Problem 11 - Basic College Mathematics

Question

Divide and simplify. \(\frac{3}{5} \div \frac{9}{4}\)

Step-by-Step Solution

Answer

\(\frac{4}{15}\)

Step 1: Rewrite Division as Multiplication

Rewrite the division problem as a multiplication problem by taking the reciprocal of the divisor. The reciprocal of \(\frac{9}{4}\) is \(\frac{4}{9}\). Thus, the equation becomes \(\frac{3}{5} \times \frac{4}{9}\).

Step 2: Multiply the Numerators

Multiply the numerators of the fractions together. \(3 \times 4 = 12\).

Step 3: Multiply the Denominators

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

Step 4: Simplify the Fraction

Simplify the fraction \(\frac{12}{45}\) by finding the greatest common divisor (GCD) of 12 and 45, which is 3. Divide both the numerator and the denominator by their GCD: \(\frac{12 \div 3}{45 \div 3} = \frac{4}{15}\).

Key Concepts

Reciprocal

To divide fractions, we use the concept of the reciprocal. The reciprocal, or multiplicative inverse, of a fraction is found by swapping the numerator and the denominator. For example, the reciprocal of \(\frac{9}{4}\) is \(\frac{4}{9}\). By using the reciprocal, we change the division problem into a multiplication problem. This makes it easier to solve.

Fraction Multiplication

After converting the division problem into a multiplication problem, the next step is to multiply the fractions. This involves multiplying the numerators (the top numbers) together and the denominators (the bottom numbers) together. For example, multiplying \(\frac{3}{5}\) by \(\frac{4}{9}\), we first multiply the numerators: \3 \times 4 = 12\. Then, we multiply the denominators: \5 \times 9 = 45\. So, \(\frac{3}{5} \times \frac{4}{9} = \frac{12}{45} \)

Simplifying Fractions

Once you've performed the multiplication, you may need to simplify the resulting fraction. Simplifying a fraction involves dividing the numerator and the denominator by their greatest common divisor (GCD). This reduces the fraction to its simplest form. For example, with \(\frac{12}{45}\), the GCD of 12 and 45 is 3. By dividing both the numerator and the denominator by 3, we simplify the fraction to \(\frac{4}{15}\).

Greatest Common Divisor (GCD)

The greatest common divisor (GCD) of two numbers is the largest number that evenly divides both of them. There are different methods to find the GCD, such as the Euclidean algorithm or prime factorization. For instance, to find the GCD of 12 and 45 using prime factorization:

• Prime factors of 12: 2, 2, and 3.
• Prime factors of 45: 3, 3, and 5.

The common prime factor is 3, thus the GCD is 3. Dividing 12 by 3 gives 4, and dividing 45 by 3 gives 15, simplifying \(\frac{12}{45}\) to \(\frac{4}{15}\).