Problem 71 - Algebra A Combined Function

Question

Find the quotient of \(-\frac{4}{9}\) and \(\frac{4}{9}\) -

Step-by-Step Solution

Answer

The quotient is \(-1\).

Step 1: Understand the Problem

We need to find the quotient of \(-\frac{4}{9}\) divided by \(\frac{4}{9}\). This is expressed mathematically as \(-\frac{4}{9} \div \frac{4}{9}\).

Step 2: Division of Fractions

To divide two fractions, we multiply the first fraction by the reciprocal of the second fraction. The reciprocal of \(\frac{4}{9}\) is \(\frac{9}{4}\). Therefore, the division \(-\frac{4}{9} \div \frac{4}{9}\) becomes \(-\frac{4}{9} \times \frac{9}{4}\).

Step 3: Multiply the Fractions

Now, multiply the numerators and the denominators: \(-4 \times 9 = -36\) and \(9 \times 4 = 36\). So, \(-\frac{4}{9} \times \frac{9}{4} = \frac{-36}{36}\).

Step 4: Simplify the Fraction

Simplify \(\frac{-36}{36}\) by dividing both the numerator and the denominator by 36. This simplifies to \(-1\).

Step 5: Conclusion

The quotient of \(-\frac{4}{9}\) and \(\frac{4}{9}\) is \(-1\).

Key Concepts

Quotients

The term "quotient" refers to the result obtained when one number is divided by another. In the context of fractions, finding a quotient involves determining how many times the second fraction fits into the first.
In our exercise, we seek the quotient of \(-\frac{4}{9}\) and \(\frac{4}{9}\). When dealing with division, particularly with fractions, the concept of quotient helps us understand the relationship between two quantities.

• The quotient can tell us how many times a fraction is contained within another.
• It helps in scaling values and understanding proportional relationships.

Finding this "how many times" effect for fractions leads us to employ multiplication through reciprocal operations, which is crucial in simplifying calculations.

Reciprocal

The reciprocal of a fraction is a key concept in the division of fractions. To find the reciprocal of a fraction, you simply swap its numerator and denominator. For instance, the reciprocal of \(\frac{4}{9}\) is \(\frac{9}{4}\).
Reciprocals are powerful tools because they allow us to convert division problems into multiplication ones. This means, instead of dividing by a fraction, we multiply by its reciprocal.
Let's look at some important points about reciprocals:

• The reciprocal of \(x\) is \(\frac{1}{x}\).
• Multiplying a number by its reciprocal results in 1 (e.g., \(\frac{4}{9} \times \frac{9}{4} = 1\)).

In our example, converting \(-\frac{4}{9} \div \frac{4}{9}\) into \(-\frac{4}{9} \times \frac{9}{4}\) simplifies the procedure and reveals that the division problem is easier to solve using multiplication.

Simplifying Fractions

Simplifying fractions makes them easier to understand and work with. This process involves reducing the numerator and denominator of a fraction by their greatest common divisor until they are as small as possible.
After performing operations such as division or multiplication, the resulting fraction might not always be in its simplest form. Consider our workout example \(\frac{-36}{36}\). Here, both the numerator and the denominator can be divided by 36, resulting in a simplified \(-1\).
Some key points about simplifying fractions are:

• Always check if both the numerator and denominator share a common factor.
• Simplifying fractions helps in achieving more understandable and manageable numbers.

In our problem, the result of the division was \(\frac{-36}{36}\). Recognizing and simplifying this to \(-1\) helps finalize our calculation neatly and correctly, illustrating the elegance of simplification.