Problem 16 - Prealgebra

Question

Find the quotient in each case by replacing the divisor by its reciprocal and multiplying. $$-\frac{25}{36} \div\left(-\frac{5}{6}\right)$$

Step-by-Step Solution

Answer

The quotient is \(\frac{5}{6}\).

Step 1: Understand the Problem

We need to find the quotient of \(-\frac{25}{36}\) divided by \(-\frac{5}{6}\). This means we need to reverse the operation by replacing the division with multiplication using the reciprocal of the divisor.

Step 2: Find the Reciprocal of the Divisor

The divisor is \(-\frac{5}{6}\). The reciprocal of a fraction \(\frac{a}{b}\) is \(\frac{b}{a}\). Therefore, the reciprocal of \(-\frac{5}{6}\) is \(-\frac{6}{5}\).

Step 3: Multiply the Dividend by the Reciprocal

Replace the division operation with multiplication using the reciprocal found in Step 2. So, we need to calculate: \[-\frac{25}{36} \times -\frac{6}{5}\]

Step 4: Perform the Multiplication

Multiply the numerators: \(-25 \times -6 = 150\). Multiply the denominators: \(36 \times 5 = 180\). So, \(-\frac{25}{36} \times -\frac{6}{5} = \frac{150}{180}\).

Step 5: Simplify the Fraction

Simplify \(\frac{150}{180}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 30. \[\frac{150 \div 30}{180 \div 30} = \frac{5}{6}\]

Step 6: Confirm the Result

Check whether the result \(\frac{5}{6}\) is in its simplest form. Since 5 and 6 have no common divisors other than 1, \(\frac{5}{6}\) is already simplified.

Key Concepts

Reciprocal

When dividing fractions, the concept of the reciprocal becomes incredibly useful. A reciprocal is simply flipping the numerator and the denominator of a fraction. For example, the reciprocal of the fraction \(-\frac{5}{6}\) is \(-\frac{6}{5}\). This flipping process is key to transforming a division problem into a multiplication problem, which is often easier to solve.

• The numerator becomes the denominator.
• The denominator becomes the numerator.
• For reciprocal operations involving negative fractions, keep the negative sign with the fraction.

Finding the reciprocal is the first step in changing a division problem into a multiplication problem, paving the way for a much simpler process. Understanding reciprocals will help you in various mathematical operations beyond just dividing fractions.

Multiplication Instead of Division

Division of fractions might feel complicated initially, but by using reciprocals, you can change the problem into something more familiar: multiplication. Once you find the reciprocal of the divisor, replace the division symbol \(\div\) with a multiplication symbol \(\times\).
Here's how it works: If you start with \(-\frac{25}{36} \div \left(-\frac{5}{6}\right)\), find the reciprocal of \(-\frac{5}{6}\) to get \(-\frac{6}{5}\). Then multiply the dividend by this reciprocal, which turns the problem into \(-\frac{25}{36} \times -\frac{6}{5}\).

• This approach eliminates the division, simplifying the calculation process.
• Multiplying fractions involves multiplying the numerators and then the denominators.
• Make sure to handle the negative signs carefully; a negative times a negative yields a positive.

This method is universal for any division of fractions, making it straightforward to obtain the quotient.

Simplifying Fractions

After multiplying fractions, it's important to simplify the resulting fraction to its simplest form. Simplification involves reducing the fraction by finding the greatest common divisor (GCD) of both the numerator and the denominator and dividing them by this number.
In the given problem, after multiplying, you arrive at \(\frac{150}{180}\). The greatest common divisor of 150 and 180 is 30. Divide both by their GCD:

• Numerator: \(150 \div 30 = 5\)
• Denominator: \(180 \div 30 = 6\)

This simplifies the fraction to \(\frac{5}{6}\), which is the simplified form. Simplification can make the final answer much clearer and easier to understand, ensuring no further reduction is possible.