Problem 98 - Elementary and Intermediate Algebra

Question

Perform the operations. $$ -\frac{15}{16} \div \frac{25}{8} $$

Step-by-Step Solution

Answer

The result is \(-\frac{3}{10}\).

Step 1: Understand the Division of Fractions

To divide fractions, we need to multiply the first fraction by the reciprocal of the second fraction. The reciprocal of \(\frac{25}{8}\) is \(\frac{8}{25}\).

Step 2: Multiply the Fractions

Multiply \(-\frac{15}{16}\) by \(\frac{8}{25}\). This involves multiplying the numerators and the denominators: \(-\frac{15 \times 8}{16 \times 25}\).

Step 3: Simplify the Fraction

Computing the products gives us \(-\frac{120}{400}\). Next, simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor.

Step 4: Finding the Greatest Common Divisor

The greatest common divisor (GCD) of 120 and 400 is 40. Divide both the numerator and the denominator by 40: \(-\frac{120 \div 40}{400 \div 40} = -\frac{3}{10}\).

Key Concepts

Understanding Reciprocals in Division of Fractions

When working with division of fractions, the concept of the "reciprocal" is crucial. The reciprocal of a fraction is simply created by swapping its numerator and denominator. For example, the reciprocal of \( \frac{25}{8} \) is \( \frac{8}{25} \). This "flipped" fraction helps in converting the division problem into a multiplication problem.

This step is vital because dividing by a fraction is the same as multiplying by its reciprocal. So, when you see a problem like \(-\frac{15}{16} \div \frac{25}{8}\), you transform it into \(-\frac{15}{16} \times \frac{8}{25}\). By understanding the meaning and application of reciprocals, you simplify the operation and make it easier to compute.

Simplifying Fractions to the Simplest Form

Simplifying fractions involves reducing them to their simplest form. This means making both the numerator and the denominator as small as possible while still maintaining the same value for the fraction.

Take, for instance, the fraction \(-\frac{120}{400}\). While this looks complex, it can be made easier by finding a common number (other than 1) that divides evenly into both the numerator and the denominator. This number is known as the greatest common divisor (GCD).

After simplification, the smaller numbers give a clearer insight into the true value of the fraction. Remember, simplifying does not change the inherent value of the fraction; it just makes it easier to work with.

Finding the Greatest Common Divisor

The greatest common divisor (GCD) is key when simplifying fractions. It is the largest number that can evenly divide both the numerator and the denominator. Finding the GCD helps in reducing fractions to their simplest terms, making them less complicated.

For instance, in \(-\frac{120}{400}\), you notice that 40 is the greatest number that divides evenly into both 120 and 400. This means that 40 is the GCD. By dividing both the numerator and the denominator by 40, you get \(-\frac{3}{10}\).

To find the GCD, you can use the prime factorization method, list all factors, or apply the Euclidean algorithm. Understanding how to find the GCD will enable you to simplify any fractions and make calculations easier.