Problem 42 - Fundamentals of Mathematics

Question

For the following problems, find each part without using a diagram. $$\frac{5}{6}\( of \)2 \frac{2}{5}$$

Step-by-Step Solution

Answer

The result is 2.

Step 1: Convert Mixed Number to Improper Fraction

First, we need to convert the mixed number in the expression into an improper fraction. The given mixed number is \( 2 \frac{2}{5} \). To convert, multiply the whole number by the denominator and add the numerator: \( (2 \times 5) + 2 = 10 + 2 = 12 \). Thus, \( 2 \frac{2}{5} \) becomes \( \frac{12}{5} \).

Step 2: Multiply the Fractions

Now, we multiply the fraction \( \frac{5}{6} \) by the improper fraction \( \frac{12}{5} \): \( \frac{5}{6} \times \frac{12}{5} \). Multiply the numerators together and the denominators together: \( 5 \times 12 = 60 \) and \( 6 \times 5 = 30 \). This gives us the fraction \( \frac{60}{30} \).

Step 3: Simplify the Fraction

Simplify the fraction \( \frac{60}{30} \) by finding the greatest common divisor (GCD) of the numerator and the denominator, which is 30. Divide both the numerator and the denominator by their GCD: \( \frac{60}{30} = \frac{60 \div 30}{30 \div 30} = \frac{2}{1} = 2 \).

Key Concepts

Mixed Numbers

A mixed number is a combination of a whole number and a fraction. For example, in the mixed number \( 2 \frac{2}{5} \), '2' is the whole number and \( \frac{2}{5} \) is the fraction.
To work with mixed numbers in calculations, it's often easier to convert them to improper fractions. This is because fractions are simpler to handle mathematically, especially when multiplying or dividing.

To convert a mixed number to an improper fraction, follow these steps:

• Multiply the whole number by the denominator of the fraction part.
• Add the numerator of the fraction part to the result.
• Place this sum over the original denominator.

Using our example \( 2 \frac{2}{5} \), calculate it as follows:

• Multiply: \( 2 \times 5 = 10 \)
• Add the numerator: \( 10 + 2 = 12 \)
• Write this over the original denominator: \( \frac{12}{5} \)

And there you have it! A simple conversion to make your math tasks easier.

Improper Fractions

An improper fraction is one where the numerator is greater than or equal to the denominator. For example, \( \frac{12}{5} \) is an improper fraction because 12 is greater than 5.
Improper fractions are useful because they allow for straightforward mathematical operations like addition, subtraction, multiplication, and division.

Remember, when multiplying two fractions:

• Multiply the numerators (top numbers) together.
• Multiply the denominators (bottom numbers) together.

Take the example where we multiplied \( \frac{5}{6} \times \frac{12}{5} \):

• Numerator: \( 5 \times 12 = 60 \)
• Denominator: \( 6 \times 5 = 30 \)

This results in \( \frac{60}{30} \). Using improper fractions, the multiplication process becomes a lot clearer and manageable.

Greatest Common Divisor

The Greatest Common Divisor (GCD) is the largest number that divides two or more numbers without leaving a remainder. It is used for simplifying fractions.
Finding the GCD helps in reducing fractions to their simplest form. In simplified form, the fraction has the smallest numbers possible for the numerator and the denominator, making it easier to understand and use in further calculations.

Here's how to simplify a fraction using the GCD:

• Identify the GCD of the numerator and denominator.
• Divide both the numerator and the denominator by their GCD.

For instance, in \( \frac{60}{30} \), the GCD of 60 and 30 is 30. Thus, dividing the numerator and the denominator by 30 gives us \( \frac{60 \div 30}{30 \div 30} = \frac{2}{1} \), which simplifies to 2.
This method ensures your fraction is as simple as possible, making it much easier to work with in any mathematical context.