Problem 32 - Prealgebra

Question

Find each quotient. Use an area model if necessary. $$-\frac{3}{10} \div-\frac{1}{5}$$

Step-by-Step Solution

Answer

\(\frac{3}{2}\)

Step 1: Understanding Division of Fractions

When dividing fractions, we multiply by the reciprocal of the divisor. Here, the problem is \(-\frac{3}{10} \div -\frac{1}{5}\). The reciprocal of \(-\frac{1}{5}\) is \(-5\). So, \(-\frac{3}{10} \div -\frac{1}{5}\) becomes \(-\frac{3}{10} \times -5\).

Step 2: Multiplication of Fractions and Integers

Multiply \(-\frac{3}{10}\) by \(-5\). We treat \(-5\) as \(-\frac{5}{1}\). Thus the multiplication becomes \(-\frac{3}{10} \times -\frac{5}{1}\). This results in \(\frac{3 \times 5}{10 \times 1} = \frac{15}{10}\).

Step 3: Simplifying the Fraction

The fraction \(\frac{15}{10}\) can be simplified. Both the numerator (15) and the denominator (10) can be divided by their greatest common divisor, which is 5. Dividing, we get \(\frac{15 \div 5}{10 \div 5} = \frac{3}{2}\).

Step 4: Conclusion on the Quotient

Since we are multiplying two negative numbers, the result is positive. Therefore, the final quotient of \(-\frac{3}{10} \div -\frac{1}{5}\) is \(\frac{3}{2}\).

Key Concepts

Reciprocal

The concept of a reciprocal is crucial when working with division of fractions. Imagine you're asked to divide one fraction by another. Instead of dividing directly, you can think of it as multiplying by a special kind of number, known as the reciprocal. The reciprocal of a fraction is just that fraction flipped upside down. The numerator (the top number) becomes the denominator (the bottom number), and the denominator becomes the numerator.
For example:

• The reciprocal of \(-\frac{1}{5}\) is \(-5\), since the fraction \(-1/5\) flips to become \(-5/1\).

When you divide by a fraction, you're essentially multiplying by its reciprocal.
This is why in the given exercise, dividing by \(-\frac{1}{5}\) becomes the same as multiplying by \(-5\).
By understanding how to find and use the reciprocal, you can make dividing fractions a breeze.

Simplifying Fractions

Simplifying fractions is about making fractions as simple or as small as possible. This helps in seeing their true value more clearly.
The goal is to reduce both the numerator and the denominator to their smallest possible whole numbers while keeping their ratio the same.To simplify, find the greatest common divisor (GCD) of the numerator and the denominator—that is, the largest number that can divide both evenly.

• For example, in the fraction \(\frac{15}{10}\), both 15 and 10 can be divided by 5 (their GCD), resulting in \(\frac{3}{2}\).

When simplifying:

• Divide the numerator and the denominator by their GCD.
• Check if there's any further simplification possible.
• Ensure the fraction is reduced to its simplest form.

This practice not only helps in expressing fractions in their neatest form but also makes calculations easier and results more readable.

Multiplication of Fractions

Multiplying fractions is simpler than it might seem. When you multiply fractions, you multiply the numerators with each other and the denominators with each other.Let's say you have two fractions \(\frac{a}{b}\) and \(\frac{c}{d}\). To multiply them, follow these steps:

• Multiply the numerators: \(a \times c\).
• Multiply the denominators: \(b \times d\).
• Rewrite the result as a single fraction: \(\frac{a \times c}{b \times d}\).

A practical application of this method was shown in the exercise: \(-\frac{3}{10} \times -5\). Here, \(-5\) can be rewritten as \(-\frac{5}{1}\).
By multiplying, you get:

• Numerators: \(3 \times 5 = 15\)
• Denominators: \(10 \times 1 = 10\)
• The result: \(\frac{15}{10}\)

After finding this product, simplifying the fraction gives you another clear solution. Understanding multiplication of fractions this way turns seemingly complex equations into manageable math.