1}{3} \), knowing which number is the numerator and which is the denominator is key to swapping them correctly for reciprocals or simplifying results.