Problem 16 - College Algebra with Modeling and Visualization

Question

Simplify. $$ \frac{7}{2} \cdot \frac{4}{21} $$

Step-by-Step Solution

Answer

The simplified result is \( \frac{2}{3} \).

Step 1: Multiply the Numerators

The first step in simplifying the expression \( \frac{7}{2} \cdot \frac{4}{21} \) is to multiply the numerators. Here, 7 and 4 are the numerators. So, multiply them together: \( 7 \times 4 = 28 \).

Step 2: Multiply the Denominators

Multiply the denominators of the fractions, which are 2 and 21. So, multiply them as follows: \( 2 \times 21 = 42 \).

Step 3: Form the New Fraction

Now, place the product of the numerators over the product of the denominators to form the new fraction: \( \frac{28}{42} \).

Step 4: Simplify the Fraction

To simplify \( \frac{28}{42} \), find the greatest common divisor (GCD) of 28 and 42. The GCD is 14. Divide both the numerator and the denominator by 14: \( \frac{28 \div 14}{42 \div 14} = \frac{2}{3} \).

Key Concepts

Fractions

Fractions are a way of expressing numbers that are not whole. They consist of two parts: a numerator and a denominator. The numerator is the top number, and it shows how many parts we have. The denominator, on the bottom, shows into how many equal parts the whole is divided. When working with fractions, a key operation is multiplication. In multiplication, the numerators are multiplied together and the denominators are multiplied together:

• Example: For \( \frac{7}{2} \times \frac{4}{21} \), multiply the numerators: 7 and 4 to get 28.
• Similarly, multiply the denominators: 2 and 21 to get 42.

This results in the new fraction \( \frac{28}{42} \). This fraction can often be simplified, which leads us to another important concept, finding the greatest common divisor.

Greatest Common Divisor

The greatest common divisor (GCD), also known as the greatest common factor, is the largest number that divides into two or more numbers without leaving a remainder. To simplify fractions, finding the GCD is crucial because it helps in reducing fractions to their simplest form.

Consider the fraction \( \frac{28}{42} \):

• First, identify the factors of each number.
• Factors of 28: 1, 2, 4, 7, 14, 28.
• Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42.

The greatest common factor between 28 and 42 is 14. When both the numerator and the denominator are divided by 14, the fraction is simplified to \( \frac{2}{3} \). Understanding how to use the GCD is a helpful algebraic technique for simplifying fractions efficiently.

Numerator and Denominator

In fractions, the numerator and denominator each serve distinct but equally important roles. The numerator, on top, signifies how many parts you have, while the denominator, beneath, tells how many parts make up a whole.
The way they interact affects the fraction's value:

• The larger the numerator compared to the denominator, the larger the fraction's value.
• A fraction like \( \frac{5}{5} \) equals 1, as the numerator and denominator are the same.

When fractions are multiplied, like in our example \( \frac{7}{2} \times \frac{4}{21} \), the numerators multiply together, and the denominators multiply together to form a new fraction. In simplification, if both the numerator and the denominator are divisible by the same number, this number is the GCD, and simplifying involves dividing both by this number, leading to a simpler, equivalent fraction.