Problem 41 - Algebra A Combined Function

Question

Divide. \(\frac{6}{7} \div\left(-\frac{1}{3}\right)\)

Step-by-Step Solution

Answer

The quotient is \(-\frac{18}{7}\).

Step 1: Understand Division of Fractions

The division of fractions involves multiplying the first fraction by the reciprocal of the second fraction. Therefore, \[\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}.\]

Step 2: Find the Reciprocal

The reciprocal of the fraction \(-\frac{1}{3}\) is \(-3\). This is obtained by swapping the numerator and denominator while keeping the negative sign in front.

Step 3: Multiply the Fractions

Multiply \(\frac{6}{7}\) by \(-3\):\[\frac{6}{7} \times \left(-3\right).\] Simplifying this multiplication gives: \[\frac{6 \times (-3)}{7} = \frac{-18}{7}.\]

Step 4: Final Result

The result of \(\frac{6}{7} \div \left(-\frac{1}{3}\right)\) after performing the multiplication is \(\frac{-18}{7}\).

Key Concepts

Reciprocal of a Fraction

Understanding the reciprocal of a fraction is fundamental when dealing with division. The reciprocal of a fraction is simply obtained by swapping its numerator and its denominator. For example, the reciprocal of a fraction given as \(\frac{a}{b}\) will be \(\frac{b}{a}\). This concept is particularly useful in fraction division because division of fractions requires us to multiply by the reciprocal of the divisor rather than performing a direct division.

It's important to note that if the original fraction is negative, its reciprocal will also be negative. Let's consider \(-\frac{1}{3}\). Its reciprocal is \(-3\) because swapping the numerator and denominator of \(-\frac{1}{3}\) gives us \(-\frac{3}{1}\), which simplifies to \(-3\).

• Step 1: Take the original fraction \(\frac{1}{3}\).
• Step 2: Swap numerator and denominator \(\rightarrow \frac{3}{1} = 3\).
• Step 3: Retain the negative sign, so the reciprocal is \(-3\).

Utilizing reciprocals is crucial in turning division exercises into multiplication problems, thus simplifying the calculation.

Multiplying Fractions

Multiplying fractions might seem complex at first, but it's really about straightforward multiplication of numerators and denominators. When you multiply two fractions, you multiply straight across: the numerators with each other and the denominators with each other.

Consider the operation \(\frac{6}{7} \times (-3)\). You should treat the integer \(-3\) as a fraction with a denominator of 1, which makes it \(-\frac{3}{1}\). When multiplying, follow these steps:

• Multiply the numerators: \(6 \times (-3) = -18\).
• Multiply the denominators: \(7 \times 1 = 7\).
• Combine the results: \(\frac{-18}{7}\).

This is your multiplied result. If necessary, simplify the fraction, though \(\frac{-18}{7}\) is already in its simplest form in this case. Remember, the main difference between multiplying fractions and whole numbers is keeping track of the numerators and denominators separately.

Negative Fractions

Negative fractions can initially feel intimidating, but they follow the same rules as positive fractions. The key difference is the negative sign, which indicates a value less than zero. Adjusting for this sign is essential in all calculations.

When working with negative fractions, especially in multiplication or division, the presence of a negative sign affects the outcome. For example, with \(\frac{6}{7} \times \left(-3\right) = \frac{-18}{7}\), the negative fraction implies a result that is flipped in its sign from what it would have been if both fractions were positive.

• A negative times a positive yields a negative result.
• A negative times a negative gives a positive result.
• Minding the sign is crucial for correct results.

Always ensure to pay attention to signs when performing operations with fractions, as overlooking them can lead to incorrect results. Understanding negative fractions thoroughly ensures precision in your calculations.