Problem 335 - Intermediate Algebra

Question

In the following exercises, simplify. $$ \frac{3}{20} \cdot \frac{49}{11} \cdot \frac{20}{3} $$

Step-by-Step Solution

Answer

\( \frac{49}{11} \)

Step 1 - Rewrite the Expression

First, rewrite the expression to make it easy to manage: \ \[ \frac{3}{20} \times \frac{49}{11} \times \frac{20}{3} \]

Step 2 - Cancel Common Factors

Identify and cancel common factors from the numerator and denominator. For instance, you can cancel a 3 from the numerators and denominators, and also a 20. This simplifies the expression: \ \[ \frac{ot{3}}{ot{20}} \times \frac{49}{11} \times \frac{ot{20}}{ot{3}} = 1 \times \frac{49}{11} \times 1 = \frac{49}{11} \]

Step 3 - Final Simplified Form

The expression is now simplified to its final form: \ \[ \frac{49}{11} \]

Key Concepts

Multiplying Fractions

To multiply fractions, we simply multiply the numerators (the top numbers) and the denominators (the bottom numbers) together. For instance, if we want to multiply \( \frac{a}{b} \times \frac{c}{d} \), we use the rule: \[ \frac{a \times c}{b \times d} \] However, it's always smart to look for common factors that might simplify our work before performing the multiplication. This makes the calculation easier and prevents dealing with large numbers unnecessarily.

Common Factors

Common factors are numbers that evenly divide both the numerator and the denominator in a fraction. For example, in our multiplication problem, \( \frac{3}{20} \times \frac{49}{11} \times \frac{20}{3} \), both 3 and 20 are common factors that appear once in the numerators and once in the denominators. By canceling out these common factors, we simplify our calculation:

• First, we cancel the 3's from \( \frac{3}{20} \) and \( \frac{20}{3} \), leaving us with \( 1 \) each.
• Next, we do the same for the 20's, also leaving us with \( 1 \) each.

This intermediate step results in \( 1 \times \frac{49}{11} \times 1 = \frac{49}{11} \).

Reducing Fractions

Reducing fractions involves simplifying them to their smallest possible numerator and denominator. This is done by dividing both the numerator and the denominator by their greatest common divisor (GCD).
In our example, after canceling the common factors, we are left with \( \frac{49}{11} \). Here, 49 and 11 do not have any common factors other than 1, which means the fraction is already in its simplest form.
So, whenever you have a fraction resulting from multiplication, always check for common factors and reduce the fraction by dividing both the numerator and the denominator by their GCD.