Problem 60 - Algebra 1: Concepts and Skills

Question

What does \(5-\left(-\frac{1}{3}\right)+\frac{2}{3}\) equal? A. 4 B. \(4 \frac{2}{3}\) C. \(5 \frac{1}{3}\) D. 6

Step-by-Step Solution

Answer

The correct answer is D. 6

Step 1: Convert Minus of Negative to Plus

The appearance of '-' sign before \(-\frac{1}{3}\) signifies subtraction of \(-\frac{1}{3}\) from 5. And subtracting a negative number is equivalent to adding its absolute value. Hence, \(5-\left(-\frac{1}{3}\right)\) is equal to \(5 + \frac{1}{3}\).

Step 2: Addition of Fractions

Now, we have an addition problem with fractions: \(5 + \frac{1}{3} + \frac{2}{3}\). We can combine the fractions, \(\frac{1}{3}\) and \(\frac{2}{3}\), as they have common denominators. The result is \(\frac{3}{3}\), which equals to 1.

Step 3: Final Addition

We can now add the fractions sum to 5: \(5 + 1 = 6\).

Key Concepts

Subtracting Negative Numbers

When dealing with subtraction in mathematics, it may at first seem counterintuitive to subtract a negative number. However, subtracting a negative is essentially the same as addition. This is because two negatives make a positive. For instance, imagine you owe someone \(5, and they decide to forgive \)1 of that debt. In a sense, it's like they've given you a dollar, so you only owe $4 now.

Coming back to our example, when we have the expression \(5 - (-\frac{1}{3})\), to subtract the negative fraction \(-\frac{1}{3}\), we instead add its absolute value, \(\frac{1}{3}\), to 5. This turns out to be a simple case of adding a positive number: \[5 + \frac{1}{3}\]. Remember, subtraction of a negative number leads to addition—keep this rule in mind when dealing with negative values to avoid confusion.

Common Denominators

Before we can simplify a fraction addition problem, we need to ensure that all the fractions involved have the same denominator—this is what we mean by finding common denominators. The denominator represents the total number of equal parts the whole is divided into, so for fractions to be combined directly, their denominators need to match. In the example \(5 + \frac{1}{3} + \frac{2}{3}\), our fractions \(\frac{1}{3}\) and \(\frac{2}{3}\) already share a common denominator of 3.

When fractions do not have a common denominator, you would need to find the least common multiple (LCM) of the denominators to create equivalent fractions that can be combined. But in this case, since we already have common denominators, we can simply add the numerators together. This foundational concept is crucial in simplifying fractions and solving addition problems involving them.

Fraction Addition

Once we've established common denominators, fraction addition becomes relatively straightforward. All that's required is to add the numerators—the top numbers of the fractions—while keeping the denominator unchanged. For instance, adding \(\frac{1}{3}\) to \(\frac{2}{3}\), you'd simply add the numerators 1 and 2, while the denominator remains as 3, yielding \(\frac{3}{3}\), which simplifies to 1 because any number divided by itself equals 1.

It's essential to simplify fractions whenever possible to ensure the final answer is in its most reduced form. In our primary example, after converting the subtraction of a negative number to addition and ensuring we have common denominators, the next and final step is simplifying the addition of fractions, which is how we find that \(\frac{3}{3} = 1\), and then add this whole number to 5, resulting in 6. This process of simplifying fraction addition problems can be applied universally and is an important skill to master for any student.