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Chapter 1: Problem 729

Simplify. $$ \frac{3}{20} \cdot \frac{49}{11} \cdot \frac{20}{3} $$

Short Answer

Expert verified
The simplified form is \[ \frac{49}{11} \].

Step by step solution

01

Write the expression

Start with the given expression: \ \[ \frac{3}{20} \times \frac{49}{11} \times \frac{20}{3} \]
02

Rewrite with common factors

Rewrite the expression by putting the numerators and denominators together: \ \[ \frac{3 \times 49 \times 20}{20 \times 3 \times 11} \]
03

Cancel common factors

Cancel the common factors from the numerator and the denominator: \ Cancel \[ 3 \] and \[ 20 \]: \[ \frac{\cancel{3} \times 49 \times \cancel{20}}{\cancel{20} \times \cancel{3} \times 11} = \frac{49}{11} \]
04

Simplify the fraction

Since \[ \frac{49}{11} \] cannot be simplified further, this is the final answer.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fraction Multiplication
Fraction multiplication might seem tricky at first, but it's quite simple once you master the steps. Just remember, when multiplying fractions, you multiply the numerators (top numbers) together and the denominators (bottom numbers) together. For example, when you multiply \(\frac{3}{20} \times \frac{49}{11} \times \frac{20}{3}\), you combine all the numerators into a single fraction numerator and all the denominators into a single fraction denominator. The calculation becomes: \(\frac{3 \times 49 \times 20}{20 \times 3 \times 11}\). Breaking up the problem like this can make the rest of the process much easier.
Common Factors
Common factors are numbers that are shared by both the numerator and the denominator in a fraction. Identifying these common factors is a key step in simplifying fractions. In our example, \(\frac{3 \times 49 \times 20}{20 \times 3 \times 11}\), the numbers 3 and 20 appear in both the numerator and the denominator. These shared numbers can help us cancel terms, making the fraction easier to simplify.
Canceling Terms
Canceling terms is a process where you remove identical factors that appear in both the numerator and the denominator. This step is crucial for simplifying fractions effectively. In our case, the equation \(\frac{3 \times 49 \times 20}{20 \times 3 \times 11}\) simplifies by canceling out the common factors. We see that we can cancel the 3 and 20 in both the numerator and denominator, leading to \(\frac{\text{cancel}(3) \times 49 \times \text{cancel}(20)}{\text{cancel}(20) \times \text{cancel}(3) \times 11} = \frac{49}{11}\). This drastically reduces the complexity of the fraction.
Simplification Process
The simplification process involves reducing the fraction to its most basic form by canceling common factors and ensuring the fraction is in its simplest terms. After canceling the common factors in our example, we're left with \(\frac{49}{11}\). Since 49 and 11 have no common factors (other than 1), this fraction cannot be simplified further. Thus, the final simplified form of \(\frac{3}{20} \times \frac{49}{11} \times \frac{20}{3}\) is \(\frac{49}{11}\).

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