Problem 97 - Elementary and Intermediate Algebra

Question

Perform the operations and, if possible, simplify. $$ 3-\frac{3}{4} $$

Step-by-Step Solution

Answer

The simplified result is \( \frac{9}{4} \) or 2 \( \frac{1}{4} \).

Step 1: Convert Whole Number to Fraction

To subtract fractions, we need a common structure. Convert the whole number 3 to a fraction with the same denominator as \( \frac{3}{4} \). We express 3 as \( \frac{12}{4} \) because \( \frac{12}{4} = 3 \).

Step 2: Subtract the Fractions

Now subtract \( \frac{3}{4} \) from \( \frac{12}{4} \). The operation is \( \frac{12}{4} - \frac{3}{4} \). Subtract the numerators: \( 12 - 3 = 9 \), which gives us \( \frac{9}{4} \).

Step 3: Simplify the Result

Check if the result can be simplified or rewritten. The fraction \( \frac{9}{4} \) cannot be simplified further, but it can be expressed as a mixed number: 2 \( \frac{1}{4} \). 9 divided by 4 gives 2 with a remainder of 1, thus \( \frac{9}{4} = 2 \frac{1}{4} \).

Key Concepts

Understanding Improper Fractions

An improper fraction is a type of fraction where the numerator (top number) is larger than the denominator (bottom number). This means the fraction is actually greater than one. For example, the fraction \( \frac{9}{4} \) is an improper fraction because 9 is larger than 4. Improper fractions are very useful in mathematics because they make certain operations, like multiplication and division, simpler than dealing with mixed numbers. Here's why you might encounter improper fractions within subtraction:

• When subtracting fractions, if the result is larger than the denominator, you will have an improper fraction.
• Improper fractions can easily convert to whole numbers or mixed numbers, which can help in understanding and visualizing results better.

Remember, converting between improper fractions and mixed numbers is a key skill when working with fractions.

Converting Improper Fractions to Mixed Numbers

Mixed numbers are a combination of a whole number and a proper fraction. They are especially helpful in expressing and interpreting results because they provide a clear separation between whole and fractional parts. If you have an improper fraction like \( \frac{9}{4} \), you can convert it into a mixed number to make it more intuitive. Here's how:

• Divide the numerator by the denominator. In this case, 9 divided by 4 equals 2.
• The whole number 2 is part of your mixed number.
• The remainder is 1, which becomes the numerator of the fraction part, over the original denominator (4). Thus, you have \( 2 \frac{1}{4} \).

Converting into mixed numbers can help in visualizing scenarios such as measurements or divisions when an outcome is needed in a more tangible form, such as two whole objects and a part of another. However, sometimes keeping results as improper fractions is more efficient, especially in further calculations.

Simplifying Fractions

Simplifying fractions is a process that involves making a fraction as simple as possible — essentially, finding an equivalent fraction that uses the smallest possible numbers. While sometimes the immediate operation doesn't allow for simplification (such as in our example \( \frac{9}{4} \)), knowing how to simplify is crucial.Here’s the simplification method:

• Find the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and the denominator by this GCD.
• If the GCD is different from 1, the fraction can be simplified further.

In cases where the result is an improper fraction, rather than simplifying further, converting to a mixed number may be the more intuitive option. Being able to simplify fractions makes calculations easier and results clearer, especially in advanced mathematics and practical applications.