Problem 53 - Elementary and Intermediate Algebra

Question

Perform the indicated operation and, if possible, simplify. If there are no variables, check using a calculator. $$ \frac{1}{2} \cdot \frac{3}{7} $$

Step-by-Step Solution

Answer

\( \frac{3}{14} \)

Step 1: Identify the operation

The operation is the multiplication of two fractions: \( \frac{1}{2} \times \frac{3}{7} \).

Step 2: Multiply the numerators

Multiply the numerators (top numbers) of both fractions. Here, 1 and 3 are the numerators.\[ 1 \times 3 = 3 \]

Step 3: Multiply the denominators

Multiply the denominators (bottom numbers) of both fractions. Here, 2 and 7 are the denominators.\[ 2 \times 7 = 14 \]

Step 4: Write the product as a fraction

The result of multiplying the numerators and the denominators gives you the new fraction:\[ \frac{3}{14} \]

Step 5: Simplify if possible

Check if the fraction can be simplified. In this case, 3 and 14 have no common factors other than 1, so \( \frac{3}{14} \) is already in its simplest form.

Key Concepts

Numerator

The numerator is the top number in a fraction. It shows how many parts we have.
For example, in the fraction \( \frac{1}{2} \), the numerator is 1.
Similarly, in \( \frac{3}{7} \), the numerator is 3.
When multiplying fractions, multiply the numerators together.
In our exercise, we multiplied 1 and 3 to get 3.
This becomes the numerator of the new fraction.

Denominator

The denominator is the bottom number in a fraction.
It shows how many equal parts the whole is divided into.
For example, in the fraction \( \frac{1}{2} \), the denominator is 2.
Similarly, in \( \frac{3}{7} \), the denominator is 7.
When multiplying fractions, multiply the denominators together.
In our exercise, we multiplied 2 and 7 to get 14.
This becomes the denominator of the new fraction.

Simplifying Fractions

Simplifying fractions means making the fraction as simple as possible.
To simplify, divide the numerator and the denominator by their greatest common factor (GCF).
For example, \( \frac{4}{8} \) can be simplified to \( \frac{1}{2} \) because the GCF of 4 and 8 is 4.
In our exercise, \( \frac{3}{14} \) is already in its simplest form.
This is because 3 and 14 have no common factors other than 1.

Math Operations

Math operations are actions we perform on numbers, like addition, subtraction, multiplication, and division.
In fractions, we often work with multiplication and division.
To multiply fractions:

• Multiply the numerators together.
• Multiply the denominators together.
• Simplify the resulting fraction, if possible.

In our exercise, we multiplied \( \frac{1}{2} \) and \( \frac{3}{7} \).
We multiplied the numerators: 1 × 3 = 3.
We multiplied the denominators: 2 × 7 = 14.
The result is \( \frac{3}{14} \).