Problem 84 - Beginning and Intermediate Algebra

Question

Divide. $$ -\frac{5}{12} \div \frac{5}{12} $$

Step-by-Step Solution

Answer

The answer is -1.

Step 1: Understand the division of fractions

When dividing fractions, you multiply by the reciprocal of the divisor. The reciprocal of a fraction is created by swapping the numerator and denominator.

Step 2: Find the reciprocal of the divisor

For the given problem \(-\frac{5}{12} \div \frac{5}{12}\), the divisor is \(\frac{5}{12}\). The reciprocal of \(\frac{5}{12}\) is \(\frac{12}{5}\).

Step 3: Multiply the fractions

Now multiply the dividend \(-\frac{5}{12}\) by the reciprocal of the divisor \(\frac{12}{5}\): \[-\frac{5}{12} \times \frac{12}{5}\]

Step 4: Simplify the multiplication

When you multiply the numerators and the denominators, you get: \[-\frac{5 \times 12}{12 \times 5}\ = \ -\frac{60}{60}\]

Step 5: Simplify the fraction

\(-\frac{60}{60}\) simplifies to \(-1\) because \(60\div60=1\). Thus, the expression simplifies to \(-1\).

Key Concepts

Reciprocal of a Fraction

Understanding the reciprocal of a fraction is fundamental when dealing with division between fractions. A reciprocal of a fraction is essentially what you multiply by to get 1.
To find the reciprocal, you simply swap the positions of the numerator and the denominator.

• For example, the reciprocal of \( \frac{5}{12} \) is \( \frac{12}{5} \).

Using the reciprocal is a key step in dividing fractions, as dividing by a number is the same as multiplying by its reciprocal. So when you see a division sign between two fractions, remember to switch to multiplication by using the reciprocal of the divisor.

Multiplication of Fractions

Once you have the reciprocal, the next step is to multiply. Multiplying fractions may seem straightforward, but it's essential to understand each part of the process.
When you multiply fractions:

• Multiply the numerators (the top numbers) together to find the new numerator.
• Multiply the denominators (the bottom numbers) together to find the new denominator.

Using our example, when we multiply \(-\frac{5}{12} \times \frac{12}{5}\), you multiply the top parts \[-5 \times 12\] and multiply the bottom parts \[12 \times 5\]. The result will be a new fraction \(\frac{-60}{60}\), which includes both the calculated numerator and denominator. It's often helpful to write these steps out clearly to avoid any confusion.

Simplifying Fractions

Finally, simplifying fractions is a crucial step to ensure your answer is in its simplest form. Simplifying involves reducing the fraction to its smallest possible form, where the numerator and denominator have no common factors other than 1.
To simplify, check if both the numerator and the denominator can be divided by a common factor. For the fraction \(-\frac{60}{60}\), both 60 and 60 can be divided by 60:

• Divide the numerator by 60: \(60 \div 60 = 1\)
• Divide the denominator by 60: \(60 \div 60 = 1\)

Thus, \(-\frac{60}{60}\) simplifies to \(-1\). Always try to simplify your fractions fully, as it makes your final answer clear and correct. Whenever you encounter a fraction where the numerator and the denominator are the same non-zero number, the fraction simplifies to 1, accounting for the sign from the original terms.