Problem 58 - Algebra 1: Concepts and Skills

Question

Simplify. $$ \frac{36}{45 a} \div \frac{-9 a}{5} $$

Step-by-Step Solution

Answer

The simplified form of the expression is \(-4 / 9a^{2}\)

Step 1: Dividing fractions by multiplying with the reciprocal

In the given expression, it can be noticed that there's a division which is \(\frac{36}{45a} \div \(-9a/5\)\). This can be rewritten as a multiplication problem by taking the reciprocal of the second fraction. Reciprocal of a fraction is obtained by interchanging the numerator and the denominator. The expression is rewritten as: \(\frac{36}{45a} * \(-\frac{5}{9a}\)\).

Step 2: Multiply nominators and denominators

Multiply the numerators together and the denominators together. This can be expressed as:\((36 * -5) / (45a * 9a)\) which is \(-180 / 405a^{2}\)

Step 3: Simplify the fraction

The obtained fraction can be simplified by dividing the numerator and the denominator by their Greatest Common Divisor (GCD). The GCD of 180 and 405 is 45. So, divide -180 and 405 by 45. This results in: \(-4 / 9a^{2}\)

Key Concepts

Reciprocal of a Fraction

Understanding the reciprocal of a fraction is pivotal when dealing with division between fractions. The reciprocal simply flips the fraction, swapping the numerator and the denominator. If you have a fraction \( \frac{a}{b} \), its reciprocal is \( \frac{b}{a} \). This concept allows us to turn division problems into multiplication problems, which are often easier to solve.

• To handle division like \( \frac{36}{45a} \div \frac{-9a}{5} \), transform it into a multiplication problem.
• Find the reciprocal of the divisor, which in this case is \(-\frac{9a}{5}\). The reciprocal becomes \(-\frac{5}{9a}\).
• The division expression is now rewritten as: \( \frac{36}{45a} \times -\frac{5}{9a} \).

By understanding and applying the reciprocal effectively, you can streamline the process of solving fraction problems, improving both your speed and accuracy.

Greatest Common Divisor (GCD)

The concept of the Greatest Common Divisor (GCD) is crucial for simplifying fractions. The GCD is the largest number that divides two integers without leaving a remainder. This allows you to reduce fractions to their simplest form. When simplifying fractions, dividing both the numerator and the denominator by their GCD will yield the simplest form of the fraction.

• Consider the fraction \(-\frac{180}{405a^{2}}\). To simplify, find the GCD of 180 and 405.
• The GCD of these two numbers is 45. This means 45 is the largest number that evenly divides both 180 and 405.
• Divide both the numerator and the denominator by 45 to simplify the fraction: \(-\frac{180 \div 45}{405a^{2} \div 45} \). This results in \(-\frac{4}{9a^{2}}\).

Using the GCD for fraction simplification helps prevent calculation errors and keeps your answers as straightforward as possible.

Multiplying Fractions

Multiplying fractions is simple and intuitive once you understand the basic rule: multiply the numerators together and the denominators together. This process can be seen in many fraction problems and appears frequently in mathematics. Here’s how it works:

• Take the fractions from any multiplication expression, such as \( \frac{36}{45a} \times -\frac{5}{9a} \).
• Multiply the numerators: 36 multiplied by -5 results in -180.
• Multiply the denominators: \(45a \times 9a\) results in \(405a^{2}\).
• Your resulting fraction is \(-\frac{180}{405a^{2}}\).

This direct multiplication of numerators and denominators is an efficient method for fraction multiplication. It lays the foundation for simplification and other operations on fractions, such as finding common denominators for addition or subtraction.