Problem 89 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. Write the answer as a fraction or mixed number in simplest form. \(\frac{1}{4}+\frac{2}{4}-\frac{3}{4}+\frac{4}{4}\)

Step-by-Step Solution

Answer

The answer is 1 as \( \frac{4}{4}\) simplifies to 1.

Step 1: Add and Subtract Fractions

Since all the fractions share the same denominator, the fractions can be added or subtracted by manipulating the numerators only: \( \frac{1}{4}+ \frac{2}{4}- \frac{3}{4}+ \frac{4}{4} \) is equal to \( \frac{1 + 2 - 3 + 4}{4}\).

Step 2: Calculate the Numerator

Perform the operations in the numerator, \(1 + 2 - 3 + 4 = 4\)

Step 3: Express the Fraction in Simplest Form

Now express the fraction as \( \frac{4}{4}\)

Key Concepts

Adding Fractions

When we talk about adding fractions, we are combining numbers that have a certain relationship to something else, often called the denominator. In this exercise, we are working with fractions like \( \frac{1}{4} \) and \( \frac{2}{4} \). Both fractions have the same denominator, making them easy to add.

Here's the key: when adding fractions with the same denominators, focus on the numerators.

• The denominator (bottom number) remains the same.
• Add the numerators (top numbers) together.

In our example, you add \( 1 \) and \( 2 \), which gives you \( 3 \). Thus, \( \frac{1}{4} + \frac{2}{4} \) equals \( \frac{3}{4} \). Remember, the denominator doesn't change because the fractions represent parts of the same whole.

Subtracting Fractions

Subtracting fractions is very similar to adding them, especially when the fractions share the same denominator. The main task is to handle the numerators while keeping the denominators unchanged.

In this exercise, after adding, we move to subtraction: \( \frac{3}{4} - \frac{3}{4} \).

• Keep the denominator the same.
• Subtract the numerators.

So, \( 3 - 3 = 0 \) results in \( \frac{0}{4} \). This process mirrors subtraction in regular arithmetic but retains the common denominator, aiding in simplifying the expression later.

Simplest Form

The simplest form of a fraction is what we aim for when we want the fraction to be as easy to understand as possible. This means the numerator and the denominator have no common factors other than 1.

For *our original expression,* simplifying was straightforward, especially after having an intermediate step that reshaped it into \( \frac{4}{4} \). Anytime a fraction's numerator and denominator are the same, it simplifies to 1, because they cancel each other out. This is why \( \frac{4}{4} = 1 \).

Going through simplification means checking for greatest common divisors between numerators and denominators and reducing them to their smallest equivalent forms, allowing a fraction to become whole numbers when applicable. This final form is often the clearest and most concise way to express the result of fraction arithmetic.