Problem 100 - Elementary and Intermediate Algebra

Question

Perform the operations and, if possible, simplify. $$ \frac{3}{4} \cdot \frac{5}{7} $$

Step-by-Step Solution

Answer

The product is \( \frac{15}{28} \).

Step 1: Multiply the Numerators

To multiply the fractions, start by multiplying the numerators. The numerator of the first fraction is 3 and the numerator of the second fraction is 5. Multiply these numbers: \[ 3 imes 5 = 15 \]

Step 2: Multiply the Denominators

Next, multiply the denominators. The denominator of the first fraction is 4 and the denominator of the second fraction is 7. Multiply these numbers: \[ 4 imes 7 = 28 \]

Step 3: Combine the Products

Form the new fraction by placing the product of the numerators over the product of the denominators. This gives:\[ \frac{15}{28} \]

Step 4: Simplify the Fraction

Look for any common factors between 15 and 28. The greatest common divisor of 15 and 28 is 1, so the fraction \( \frac{15}{28} \) is already in its simplest form.

Key Concepts

The Role of Numerators

A numerator is the top number in a fraction. It tells us how many parts of a whole we have. For example, in the fraction \( \frac{3}{4} \), 3 is the numerator, indicating 3 parts out of 4.
When multiplying fractions, as in the exercise \( \frac{3}{4} \times \frac{5}{7} \), we focus on the numerators first. Multiply the numerators together: 3 from the first fraction and 5 from the second. This multiplication gives us 15.

• This new number, 15, becomes the numerator of our answer.
• The multiplication of numerators is straightforward: Multiply top by top.

Understanding the numerator's role helps you see what portion of the whole is being represented after the fractions are multiplied.

The Importance of Denominators

The denominator is found at the bottom of the fraction. It shows the total number of equal parts in a whole. For example, in \( \frac{3}{4} \), 4 is the denominator, indicating 4 total parts.
In multiplying fractions like \( \frac{3}{4} \times \frac{5}{7} \), after dealing with the numerators, we turn to denominators.

• Multiply them in a similar straightforward way: 4 from the first fraction and 7 from the second.
• This multiplication gives us 28, which becomes the denominator of our resulting fraction.

Every time, ensure you multiply the bottom numbers together directly. This is crucial because it maintains the proportionality and size of the pieces represented by the numerators.

Understanding Simplifying Fractions

Simplifying fractions is about finding the simplest form of a fraction. This means reducing the fraction so that the numerator and denominator have no common factors except 1.
It involves dividing both the numerator and the denominator by their greatest common divisor (GCD).Here’s how to simplify:

• Find the greatest common factor of the two numbers. For \( \frac{15}{28} \), identify factors of both 15 and 28.
• The only common factor here is 1, meaning \( \frac{15}{28} \) is already as simple as it gets.
  This step ensures the fraction is easy to read and understand, helping in comparing sizes of fractions effectively.

By understanding how to simplify, we create cleaner mathematical expressions, making it easier to analyze and compare values in future problems.