Problem 28 - Prealgebra

Question

Add or subtract as indicated. $$3-\frac{4}{y}$$

Step-by-Step Solution

Answer

\( \frac{3y - 4}{y} \).

Step 1: Identify the Operation

We need to perform the subtraction of the fraction \( \frac{4}{y} \) from the number 3. Our goal is to express this difference in a single fractional expression.

Step 2: Rewrite the Whole Number as a Fraction

For addition or subtraction with fractions, it's helpful to express whole numbers as fractions. Convert 3 to a fraction with a denominator of 1: \( \frac{3}{1} \).

Step 3: Find a Common Denominator

We need a common denominator to combine the fractions. The denominators in the problem are 1 and \( y \). The least common denominator (LCD) is \( y \).

Step 4: Express Each Fraction with the Common Denominator

Rewrite \( \frac{3}{1} \) as \( \frac{3y}{y} \) so that it has the same denominator as \( \frac{4}{y} \). Now the expression is: \( \frac{3y}{y} - \frac{4}{y} \).

Step 5: Perform the Subtraction

Subtract the numerators: \( 3y - 4 \), and keep the common denominator \( y \). The expression becomes \( \frac{3y - 4}{y} \).

Step 6: Simplify if Possible

Check if the fraction can be simplified. Since 3y - 4 and y are already in simplest form with no common factors, no further simplification is possible.

Key Concepts

Common Denominator

A common denominator is essential when you are dealing with the subtraction or addition of fractions. To subtract fractions like the ones given in the exercise, having a similar base or bottom part of the fraction—the denominator—is necessary. When fractions have the same denominator, it is easy to directly subtract the numerators (the top parts) without messing with the whole numbers. If you have a fraction with denominator 1, like the number 3 written as \(\frac{3}{1}\), it's crucial to convert it to have a denominator that matches with the fraction you are working with.For example, in the exercise, to subtract \(\frac{4}{y}\) from 3, you first convert the whole number 3 to a fraction \(\frac{3}{1}\). Then, you identify a denominator that both fractions could share. In this case, the common denominator is \(y\) because it contains both 1 and \(y\). Transform the whole number to \(\frac{3y}{y}\), and you are ready for the subtraction task! Remember, the main goal is to bring both fractions to a common base to ease the subtraction process.

Least Common Denominator (LCD)

Finding the least common denominator (LCD) is like constructing a bridge between separate worlds of fractions. It’s the smallest number that each of your denominators can divide evenly into.In our exercise, the denominators in question are 1 and \(y\). Since \(y\) is a variable, it conveniently becomes the least common denominator.Here's how to find it:

• List the denominators: 1, \(y\).
• Identify the smallest expression that includes both 1 and \(y\). That's \(y\).

By using \(y\) as the LCD, you can adjust fractions into like terms, allowing you to subtract them without any fuss. The goal is to get each fraction working in harmony with a single common denominator. After achieving this, you're one step closer to completing your subtraction or addition without hitches.

Simplifying Fractions

After completing subtraction or addition, you might ask, "Can I make this fraction simpler?" That's where simplifying fractions comes into play.Simplifying a fraction means reducing it to its simplest form, where the numerator and the denominator have no common factors other than 1.In the solution for our exercise, we reach a result of \(\frac{3y - 4}{y}\). It's important to check if you can factor or rule out any commonality between the numerator and denominator:

• Is there any common factor between \(3y - 4\) and \(y\)?
• Can any parts of the numerator cancel out with the denominator?

In this case, the expression \(3y - 4\) and \(y\) have no common factors, so \(\frac{3y - 4}{y}\) is already simplified. Remember, the aim of simplifying is to make the expression as concise and clean as possible, without changing its value.