Problem 336 - Intermediate Algebra

Question

In the following exercises, simplify. $$ \frac{13}{18} \cdot \frac{25}{7} \cdot \frac{18}{13} $$

Step-by-Step Solution

Answer

The simplified result is \( \frac{25}{7} \).

Step 1: Write the Expression

Start by writing the given expression: \[ \frac{13}{18} \times \frac{25}{7} \times \frac{18}{13} \]

Step 2: Simplify Numerators and Denominators

Observe that the numerator 13 and the denominator 13 can cancel each other out. Similarly, the numerator 18 and the denominator 18 can also cancel each other out: \[ \frac{13}{18} \times \frac{25}{7} \times \frac{18}{13} = \frac{1}{1} \times \frac{25}{7} \times \frac{1}{1} \]

Step 3: Multiply Remaining Fractions

After the cancellation, multiply the resulting simplified fractions: \[ 1 \times \frac{25}{7} \times 1 = \frac{25}{7} \]

Step 4: Write the Final Answer

The final simplified answer is: \[ \frac{25}{7} \]

Key Concepts

Numerator and Denominator

In a fraction, the **numerator** is the top part and the **denominator** is the bottom part. Think of the numerator as the number of parts you have. The denominator tells you how many parts make up a whole. For example, with the fraction \(\frac{13}{18}\), you have 13 parts out of 18 total parts.

Understanding these components is crucial because they allow you to perform various operations such as addition, subtraction, multiplication, and division of fractions. Always remember: To simplify a fraction, you need to make the numerator and denominator as small as possible while keeping the value of the fraction the same.

Cancellation

Cancellation is a technique where you can reduce a fraction by removing common factors from the numerator and the denominator. This is very useful in fraction multiplication.

When you multiply fractions, look for numbers that appear in both the numerator and the denominator across the fractions. You can 'cancel' them out because their ratio is 1. In this exercise: \(\frac{13}{18} \times \frac{25}{7} \times \frac{18}{13}\),
you can see that 13 in the numerator of the first fraction and the denominator of the third can cancel each other out to 1. The same goes for the 18 in the denominator of the first and the numerator of the third. After canceling, you're left with \(\frac{1}{1}\times\ \frac{25}{7}\times\ \frac{1}{1}\).

Multiplication of Fractions

Multiplying fractions is straightforward. You simply multiply the numerators together and the denominators together. After any cancellation, you can multiply across. In our example, once you've simplified: \(\frac{13}{18} \times \frac{25}{7} \times \frac{18}{13} = \frac{1}{1} \times \frac{25}{7} \times \frac{1}{1}\),
it's a simple task of multiplying the numerators (1 \times 25 \times 1 = 25) and the denominators (1 \times 7 \times 1 = 7). This gives us \(\frac{25}{7}\) as the final answer.

The steps are easy to follow, but always remember to simplify first by canceling out common factors. It makes multiplication much cleaner and simpler.