Problem 17 - College Algebra with Modeling and Visualization

Question

Simplify. $$ \frac{5}{6} \cdot \frac{3}{10} \cdot \frac{8}{3} $$

Step-by-Step Solution

Answer

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

Step 1: Identify and Write Down the Fractions

The problem requires you to multiply three fractions: \( \frac{5}{6} \), \( \frac{3}{10} \), and \( \frac{8}{3} \). Write these fractions clearly so you know what numbers you're working with.

Step 2: Multiply the Numerators

Multiply the numerators of each fraction together. This means multiplying 5, 3, and 8. \[ 5 \times 3 \times 8 = 120 \].

Step 3: Multiply the Denominators

Next, multiply the denominators of each fraction: 6, 10, and 3. \[ 6 \times 10 \times 3 = 180 \].

Step 4: Form the New Fraction

Create a new fraction using the results from your multiplications. You should now have \( \frac{120}{180} \).

Step 5: Simplify the Fraction

To simplify \( \frac{120}{180} \), find the greatest common divisor (GCD) of 120 and 180, which is 60. Divide both the numerator and the denominator by 60. \[ \frac{120 \div 60}{180 \div 60} = \frac{2}{3} \].

Key Concepts

Numerators and Denominators

Understanding the basics of fractions is essential for fraction multiplication. A fraction consists of two parts:

• Numerator: This is the top number of the fraction and represents how many parts you have.
• Denominator: This is the bottom number of the fraction and tells you into how many parts the whole is divided.

When multiplying fractions, you multiply the numerators together and then multiply the denominators together. It's like handling two separate multiplications at once, one at the top and one at the bottom. For instance, in our example: 5, 3, and 8 are numerators that multiply to form 120. Meanwhile, 6, 10, and 3 are denominators that multiply to form 180.
Once this multiplication is done, we create a new fraction \( \frac{120}{180} \) ready for simplification.

Simplifying Fractions

Simplifying fractions is a skill that makes things much clearer and more manageable. It involves reducing the fraction to its simplest form so that it is easier to read and work with in calculations. The process means dividing both the numerator and the denominator by their greatest common divisor (GCD).
For instance, the new fraction \( \frac{120}{180} \) has larger numbers, and it's usually best to simplify these wherever possible. By finding the GCD of 120 and 180—which is 60—we can reduce the fraction in size to make it much simpler to deal with. It's like making the fraction tidier by cutting out extra fluff.By dividing 120 and 180 each by 60, \( \frac{120}{180} \) simplifies down to \( \frac{2}{3} \). This simplified form is cleaner and straightforward, making it easier to comprehend and use in further calculations.

Greatest Common Divisor

The Greatest Common Divisor (GCD) is a key tool in simplifying fractions. It is the largest number that divides both the numerator and the denominator without leaving a remainder. Finding the GCD of two numbers helps you to reduce a fraction to its simplest form.To find the GCD of 120 and 180 in our fraction \( \frac{120}{180} \), follow these steps:

• List the factors of each number. For 120, they are: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120.
• List the factors of 180: 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180.
• Identify the largest common factor in both lists, which is 60.

By simplifying the fraction using its GCD, we ensure that it remains equivalent to the original fraction but is much easier to work with. When everything is clear and simple, math can actually be quite fun!