Problem 35 - Prealgebra

Question

Find each quotient. Use an area model if necessary. $$-\frac{5}{8} \div(-4)$$

Step-by-Step Solution

Answer

The quotient is \(\frac{5}{32}\).

Step 1: Identify the Division Problem

The given problem is to find the quotient of \(-\frac{5}{8} \div (-4)\). This is dividing a fraction by a whole number.

Step 2: Convert Whole Number to a Fraction

Rewrite the whole number \(-4\) as a fraction: \(-4 = \frac{-4}{1}\). Now the expression is: \(-\frac{5}{8} \div \frac{-4}{1}\).

Step 3: Division Rule for Fractions

Recall that dividing by a fraction is equivalent to multiplying by its reciprocal. Thus, \(-\frac{5}{8} \div \frac{-4}{1} = -\frac{5}{8} \times \left(-\frac{1}{4}\right)\).

Step 4: Multiply the Numerators

Multiply the numerators of the fractions: \(-5 \times (-1) = 5\). Since a negative times a negative is a positive.

Step 5: Multiply the Denominators

Multiply the denominators: \(8 \times 4 = 32\).

Step 6: Put it Together

Combine the results to get the product: \[\frac{5}{32}\]. Thus, \(-\frac{5}{8} \div -4 = \frac{5}{32}\).

Key Concepts

Area Model

The area model is a helpful way to visually represent the division of fractions. It works by drawing rectangles to represent the quantities involved. Each smaller rectangle within corresponds to a part of the whole. This model breaks down abstract calculations into tangible visual aids. To illustrate with our example, consider the fraction \(-\frac{5}{8}\). You can visualize this as a rectangle divided into 8 equal parts, where 5 parts are shaded. We want to see how many times a value equivalent to \(-4\) fits into this shaded area. However, due to the involvement of negative numbers, using the area model can be a bit tricky. Yet, this visualization remains useful to make initial sense of fractional division tasks, especially when signs are not involved.

Reciprocal

The reciprocal of a number is what you multiply the number by to get one. For a fraction, it's quite simple: you swap the numerator and the denominator. For example, the reciprocal of \(\frac{-4}{1}\) is \(-\frac{1}{4}\). This concept is crucial because dividing by a fraction is the same as multiplying by its reciprocal. It fundamentally changes the operation but keeps our expressions equivalent. In our exercise, converting division into multiplication helps simplify solving the problem.

Multiplication of Fractions

Multiplying fractions involves simply multiplying the numerators together to get the new numerator and multiplying the denominators together to reach the new denominator. Let's break it down.In our example:

• The numerators are \(-5\) and \(-1\). When we multiply these, \(-5\times(-1)=5\).
• The denominators are \(8\) and \(4\). So, we find \(8\times4=32\).

Our final fraction after multiplication is \(\frac{5}{32}\). Multiplication here has altered the negatives into a positive, as a negative times a negative equals a positive. This particular property is incredibly useful when solving problems involving negative fractions.

Negative Numbers

Negative numbers, denoted with a "-" symbol, broadly represent values less than zero. These are vital for representing debts or reductions. Understanding multiplication and division with these numbers requires knowing their unique properties. One important rule is that multiplying or dividing two negative numbers results in a positive number. This is because two negatives offset each other. For example, in our exercise, \(-5\) times \(-1\) gives \(5\). Remember this rule:

• Negative times a negative = Positive
• Negative times a positive = Negative

Understanding these interactions is essential when working across various mathematical operations.