Problem 83 - Prealgebra

Question

The problems below review material involving fractions and mixed numbers. Perform the indicated operations. Write your answers as whole numbers, proper fractions, or mixed numbers. $$\frac{5}{7} \cdot \frac{14}{25}$$

Step-by-Step Solution

Answer

The product of \( \frac{5}{7} \) and \( \frac{14}{25} \) is \( \frac{2}{5} \).

Step 1: Understand the Problem

We need to multiply two fractions: \( \frac{5}{7} \) and \( \frac{14}{25} \). To do this, we'll multiply the numerators together and the denominators together.

Step 2: Multiply the Numerators

The numerators of the two fractions are \(5\) and \(14\). So we multiply them together: \(5 \times 14 = 70\).

Step 3: Multiply the Denominators

The denominators of the two fractions are \(7\) and \(25\). So we multiply them together: \(7 \times 25 = 175\).

Step 4: Form the Product Fraction

After multiplying the numerators and denominators, the resulting fraction is \( \frac{70}{175} \).

Step 5: Simplify the Fraction

To simplify \( \frac{70}{175} \), find the greatest common divisor (GCD) of \(70\) and \(175\), which is \(35\). Divide both the numerator and the denominator by their GCD: \( \frac{70 \div 35}{175 \div 35} = \frac{2}{5}\).

Key Concepts

Numerators

In fractions, the numerator is the number located above the fraction line. It represents how many parts of a whole you have. For example, in the fraction \( \frac{5}{7} \), the numerator is 5. Similarly, in \( \frac{14}{25} \), the numerator is 14.
When you multiply fractions, you start by multiplying the numerators together. This step gives you the numerator of the resulting fraction. For instance, multiplying the numerators 5 and 14 gives us \( 5 \times 14 = 70 \).
Remember, the numerator can be any whole number, and the larger the numerator (while keeping the denominator constant), the greater the fraction. Understanding numerators is essential to working with fractions efficiently.

Denominators

The denominator is the number beneath the fraction line. It tells you into how many parts the whole is divided. In the fraction \( \frac{5}{7} \), the denominator is 7, indicating the whole is split into 7 parts. Similarly, the fraction \( \frac{14}{25} \) has a denominator of 25.
When multiplying fractions, after dealing with the numerators, you multiply the denominators. For example, \( 7 \times 25 = 175 \). This product becomes the denominator of the new fraction: \( \frac{70}{175} \).
A key point to keep in mind is that while the numerator tells us "how many," the denominator tells us "of what." A lower denominator value, for the same numerator, represents larger parts.

Greatest Common Divisor (GCD)

The greatest common divisor (GCD) is a valuable tool for simplifying fractions. It represents the largest number that divides both the numerator and the denominator without leaving a remainder. Simplifying a fraction means reducing it to its simplest form.
To simplify the fraction \( \frac{70}{175} \), you find the GCD of 70 and 175. Here, the GCD is 35. You then divide both the numerator and the denominator by this number:

• For the numerator: \( 70 \div 35 = 2 \)
• For the denominator: \( 175 \div 35 = 5 \)

This division simplifies \( \frac{70}{175} \) to \( \frac{2}{5} \), which is easier to understand and work with in further calculations.
Using the GCD not only helps simplify calculations, but it also provides clearer insights into the size and relationship of the parts of a fraction to its whole. Understanding how to find and use the GCD is crucial for simplifying complex fractions.