Problem 162 - Intermediate Algebra

Question

In the following exercises, perform the indicated operation. $$ \frac{7}{18} \div\left(-\frac{14}{27}\right) $$

Step-by-Step Solution

Answer

The result of the operation is \( \frac{-3}{4} \).

Step 1: Identify the operation

The given problem requires division of two fractions: \( \frac{7}{18} \) divided by \( -\frac{14}{27} \).

Step 2: Reciprocal of the divisor

To divide by a fraction, multiply by its reciprocal. The reciprocal of \( -\frac{14}{27} \) is \( -\frac{27}{14} \).

Step 3: Rewrite the problem as a multiplication

Rewrite the division problem as a multiplication problem: \( \frac{7}{18} \times -\frac{27}{14} \).

Step 4: Multiply the numerators

Multiply the numerators: \( 7 \times -27 = -189 \).

Step 5: Multiply the denominators

Multiply the denominators: \( 18 \times 14 = 252 \).

Step 6: Simplify the fraction

Simplify \( \frac{-189}{252} \) by finding the greatest common divisor of 189 and 252, which is 63. Then, \( \frac{-189 \/ 63}{252 \/ 63} = \frac{-3}{4} \).

Key Concepts

Reciprocal

The concept of a reciprocal is crucial when dividing fractions. A reciprocal of a fraction is simply that fraction flipped upside down. For example, the reciprocal of \(-\frac{14}{27}\) is \(-\frac{27}{14}\). When dividing by a fraction, you multiply by its reciprocal. This step transforms the division problem into a simpler multiplication problem. Always remember: turning a division into a multiplication makes the problem easier to handle.

Multiplication of Fractions

Once we have the reciprocal, we change our division problem into a multiplication problem. Here, \(\frac{7}{18} \div -\frac{14}{27}\) becomes \(\frac{7}{18} \times -\frac{27}{14}\). When multiplying fractions, you simply multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together. So, for our example: numerators: \(7 \times -27 = -189\) and denominators: \(18 \times 14 = 252\). This gives us the fraction \(\frac{-189}{252}\).

Simplifying Fractions

After performing the multiplication, you often get a fraction that needs simplifying. Simplifying makes the fraction easier to understand. To simplify \(\frac{-189}{252}\), we find the Greatest Common Divisor (GCD) of the numerator and the denominator. Using the GCD helps us reduce the fraction to its simplest form. Divide both the numerator and the denominator by their GCD to get the simplified fraction.

Greatest Common Divisor

The Greatest Common Divisor (GCD) is the largest number that can divide both the numerator and the denominator without leaving a remainder. For \(\frac{-189}{252}\), the GCD is 63. We divide both the numerator and the denominator by 63 to simplify the fraction. This gives us: \(\frac{-189 \/ 63}{252 \/ 63} = \frac{-3}{4}\). Finding the GCD can be done through methods like prime factorization or the Euclidean algorithm. Simplification helps make fractions much easier to work with and understand.