Problem 47 - Algebra 1: Concepts and Skills

Question

Find the sum. $$4+\frac{1}{10}+\left(-\frac{1}{10}\right)$$

Step-by-Step Solution

Answer

The sum is \(4\).

Step 1: Identify the Numbers

The numbers to be added are \(4\), \(\frac{1}{10}\), and \(-\frac{1}{10}\).

Step 2: Add the Fractions

Add the fractions \(\frac{1}{10} + -\frac{1}{10}\). The sum of these two fractions is \(0\) because the negative fraction cancels out the positive fraction.

Step 3: Add the Remainder

Now, add this sum to the whole number \( 4 + 0\).

Key Concepts

Simple Fractions Addition

When adding simple fractions, the first step is to ensure that the fractions have a common denominator, which is necessary to combine the values correctly. In our example, the fractions \(\frac{1}{10}\) and \(\left(-\frac{1}{10}\right)\) already share the same denominator, making them easy to add together.

Consider the numerators when combining these fractions. Here, adding the positive numerator to the negative one \(1 + (-1)\) results in zero. This effectively eliminates the fraction part of the problem because \(\frac{0}{10}\) simplifies to zero. Understanding the concept of numerator arithmetic is essential when dealing with simple fractions addition.

Combining Whole Numbers and Fractions

When a math problem involves a mix of whole numbers and fractions, as in \(4 + \frac{1}{10}\), the approach is typically to address the fractions separately and then combine them with the whole number. In our example, however, the sum of the fractions equated to zero, simplifying the task.

Generally, if the fractions summed to a value other than zero, you would then add that result to the whole number. For instance, if the fraction sum had been \(\frac{2}{10}\), which simplifies to \(\frac{1}{5}\), you would then calculate \(4 + \frac{1}{5}\) to find the final result. Remember, whole numbers can be treated as fractions too, by using 1 as the denominator, to make combining easier in more complex scenarios.

Subtracting Negative Fractions

Subtracting negative fractions can seem intimidating at first, but it's based on the rule that two negatives make a positive. Therefore, subtracting a negative fraction is the same as adding its positive counterpart. For instance, \( - \frac{1}{10}\) subtracted from any number is the same as adding \(\frac{1}{10}\).

In our solution, the negative fraction \(\left(-\frac{1}{10}\right)\) is being added, which at first might look like a contradiction. However, it’s an application of this subtracting-negatives rule implicitly. When you add \(\frac{1}{10}\) and \(\left(-\frac{1}{10}\right)\), they cancel each other out, resulting in zero. Grasping this concept is vital for understanding operations involving negative fractions.