Problem 20 - College Algebra with Modeling and Visualization

Question

Simplify. $$ \frac{5}{12} \div \frac{10}{9} $$

Step-by-Step Solution

Answer

The simplified form is \( \frac{3}{8} \).

Step 1: Reciprocal of the Second Fraction

When dividing fractions, we multiply by the reciprocal of the second fraction. The reciprocal of \( \frac{10}{9} \) is \( \frac{9}{10} \).

Step 2: Rewrite the Division as Multiplication

Rewrite the original division problem as multiplication by the reciprocal. This changes \( \frac{5}{12} \div \frac{10}{9} \) to \( \frac{5}{12} \times \frac{9}{10} \).

Step 3: Multiply the Numerators

To multiply fractions, multiply the numerators together: \( 5 \times 9 = 45 \). Thus, the new numerator is 45.

Step 4: Multiply the Denominators

Next, multiply the denominators: \( 12 \times 10 = 120 \). The new denominator is 120.

Step 5: Form the New Fraction

Combine the results of the previous steps to form the new fraction: \( \frac{45}{120} \).

Step 6: Simplify the Fraction

Find the greatest common divisor (GCD) of 45 and 120 to simplify the fraction. The GCD is 15. Divide both the numerator and the denominator by 15: \( \frac{45}{15} = 3 \) and \( \frac{120}{15} = 8 \).

Step 7: Write the Simplified Fraction

After simplifying, the fraction becomes \( \frac{3}{8} \). This is the final simplified form of the original expression.

Key Concepts

Reciprocal

When working with fraction division, one of the key steps involves the concept of a reciprocal. The reciprocal of a fraction is essentially what you get when you flip the fraction upside down. This means the numerator and the denominator swap places. For instance, the reciprocal of \(\frac{10}{9}\) is \(\frac{9}{10}\). Reciprocals are important because they allow us to turn a division problem into a multiplication problem, which is typically easier to manage. By anchoring the division of fractions to reciprocal operations, calculations become more straightforward and follow clearer rules.

Multiplying Fractions

Once you've got the reciprocal of the second fraction, the next step in fraction division involves multiplying the fractions. Multiplying fractions is quite simple. You multiply the numerators (the top numbers) together to get the new numerator and multiply the denominators (the bottom numbers) together to get the new denominator.

For example:

• If our fractions are \(\frac{5}{12}\) and its reciprocal pair \(\frac{9}{10}\), we multiply 5 (the numerator of the first) by 9 (the numerator of the second), giving us 45.
• Then, multiply 12 (the denominator of the first) by 10 (the denominator of the second), equaling 120.

So, the multiplication of these fractions results in a new fraction \(\frac{45}{120}\). Multiplication of fractions is a direct process where understanding turns perceived complexity into simplicity.

Simplifying Fractions

Once you've multiplied the fractions, the final critical step is simplifying the result. Simplifying is all about finding the simplest form of a fraction or the form where the numerator and denominator have no common factors other than 1.

The process involves:

• Finding the greatest common divisor (GCD) of both the numerator and the denominator. For our current example of \(\frac{45}{120}\), the GCD is 15.
• Next, you divide both the numerator and the denominator by this GCD. So, \(\frac{45}{15} = 3\) and \(\frac{120}{15} = 8\).

The simplified fraction in this example becomes \(\frac{3}{8}\). Simplicity is achieved when the fraction is in its lowest terms, allowing for comparisons or additional calculations to be more efficient and clear.