Problem 14 - Intermediate Algebra

Question

Multiply, and then simplify, if possible. See Objective 1. $$ \frac{49}{36} \cdot \frac{18}{35} $$

Step-by-Step Solution

Answer

The simplified product of the fractions is \( \frac{7}{10} \).

Step 1: Multiply Numerators

To find the product of two fractions, multiply the numerators together. Here, multiply the numerators 49 and 18:\[ 49 imes 18 = 882 \]

Step 2: Multiply Denominators

Next, multiply the denominators 36 and 35 together:\[ 36 imes 35 = 1260 \]

Step 3: Write the Product of Fractions

Combine the results from the previous steps into a single fraction:\[ \frac{882}{1260} \]

Step 4: Simplify the Fraction

Find the greatest common divisor (GCD) of 882 and 1260, which is 126, and use it to simplify the fraction by dividing both numerator and denominator by 126:\[ \frac{882 \div 126}{1260 \div 126} = \frac{7}{10} \]

Step 5: Final Step: Conclude with Simplified Fraction

The simplified result of multiplying and simplifying the given fractions is:\( \frac{7}{10} \)

Key Concepts

Simplifying Fractions

Simplifying fractions is an essential skill in mathematics, which helps to make fractions easier to understand and work with. When we simplify fractions, we are essentially expressing them in their lowest or simplest terms. This involves reducing the fraction by dividing both the numerator and the denominator by their greatest common factor (GCF). Simplification is especially useful because it reveals if the fraction can be expressed more simply.

To simplify a fraction:

• Identify the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and the denominator by this GCD.

For example, in the fraction \( \frac{882}{1260} \), after identifying the GCD as 126, we can simplify the fraction to \( \frac{7}{10} \). This means that both 882 and 1260 share 126 as a factor, and dividing by it gives us the fraction in its simplest form.

Greatest Common Divisor

The greatest common divisor (GCD) is a critical concept in fraction simplification. The GCD of two numbers is the largest positive integer that divides both numbers evenly, without leaving a remainder. Identifying the GCD allows us to reduce fractions to their simplest form, making them easier to compare or use in further calculations.

Here is how you find the GCD:

• List out the factors of each number.
• Identify the largest factor that appears in both lists.

The GCD can also be found through the Euclidean algorithm, which is more efficient for larger numbers. In our example, the GCD of 882 and 1260 is 126. Dividing both numbers by this GCD simplifies the fraction efficiently.

Numerator and Denominator

Understanding the terms numerator and denominator is fundamental when dealing with fractions. A fraction resembles a division and consists of two parts: a numerator and a denominator. The numerator is the top number, representing the part of the whole you have, while the denominator is the bottom number, showing how many equal parts the whole is divided into.

Here's a quick breakdown:

• Numerator: The number above the fraction bar. It tells us how many parts of the whole we are considering.
• Denominator: The number below the fraction bar. It tells us into how many parts the whole is divided.

For instance, in the fraction \( \frac{49}{36} \), 49 is the numerator, and 36 is the denominator. Knowing these terms helps us understand operations on fractions like addition, subtraction, multiplication, and division. When you multiply numerators and denominators as separate steps, it ensures accurate computation before any simplification.