Understanding the Problem: Adding Fractions to Find the Nearest Natural Number

The question asks us to find a number that, when added to \(\frac{-1}{2}\), results in the nearest natural number. First, let's understand what natural numbers are and which one is nearest to \(\frac{-1}{2}\).

Natural Numbers: Natural numbers are positive integers starting from 1. They are 1, 2, 3, 4, and so on. \( \{1, 2, 3, ...\} \)

Nearest Natural Number to \(\frac{-1}{2}\): The number \(\frac{-1}{2}\) is equal to -0.5 in decimal form. Let's look at the natural numbers on a number line relative to -0.5.

• The natural number 1 is at a distance of \(|1 - (-0.5)| = |1 + 0.5| = 1.5\) from -0.5.
• The natural number 2 is at a distance of \(|2 - (-0.5)| = |2 + 0.5| = 2.5\) from -0.5.

Comparing the distances, 1.5 is less than 2.5. Also, natural numbers like 0 (if considered), -1, etc., are further away or not natural numbers. Therefore, the nearest natural number to \(\frac{-1}{2}\) is 1.

Setting up the Equation for Adding Fractions

Now we know that adding the unknown number to \(\frac{-1}{2}\) should give us 1. Let the unknown number be \(x\). We can write this as an equation:

\[ \frac{-1}{2} + x = 1 \]

We need to solve this equation for \(x\) to find the value that needs to be added. This involves adding fractions.

Solving the Equation to Find the Required Fraction

To find \(x\), we need to isolate it on one side of the equation. We can do this by adding \(\frac{1}{2}\) to both sides of the equation:

\[ \frac{-1}{2} + x + \frac{1}{2} = 1 + \frac{1}{2} \] \[ x = 1 + \frac{1}{2} \]

Now, we need to perform the addition of 1 and \(\frac{1}{2}\). To add 1 to a fraction, we can express 1 as a fraction with the same denominator as the other fraction. The denominator is 2, so \(1 = \frac{2}{2}\).

\[ x = \frac{2}{2} + \frac{1}{2} \]

Now that both fractions have the same denominator, we can add the numerators:

\[ x = \frac{2 + 1}{2} \] \[ x = \frac{3}{2} \]

So, the number that should be added to \(\frac{-1}{2}\) to obtain the nearest natural number (which is 1) is \(\frac{3}{2}\).

Verifying the Result of Adding Fractions

Let's check if adding \(\frac{3}{2}\) to \(\frac{-1}{2}\) gives us 1:

\[ \frac{-1}{2} + \frac{3}{2} \]

Since the denominators are already the same, we can add the numerators:

\[ \frac{-1 + 3}{2} = \frac{2}{2} = 1 \]

The result is 1, which is indeed the nearest natural number to \(\frac{-1}{2}\). This confirms our calculation for adding fractions is correct.

Conclusion

To obtain the nearest natural number from \(\frac{-1}{2}\), which is 1, we need to add the value of \(x\) we found by solving the equation \(\frac{-1}{2} + x = 1\). Our calculation showed that \(x = \frac{3}{2}\). Therefore, adding fractions, specifically \(\frac{3}{2}\) to \(\frac{-1}{2}\), gives the desired result.