Problem 36 - Algebra 1: Concepts and Skills

Question

Find the quotient. $$ \frac{-\frac{8}{9}}{-2} $$

Step-by-Step Solution

Answer

The quotient of the given exercise is \(\frac{16}{9}\)

Step 1: Rewrite the Division as Multiplication

The expression can be rewritten from \(\frac{-\frac{8}{9}}{-2}\) to \( -\frac{8}{9} \times \frac{1}{-2} \). This is because dividing by a number is the same as multiplying by its reciprocal. Here, the reciprocal of \(-2\) is \(\frac{1}{-2}\).

Step 2: Apply the Multiplication Rule for Fractions

To multiply fractions, multiply the numerators together to get the new numerator, and multiply the denominators together to get the new denominator. Hence, \(-\frac{8}{9} \times \frac{1}{-2} = -8 \times 1/9 \times -2 = \frac{8 \times 2}{9}\).

Step 3: Simplify the Expression

Multiplying the numbers in the numerator results in \(\frac{16}{9}\).

Key Concepts

Division of Fractions

Dividing fractions can be tricky for some, but understanding it will make math much easier. Division of fractions is based on a simple rule: when you divide by a fraction, you essentially multiply by its reciprocal. This means instead of dividing, you flip the second fraction and multiply the two fractions together.

• Find the reciprocal of the divisor (the second number in division).
• Change the division sign into a multiplication sign.
• Multiply the fractions as they now appear.

For instance, in the exercise \[\frac{-\frac{8}{9}}{-2}\], instead of dividing \(-\frac{8}{9}\) by \(-2\), you multiply \(-\frac{8}{9}\) by the reciprocal of \(-2\), which is \(\frac{1}{-2}\). This makes the problem much simpler, as it turns into a multiplication problem.

Multiplication of Fractions

Multiplying fractions is quite straightforward and involves a few simple steps. The key is to multiply the numerators and the denominators separately.

• Multiply the numerators (top numbers) together for your new numerator.
• Multiply the denominators (bottom numbers) together for your new denominator.

Remember, signs matter. If you multiply two negative numbers, the result is positive. Similarly, in the exercise with fractions \(-\frac{8}{9} \times \frac{1}{-2}\), you multiply the numerators to get \(-8 \times 1\), and the denominators to get \(9 \times 2\). The negative signs cancel each other out, resulting in a positive fraction, yielding \(\frac{16}{9}\).

Reciprocal in Mathematics

The concept of reciprocal is crucial when dividing fractions. The reciprocal of a number is simply 1 divided by that number. Flip the numerator and the denominator to find it. For a whole number, like 2, its reciprocal is \(\frac{1}{2}\).

• If you have a fraction \(\frac{a}{b}\), its reciprocal is \(\frac{b}{a}\).
• For a negative number like -2, the reciprocal is \(\frac{1}{-2}\).

Reciprocals are particularly useful because multiplying a number by its reciprocal always equals one. When dividing by a number, think of instead multiplying by its reciprocal to simplify the fraction division, just like in the original exercise. This concept turns any division of fractions into a multiplication problem, making it easier to solve.