Problem 45 - Fundamentals of Mathematics

Question

Convert \(3.2 \frac{3}{5}\) to a fraction.

Step-by-Step Solution

Answer

The fraction is \(\frac{19}{5}\).

Step 1: Convert Mixed Number to Improper Fraction

First, let's convert the mixed number \(3.2\frac{3}{5}\) into an improper fraction. The whole number part is 3.2, and the fractional part is \(\frac{3}{5}\). We can write this as \(3.2 + \frac{3}{5}\). To add these together, we first convert \(3.2\) into a fraction. We express \(3.2\) as \(\frac{32}{10}\) because 3.2 is equivalent to \(\frac{32}{10}\).

Step 2: Find a Common Denominator

Now, to add \(\frac{32}{10}\) and \(\frac{3}{5}\), we need a common denominator. The least common multiple of 10 and 5 is 10. \(\frac{32}{10}\) already has a denominator of 10, but \(\frac{3}{5}\) needs to be converted. Multiply both the numerator and the denominator of \(\frac{3}{5}\) by 2 to get \(\frac{6}{10}\).

Step 3: Add the Fractions

With both fractions having the same denominator, add the numerators: \(\frac{32}{10} + \frac{6}{10} = \frac{32 + 6}{10} = \frac{38}{10}\).

Step 4: Simplify the Fraction

Finally, simplify \(\frac{38}{10}\) by dividing the numerator and the denominator by their greatest common divisor, which is 2. This gives \(\frac{38 \div 2}{10 \div 2} = \frac{19}{5}\).

Key Concepts

Mixed Numbers

A mixed number is a combination of a whole number and a fraction. For example, in the mixed number \(3.2\frac{3}{5}\), \(3.2\) is the whole number, and \(\frac{3}{5}\) is the fraction. Mixed numbers are useful in representing quantities that are more than a whole but not quite reaching the next whole number.
To convert a mixed number to an improper fraction, you have to deal with both parts separately. Get the whole number part and convert it to a fraction. Then, look at the fractional part as well.

• Convert the whole number to a fraction with a denominator of 1.
• Then, find a common denominator to combine it with the fractional part.
• Finally, sum up the fractions to get an improper fraction.

This process turns complex numbers into a simple fraction format that is easier to work with in calculations.

Improper Fractions

Improper fractions are fractions where the numerator (the top number) is larger than the denominator (the bottom number). This is different from a proper fraction, where the numerator is smaller than the denominator.
Improper fractions can seem a bit intimidating at first, but they are quite easy to handle. In the example from the exercise, once the mixed number \(3.2\frac{3}{5}\) was converted, the resulting improper fraction was \(\frac{38}{10}\).

• This fraction allows us to represent the same value but is more versatile for calculations.
• With improper fractions, you can easily multiply or divide fractions, which isn't as straightforward with mixed numbers.
• It's also easier to simplify or find a common denominator when working with improper fractions.

By using improper fractions, you can keep your calculations accurate and consistent, especially when adding or subtracting fractions with different denominators.

Simplifying Fractions

Simplifying fractions is the process of making a fraction as simple or as small as possible. This means reducing the fraction to its lowest terms where both the top (numerator) and bottom (denominator) numbers have no common factors other than 1.
In the solution, the fraction \(\frac{38}{10}\) was simplified to \(\frac{19}{5}\).
Here's how you do it:

• Find the greatest common divisor (GCD) of both the numerator and the denominator.
• For \(\frac{38}{10}\), both numbers can be divided by 2, which is their GCD.
• Divide both the numerator and the denominator by this number to simplify the fraction.

Simplifying fractions helps in better understanding and comparing the sizes of fractions. It also makes working with fractions easier, especially in arithmetic operations such as addition, subtraction, multiplication, and division.

Common Denominator

Finding a common denominator is crucial when you need to add, subtract, or compare fractions. It means finding a common multiple for the denominators involved so they share the same denominator.
In the example given, \(\frac{32}{10}\) and \(\frac{3}{5}\) were combined. The least common multiple of 10 and 5 was chosen as the common denominator, which is 10.

• Ensure both fractions have the same denominator.
• Adjust the numerator accordingly by multiplying the numerator and the denominator by the same number.
• This step is critical for accurate addition and subtraction of fractions.

Using a common denominator simplifies the process of working with different fractions and ensures that the result is accurate and understandable.