Problem 20 - Fundamentals of Mathematics

Question

Perform the following multiplications. Convert improper fractions to mixed numbers. $$2 \frac{2}{5} \cdot 3 \frac{3}{4} \cdot 3 \frac{1}{3}$$

Step-by-Step Solution

Answer

The result is 30.

Step 1: Convert Mixed Numbers to Improper Fractions

First, we need to convert each mixed number into an improper fraction. For \(2 \frac{2}{5}\), multiply the whole number \(2\) by \(5\) and add \(2\): \(2 \cdot 5 + 2 = 12\). So, \(2 \frac{2}{5} = \frac{12}{5}\). Next, for \(3 \frac{3}{4}\), multiply the whole number \(3\) by \(4\) and add \(3\): \(3 \cdot 4 + 3 = 15\). So, \(3 \frac{3}{4} = \frac{15}{4}\). Finally, for \(3 \frac{1}{3}\), multiply the whole number \(3\) by \(3\) and add \(1\): \(3 \cdot 3 + 1 = 10\). So, \(3 \frac{1}{3} = \frac{10}{3}\).

Step 2: Perform the Multiplication of Improper Fractions

Now, multiply all three improper fractions: \(\frac{12}{5} \cdot \frac{15}{4} \cdot \frac{10}{3}\). First, multiply the numerators: \(12 \cdot 15 \cdot 10 = 1800\). Next, multiply the denominators: \(5 \cdot 4 \cdot 3 = 60\). So, \(\frac{12}{5} \cdot \frac{15}{4} \cdot \frac{10}{3} = \frac{1800}{60}\).

Step 3: Simplify the Fraction

Simplify \(\frac{1800}{60}\). Divide both the numerator and the denominator by 60: \(1800 \div 60 = 30\) and \(60 \div 60 = 1\). So, \(\frac{1800}{60} = 30\).

Step 4: Convert the Result to a Mixed Number (if necessary)

Since the result \(30\) is a whole number, there is no need for further conversion to a mixed number.

Key Concepts

Mixed Numbers

Mixed numbers are numbers that have both a whole number and a fraction part, like \(2 \frac{2}{5}\). These numbers can be helpful in many situations as they clearly separate the whole from the fractional part.
To understand mixed numbers thoroughly, it's crucial to practice converting them into improper fractions, especially for mathematical operations like multiplication.
This involves multiplying the whole number by the denominator of the fraction and then adding the numerator. The output is an improper fraction where the numerator is greater than the denominator.

Improper Fractions

Improper fractions have numerators that are greater than or equal to the denominator. They are practical for multiplication and division because they simplify these operations.
When converting mixed numbers to improper fractions, you follow a few simple steps:

• Multiply the whole number by the fraction's denominator.
• Add the original numerator to the result.
• Place this new total as the numerator over the original denominator.

An example is converting \(3 \frac{3}{4}\) into \(\frac{15}{4}\). This makes calculations easier and more straightforward.
After completing the conversion, calculations like multiplication of fractions become simple.

Multiplication of Fractions

When multiplying fractions, the process is easy: multiply the numerators together and the denominators together.
This means that if you have fractions like \(\frac{12}{5} \cdot \frac{15}{4} \cdot \frac{10}{3}\), you simply do:

• \(12 \times 15 \times 10\) for the numerators.
• \(5 \times 4 \times 3\) for the denominators.

Giving you \(\frac{1800}{60}\).
This simple yet effective method allows you to handle complex problems with ease by breaking down the task of multiplying into clear, manageable segments.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form, where the numerator and denominator are as small as possible while still having the same value.
The process involves finding a common factor of both the numerator and the denominator, and dividing by it. For example, with \(\frac{1800}{60}\), both numbers divide evenly by 60, simplifying the fraction to \(\frac{30}{1}\), or just 30.
This step is essential after performing operations with fractions, like addition, subtraction, or multiplication, to ensure the final answer is clear and as simplified as possible.