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Chapter 2: Problem 62

Evaluate. Half of \(-18,\) added to the reciprocal of \(\frac{1}{5}\)

Short Answer

Expert verified
-4

Step by step solution

01

- Calculate Half of -18

To find half of -18, divide -18 by 2. So, \(\text{Half of } -18 = \frac{-18}{2} = -9\).
02

- Find the Reciprocal of \(\frac{1}{5}\)

The reciprocal of \(\frac{1}{5}\) is the number that, when multiplied by \(\frac{1}{5}\), gives 1. The reciprocal of \(\frac{1}{5}\) is 5.
03

- Add the Two Results

Now, add the two results obtained in the previous steps: \(-9\) and 5. \(-9 + 5 = -4\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Reciprocal
The term 'reciprocal' is essential in mathematics, especially when dealing with fractions. A reciprocal of a number is defined as the number that, when multiplied by the original number, yields 1. For example, the reciprocal of \(\frac{1}{5}\) is simply 5, because \(\frac{1}{5} \times 5 = 1\). Remember, the reciprocal of a fraction \(\frac{a}{b}\) is \(\frac{b}{a}\). This concept helps in various computations like division of fractions and simplifying expressions. In the exercise, we found that the reciprocal of \(\frac{1}{5}\) is 5.
Addition of Integers
Adding integers can sometimes be confusing, especially when dealing with negative numbers. Let's break it down. When adding a positive number to a negative number, you can think of it as subtracting the smaller number from the larger number and keeping the sign of the larger number. For instance, in the exercise, we added \(-9\) and 5. Here's the process step-by-step:
  • Identify the signs: \(-9\) is negative, 5 is positive.
  • Subtract the smaller number from the larger number ignoring the signs: \| -9 | - | 5 | = 4\.
  • Keep the sign of the larger number in magnitude: Since \| -9 | > | 5 |\, the result is -4.
Lo and behold, \(-9 + 5 = -4\). This method helps in understanding and securing accurate results.
Division
Division is another fundamental arithmetic operation. It essentially breaks a number into equal parts. When we divide \(-18\) by 2 to find half of \(-18\), we use basic division rules, which state:
  • If you divide a negative number by a positive number, the result is negative.
  • If dividing a negative number, think of it as dividing the absolute values, and then assign a negative sign to the answer.
So, \(\frac{-18}{2} = -9\), accomplished by simply halving 18, and then considering the negative sign. Consequently, you get the half of \(-18\) as \-9\.
This concept is valuable not just in evaluating expressions but also in real-life scenarios where equitable distribution or equal sharing is involved.

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