Problem 27 - Prealgebra

Question

Add or subtract as indicated. $$2-\frac{5}{y}$$

Step-by-Step Solution

Answer

\( \frac{2y - 5}{y} \)

Step 1: Identify the Operation

The given expression is \( 2 - \frac{5}{y} \). Here, you need to subtract the fraction \( \frac{5}{y} \) from the whole number 2.

Step 2: Rewrite as a Single Fraction

To combine the terms into a single expression, recognize that 2 can be written as \( \frac{2y}{y} \) to have a common denominator with \( \frac{5}{y} \).

Step 3: Subtract the Fractions

Now subtract the fractions: \( \frac{2y}{y} - \frac{5}{y} = \frac{2y - 5}{y} \).

Step 4: Simplify if Possible

Check if the expression \( \frac{2y - 5}{y} \) can be simplified further. In this case, it cannot be simplified further as there are no common factors between the numerator and the denominator.

Key Concepts

Common Denominator

Understanding the concept of a common denominator is essential for subtracting fractions. In fractional operations, like addition or subtraction, both fractions must share a common denominator to be combined properly.

To achieve this, you must first identify if the denominators of the fractions involved are the same. If they aren't, you'll need to find a common denominator. This typically involves finding the lowest common multiple between the denominators. However, in our example, since we are subtracting from a whole number, we can use the same denominator as the given fraction.

By rewriting the whole number 2 as \( \frac{2y}{y} \), you set both terms to have the denominator \( y \). This common denominator \( y \) allows for direct fraction subtraction.

Fraction Subtraction

Fraction subtraction involves subtracting the numerators while keeping the common denominator unchanged. In our example, after finding a common denominator, we change the expression from \( 2 - \frac{5}{y} \) to \( \frac{2y}{y} - \frac{5}{y} \).

The subtraction process involves segregating the numerators: here, it's \( 2y - 5 \). This step is straightforward because the denominator remains consistent (\( y \) stays as is), enabling the subtraction of numerators directly.

Thus, the operation simplifies to \( \frac{2y - 5}{y} \). Always ensure to subtract correctly and line up numerators accurately, keeping signs in check.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form. While dealing with the result of a fraction subtraction, check for any possible common factors between the numerator and the denominator.

In our situation, the fraction is \( \frac{2y - 5}{y} \). Determine if any number or variable divides both the numerator and the denominator evenly. After inspection, you'll see there are no such common factors in this example. Consequently, the expression \( \frac{2y - 5}{y} \) is already in its simplest form.

Simplification is a key final step in fraction operations to ensure the result is as reduced as possible, providing a clean and understandable answer.