Problem 172 - Intermediate Algebra

Question

In the following exercises, simplify. $$ \frac{-\frac{3}{8}}{-\frac{y}{12}} $$

Step-by-Step Solution

Answer

Simplified expression is \ \( \frac{9}{2y} \).

Step 1: Understand the problem

The given expression is \ \( \frac{-\frac{3}{8}}{-\frac{y}{12}} \). You need to simplify this fraction.

Step 2: Invert the denominator

To simplify \ \( \frac{A}{B} \), where \ \( A \) and \ \( B \) are fractions, multiply \ \( A \) by the reciprocal of \ \( B \). Here, the reciprocal of \ \( -\frac{y}{12} \) is \ \( -\frac{12}{y} \). So rewrite the expression as \ \( \frac{-\frac{3}{8} \times -\frac{12}{y}}{\underline{\phantom{xx}}} \).

Step 3: Multiply the fractions

Multiply the numerators and denominators: \ \( -\frac{3}{8} \times -\frac{12}{y} = \frac{(-3) \times (-12)}{8 \times y} = \frac{36}{8y} \).

Step 4: Simplify the fraction

Finally, simplify \ \( \frac{36}{8y} \) by finding the greatest common divisor (GCD) of 36 and 8, which is 4. Divide the numerator and the denominator by 4 to get \ \( \frac{9}{2y} \).

Key Concepts

reciprocal

A reciprocal of a number is essentially its 'flipped' version. For example, if you have the fraction \(\frac{a}{b}\), its reciprocal is \(\frac{b}{a}\). When it comes to simplifying fractions where the denominator is also a fraction, finding the reciprocal is crucial. In our example, the denominator is \(-\frac{y}{12}\). The reciprocal of \(-\frac{y}{12}\) is \(-\frac{12}{y}\).

multiplying fractions

Multiplying fractions is straightforward. You multiply the numerators with each other and the denominators with each other. For example, to multiply \(\frac{a}{b}\) and \(\frac{c}{d}\), you would do \(\frac{a \times c}{b \times d}\). In our step-by-step solution, after substituting the reciprocal of the denominator, the problem becomes \(\frac{-\frac{3}{8} \times -\frac{12}{y}}{\underline{\phantom{xx}}}\). Multiplying these gives us \(\frac{(-3) \times (-12)}{8 \times y} = \frac{36}{8y}\).

greatest common divisor (GCD)

The greatest common divisor (GCD) of two numbers is the largest number that can divide both of them without leaving a remainder. Finding the GCD is an important step in simplifying fractions. For instance, to simplify \(\frac{36}{8y}\), first find the GCD of 36 and 8, which is 4. Divide both the numerator and the denominator by this GCD for simplification. So, \(\frac{36 \/ 4}{8y \/ 4}\) simplifies to \(\frac{9}{2y}\).

fraction simplification

Fraction simplification involves expressing the fraction in its simplest form. This often entails finding the GCD and dividing the numerator and the denominator by this number. For our solution, we simplified \(\frac{36}{8y}\) to \(\frac{9}{2y}\) by dividing both the numerator and the denominator by their GCD, 4. Simplification makes fractions easier to understand and work with.