Problem 43 - Elementary Algebra

Question

For exercises 39-82, simplify. $$ \frac{3 a}{5} \div \frac{9}{10 a^{2}} $$

Step-by-Step Solution

Answer

\( \frac{2a^3}{3} \)

Step 1: Rewrite the Division as Multiplication

To simplify the division of fractions, rewrite the division problem as a multiplication problem by multiplying by the reciprocal of the divisor. The given problem is: \( \frac{3a}{5} \div \frac{9}{10a^2} \).Rewrite it as: \( \frac{3a}{5} \times \frac{10a^2}{9} \).

Step 2: Multiply the Numerators

Multiply the numerators of the fractions together:\( 3a \times 10a^2 = 30a^3 \).

Step 3: Multiply the Denominators

Multiply the denominators of the fractions together:\( 5 \times 9 = 45 \).

Step 4: Combine the Results

Combine the results from steps 2 and 3 into one fraction:\( \frac{30a^3}{45} \).

Step 5: Simplify the Fraction

Simplify the fraction. Find the greatest common divisor (GCD) of 30 and 45, which is 15, and divide both the numerator and denominator by this GCD:\( \frac{30a^3 \div 15}{45 \div 15} = \frac{2a^3}{3} \).

Key Concepts

Multiplying Fractions

To simplify algebraic fractions, one core concept to understand is how to multiply fractions. When multiplying fractions, you simply:

• Multiply the numerators together.
• Multiply the denominators together.

This process involves straightforward steps:
Consider two fractions, \( \frac{a}{b} \) and \( \frac{c}{d} \). When you multiply them, the result is \( \frac{a \times c}{b \times d} \). It's important to remember to perform these multiplications separately.

In our exercise, we reframe the given problem to a multiplication scenario. This results in \( \frac{3a}{5} \times \frac{10a^2}{9} \). We then multiply the numerators: \( 3a \times 10a^2 = 30a^3 \), and the denominators: \( 5 \times 9 = 45 \). Finally, we combine them to form \( \frac{30a^3}{45} \).

Greatest Common Divisor

Simplifying a fraction often involves finding the Greatest Common Divisor (GCD). The GCD is the largest number that can exactly divide both the numerator and the denominator.

• Firstly, find the prime factors of both numbers.
• Identify the common factors.
• Select the highest common factor, which is the GCD.

In our example, we need to simplify \( \frac{30a^3}{45} \).
The prime factors of 30 are 2, 3, and 5, and for 45, they are 3, 3, and 5.
The common factors are 3 and 5, which multiply to give the GCD of 15.
We simplify the fraction by dividing both the numerator and the denominator by 15:
\[ \frac{30a^3 \div 15}{45 \div 15} = \frac{2a^3}{3} \]. Hence, the simplified fraction is \( \frac{2a^3}{3} \).

Reciprocal

The concept of the reciprocal is key in simplifying fractions involving division. The reciprocal of a fraction is obtained by swapping the numerator and the denominator.
For example, the reciprocal of \( \frac{a}{b} \) is \( \frac{b}{a} \).

To solve a division of fractions, multiply by the reciprocal of the divisor.
In our exercise, converting \( \frac{3a}{5} \div \frac{9}{10a^2} \) involves using the reciprocal of \( \frac{9}{10a^2} \), which is \( \frac{10a^2}{9} \).
Thus, we rewrite it as \( \frac{3a}{5} \times \frac{10a^2}{9} \). This conversion simplifies the division problem into a multiplication problem,
which we can solve using the multiplication rules discussed earlier.