Problem 15 - Fundamentals of Mathematics

Question

Find each value. \((\) Section 4.5\() 3 \frac{3}{4} \cdot 2 \frac{2}{9} \cdot 6 \frac{3}{5}\)

Step-by-Step Solution

Answer

The result of the multiplication is 55.

Step 1: Convert Mixed Numbers to Improper Fractions

Convert each mixed number into an improper fraction.\[3 \frac{3}{4} = \frac{3 \times 4 + 3}{4} = \frac{15}{4}\]\[2 \frac{2}{9} = \frac{2 \times 9 + 2}{9} = \frac{20}{9}\]\[6 \frac{3}{5} = \frac{6 \times 5 + 3}{5} = \frac{33}{5}\]

Step 2: Multiply Improper Fractions

Multiply the improper fractions obtained in the previous step.\[\frac{15}{4} \cdot \frac{20}{9} \cdot \frac{33}{5}\]

Step 3: Simplify Numerator and Denominator

First, multiply the numerators together and the denominators together:- Numerator: \(15 \times 20 \times 33 = 9900\)- Denominator: \(4 \times 9 \times 5 = 180\)

Step 4: Division of Fraction

Divide the result from Step 3:\[\frac{9900}{180}\]

Step 5: Further Simplification

Simplify the fraction by dividing the numerator and the denominator by their greatest common divisor (GCD), which is 180.\[\frac{9900 \div 180}{180 \div 180} = \frac{55}{1} = 55\]

Key Concepts

Improper Fractions

Fractions can sometimes look a bit intimidating, especially when they appear as mixed numbers. A mixed number consists of a whole number and a proper fraction. To perform calculations, it's often easier to convert mixed numbers into improper fractions. An improper fraction is simply a fraction where the numerator is greater than or equal to the denominator.
Imagine you have the mixed number 3 3/4. The way to convert this into an improper fraction is to multiply the whole number by the denominator, add the numerator, and place this result over the original denominator:

• Multiply the whole number: 3 x 4 = 12
• Add the numerator: 12 + 3 = 15
• Write the fraction: \(\frac{15}{4}\)

You'll do the same for other mixed numbers. By converting mixed numbers to improper fractions, you set up a simpler path for further calculations like multiplication or division of fractions.

Multiplication of Fractions

Multiplying fractions is a straightforward process once you know the steps. Unlike adding or subtracting fractions, you don't need a common denominator. Instead, you multiply directly across numerators and denominators to get the product. Here's a simple guide to help you understand the process:

• Take the first fraction: \(\frac{15}{4}\), the second: \(\frac{20}{9}\), and the third: \(\frac{33}{5}\).
• Multiply the numerators: 15 x 20 x 33 = 9900.
• Multiply the denominators: 4 x 9 x 5 = 180.

Once you have your products, you can form a new fraction \(\frac{9900}{180}\). But don't stop here – sometimes a simplification is possible, making it easier and clearer to understand your answer.

Simplifying Fractions

After multiplying fractions, you often get a large fraction that can be simplified. Simplifying makes fractions easier to understand and work with. To simplify a fraction, divide both the numerator and the denominator by their greatest common divisor (GCD). Here’s how you would simplify the fraction \(\frac{9900}{180}\):

• Find the GCD of 9900 and 180, which is 180.
• Divide the numerator by the GCD: 9900 ÷ 180 = 55.
• Divide the denominator by the GCD: 180 ÷ 180 = 1.

Now, the fraction becomes \(\frac{55}{1}\), which simplifies to just 55. In essence, simplifying a fraction boils down to finding factors common to both the numerator and the denominator, and this will give you the simplest form of the fraction possible.