Problem 25 - Prealgebra

Question

Find each product. Use an area model if necessary. $$\frac{8}{9} \cdot \frac{27}{28}$$

Step-by-Step Solution

Answer

\( \frac{8}{9} \cdot \frac{27}{28} = \frac{6}{7} \)

Step 1: Multiply the Numerators

To begin solving the fraction multiplication problem \( \frac{8}{9} \cdot \frac{27}{28} \), first multiply the numerators of the two fractions. Here, the numerators are 8 and 27. Thus, we calculate: \( 8 \cdot 27 = 216 \).

Step 2: Multiply the Denominators

Next, multiply the denominators of the fractions, which are 9 and 28 in this case. Compute: \( 9 \cdot 28 = 252 \).

Step 3: Write the Initial Product

The result of multiplying the numerators and denominators gives us the initial product of the fractions: \( \frac{216}{252} \).

Step 4: Simplify the Fraction

The final step is to simplify \( \frac{216}{252} \). First, find the greatest common divisor (GCD) of 216 and 252. The GCD is 36, so divide both the numerator and the denominator by 36: \( \frac{216 \div 36}{252 \div 36} = \frac{6}{7} \).

Step 5: Conclusion

The simplified product of \( \frac{8}{9} \cdot \frac{27}{28} \) is \( \frac{6}{7} \).

Key Concepts

Understanding Numerator and Denominator

In a fraction, the top number is called the numerator and the bottom number is called the denominator. They represent a part of a whole. The numerator tells us how many parts we have, while the denominator tells us how many equal parts the whole is divided into.
For example, in the fraction \( \frac{8}{9} \), 8 is the numerator and 9 is the denominator. This means you have 8 parts out of 9 equal parts. When multiplying fractions, you multiply the numerators together and the denominators together to get a new fraction.

• The product of the numerators from the original fractions becomes the new numerator
• The product of the denominators becomes the new denominator

This is why for \( \frac{8}{9} \cdot \frac{27}{28} \), we first multiply 8 and 27 to get the new numerator (216), and then 9 and 28 for the new denominator (252). Each step maintains the relationship between the fractions' parts in respect to the whole.

Simplifying Fractions

Once you have multiplied the fractions, you'll often end up with a fraction that is not in its simplest form, like \( \frac{216}{252} \). Simplifying a fraction means making it as small as possible when both the numerator and the denominator are divided by the largest number that they both can share without leaving a remainder.
Simplifying helps in better understanding the proportion the fraction represents. It also makes calculations more manageable and easier to compare with other fractions. Remember:

• Simplification makes a fraction more intuitive for understanding relative sizes.
• This is done by dividing both numerator and denominator by their greatest common divisor (GCD).

The process begins with identifying the GCD of the numbers involved.

Finding the Greatest Common Divisor

The greatest common divisor (GCD) of two numbers is the largest number that divides both numbers without leaving any remainder. Finding the GCD is a key step in simplifying fractions since it reveals the largest factor shared by the numerator and the denominator.
For example, to simplify \( \frac{216}{252} \), the GCD of 216 and 252 is found. We determine this by identifying common factors of both numbers. This can be done through factorization or using the Euclidean algorithm.

• Factorization: List all the factors of each number to find the largest common one.
• Euclidean algorithm: Continuously divide and look for remainders until you reach zero.

For 216 and 252, the GCD is 36. Dividing both 216 and 252 by 36 results in the simplified fraction \( \frac{6}{7} \). This reduced fraction maintains the same value and proportion as the original product fraction.