Problem 43 - Fundamentals of Mathematics

Question

For the following problems, find each part without using a diagram. $$\frac{3}{4}\( of \)3 \frac{3}{5}$$

Step-by-Step Solution

Answer

\(\frac{3}{4} \times 3 \frac{3}{5} = \frac{27}{10} = 2 \frac{7}{10}\)

Step 1: Convert Mixed Number to Improper Fraction

First, we need to convert the mixed number \(3 \frac{3}{5}\) into an improper fraction. To do this, multiply the whole number by the denominator of the fraction and add the numerator.\[3 \times 5 + 3 = 15 + 3 = 18\]So, \(3 \frac{3}{5}\) becomes \(\frac{18}{5}\).

Step 2: Multiplication of Fractions

Now, multiply the fraction \(\frac{3}{4}\) by the improper fraction \(\frac{18}{5}\).\[\frac{3}{4} \times \frac{18}{5} = \frac{3 \times 18}{4 \times 5} = \frac{54}{20}\]

Step 3: Simplify the Fraction

Finally, simplify the fraction \(\frac{54}{20}\). Find the greatest common divisor of 54 and 20, which is 2, and divide both the numerator and the denominator by 2.\[\frac{54}{20} = \frac{54 \div 2}{20 \div 2} = \frac{27}{10}\]

Step 4: Convert Back to Mixed Number (Optional)

If you want the result as a mixed number, divide 27 by 10. The quotient is 2, and the remainder is 7.So, \(\frac{27}{10}\) can be expressed as the mixed number \(2 \frac{7}{10}\).

Key Concepts

Mixed Number Conversion

When you're faced with a mixed number, such as \(3 \frac{3}{5}\), one of the first steps in solving a problem is to convert it into an improper fraction. This is crucial because improper fractions make multiplication with other fractions straightforward. To perform this conversion:

• Multiply the whole number, in this case, 3, by the denominator of the fraction, which is 5.
• Add the numerator, 3, to the product above. This gives you \(15 + 3 = 18\).

Thus, \(3 \frac{3}{5}\) is equivalent to \(\frac{18}{5}\). Converting mixed numbers in this way ensures that subsequent fraction operations are easy to handle.

Improper Fractions

Improper fractions are fractions where the numerator is greater than or equal to the denominator. They often arise from mixed numbers, like our example which resulted in \(\frac{18}{5}\). Such fractions can seem intimidating, but they're quite manageable:

• They allow straightforward multiplication or division with other fractions.
• Eventually, they can be converted back to mixed numbers if a more intuitive representation is needed.

In our example, we multiplied \(\frac{3}{4}\) and \(\frac{18}{5}\) to set up our problem to be solved. Embracing improper fractions simplifies working with mixed number calculations.

Simplifying Fractions

After performing operations with fractions, the result is often a fraction that can be simplified. Simplification helps in representing fractions in their simplest form, making them easier to interpret. For instance, when you obtain \(\frac{54}{20}\) from multiplying fractions, simplifying is the next logical step:

• Find the greatest common divisor (GCD) of the numerator and the denominator. Here, the GCD of 54 and 20 is 2.
• Divide both by their GCD: \(\frac{54}{20} = \frac{54 \div 2}{20 \div 2} = \frac{27}{10}\).

By simplifying, the fraction becomes more straightforward to interpret and work with, especially if further use is needed.

Greatest Common Divisor

The greatest common divisor (GCD) is essential when simplifying fractions. It is the largest number that can evenly divide both the numerator and the denominator. In our example, calculating the GCD of 54 and 20 helps to simplify \(\frac{54}{20}\). Here's how you can find the GCD:

• List or find the factors of each number.
• Determine the largest factor that both numbers share.

For 54 and 20, this factor is 2. Dividing each term of the fraction by 2 gives a simpler form: \(\frac{27}{10}\). Finding the GCD makes fraction calculations more manageable and comprehensible.