Problem 40 - Prealgebra

Question

Find each product. Write in simplest form. $$\frac{12}{j k} \cdot \frac{3 k}{4}$$

Step-by-Step Solution

Answer

The product in simplest form is \( \frac{9}{j} \).

Step 1: Multiply the Numerators

To find the product of two fractions, start by multiplying the numerators together. Here, the numerators are 12 and 3k. Multiply them: \[ 12 imes 3k = 36k \]

Step 2: Multiply the Denominators

Next, multiply the denominators together. In this case, the denominators are jk and 4. Multiply them:\[ j k imes 4 = 4jk \]

Step 3: Write the Fraction

Use the results from Steps 1 and 2 to form the new fraction. The numerator is 36k and the denominator is 4jk:\[ \frac{36k}{4jk} \]

Step 4: Simplify the Fraction

Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor. Since 36 and 4 have a common factor of 4, divide both by 4:\[ \frac{36k}{4jk} = \frac{(36k \div 4)}{(4jk \div 4)} = \frac{9k}{jk} \]

Step 5: Simplify Further

Continue simplifying by canceling out common factors in the numerator and denominator. Here, the k in 9k and the k in jk can be canceled:\[ \frac{9k}{jk} = \frac{9}{j} \]

Step 6: Write the Simplest Form

Now that we have canceled out all possible terms, the fraction in its simplest form is:\[ \frac{9}{j} \]

Key Concepts

Simplifying Fractions

When working with fractions, simplifying them into their simplest form is a crucial step. Simplifying a fraction means finding an equivalent fraction where the numerator and denominator have the smallest possible whole numbers. This makes the fraction easier to understand and use.

To simplify a fraction:

• Identify any common factors in the numerator and the denominator.
  
• Divide both the numerator and the denominator by their greatest common divisor (GCD). This reduces the fraction to its simplest form.
  

For example, in the fraction \(\frac{36k}{4jk}\), simplifying involved removing common factors. We divided both 36k and 4jk by 4, the GCD of the numbers in the numerator and denominator. This left us with \(\frac{9k}{jk}\). We noted that 'k' was common in both terms, so we canceled them out for the final simplest form, \(\frac{9}{j}\).

Numerator and Denominator

The terms numerator and denominator are fundamental to understanding fractions. They are the two parts that make up a fraction. The numerator is the top number above the line in a fraction, representing how many parts of a whole are being considered. The denominator is the bottom number, showing into how many parts the whole is divided.

Here's how they were used in our exercise:

• The numerator in our problem began as 12 (from the first fraction) and 3k (from the second fraction). By multiplying them together, we obtained 36k as the new numerator.
  
• The denominator started with jk (from the first fraction) and 4 (from the second). After multiplying, the new denominator was 4jk.
  

Once multiplied, these two parts formed \(\frac{36k}{4jk}\). Understanding these roles helps in the multiplication and eventual simplification of fractions.

Greatest Common Divisor

The greatest common divisor (GCD) is the largest number that divides evenly into two or more numbers. Finding the GCD is essential in simplifying fractions because it helps identify which number can fully reduce both the numerator and the denominator.

Steps to find the GCD:

• List the factors of both numbers involved (in the numerator and denominator).
  
• Identify the highest number common to both lists of factors.
  

In the problem \(\frac{36k}{4jk}\), we needed to simplify by dividing by the GCD. The factors of 36 and 4 helped us identify the GCD as 4. Dividing both 36k and 4jk by this GCD allowed us to reduce the fraction to \(\frac{9k}{jk}\), which was then further simplified by removing the common 'k'. Recognizing and using the GCD is a key step in ensuring your fractions are as simplified as possible.