Problem 210 - Intermediate Algebra

Question

In the following exercises, simplify. $$ -\frac{3}{12} \div\left(-\frac{5}{9}\right) $$

Step-by-Step Solution

Answer

\( \frac{9}{20} \)

Step 1 - Simplify the First Fraction

Simplify \(-\frac{3}{12}\) by dividing both the numerator and the denominator by 3. This yields \(-\frac{1}{4}\).

Step 2 - Reciprocate the Second Fraction

Reciprocate the second fraction \(-\frac{5}{9}\) to get its multiplicative inverse. The reciprocal is \(-\frac{9}{5}\).

Step 3 - Multiply the Simplified First Fraction by the Reciprocal of the Second Fraction

Now, multiply \(-\frac{1}{4}\) by the reciprocal of the second fraction \(-\frac{9}{5}\): \(-\frac{1}{4} \times -\frac{9}{5}\).

Step 4 - Perform the Multiplication

Multiply the numerators and the denominators: \(-1 \times -9 = 9\ and \4 \times 5 = 20\). Therefore, the product is \frac{9}{20}\.

Key Concepts

fraction multiplication

When you multiply fractions, you multiply the numerators together and the denominators together. For example, to multiply \(\frac{a}{b} \times \frac{c}{d}\), you do: \[ \frac{a \times c}{b \times d} \]. This way, everything is kept simple by focusing on the top numbers (numerators) and the bottom numbers (denominators) separately. Make sure you handle any negative signs properly – if both fractions are negative, the result will be positive.

reciprocals

The reciprocal of a fraction is just flipping the numerator and the denominator. For instance, the reciprocal of \(\frac{a}{b}\) is \(\frac{b}{a}\). When dividing by a fraction, you multiply by its reciprocal instead. For example, dividing by \(\frac{5}{9}\) is the same as multiplying by \(\frac{9}{5}\). Finding reciprocals is very important in fraction division.

numerator and denominator simplification

Before multiplying or dividing fractions, always look to simplify the fractions first. Simplification involves dividing the numerator and the denominator by their greatest common divisor (GCD). For \(\frac{3}{12}\), both 3 and 12 can be divided by 3, simplifying it to \(\frac{1}{4}\). Simplifying fractions beforehand makes the math smaller and more manageable.

basic algebra

Algebra involves working with variables and constants to solve for unknowns or simplify expressions. When fractions get involved, you use the same principles. Multiply and divide fractions just as you do with numbers. Remember the order of operations. When simplifying \(-\frac{3}{12} \/ ((-\frac{5}{9}))\), you simplify the fractions, flip the second one (reciprocal), then multiply. Always follow these steps, and you will find it makes algebra easier.