Problem 19 - Prealgebra

Question

Complete the following tables. $$\begin{array}{|c|c|c|} \hline\begin{array}{c}\text { First } \\\\\text { Number } \\\a\end{array} & \begin{array}{c}\text { Second } \\\\\text { Number } \\\b\end{array} & \begin{array}{c}\text { The Quotient } \\ \text { of } a \text { and } b\\\ \frac{a}{b} \end{array} \\ \hline -100 & -5 \\\\\hline -100 & 5 \\\\\hline 100 & -5 \\ \hline 100 & 5 \\\\\hline\end{array}$$

Step-by-Step Solution

Answer

The completed table of quotients is: 20, -20, -20, 20.

Step 1: Understand the Problem

We are given a table that lists pairs of numbers "a" and "b". The task is to complete the table by finding the quotient \( \frac{a}{b} \) for each pair of numbers \( a \) and \( b \).

Step 2: Compute the Quotient for the First Row

For the first row, the numbers given are \( a = -100 \) and \( b = -5 \). The quotient is \( \frac{-100}{-5} \). We calculate this as follows:\[ \frac{-100}{-5} = 20 \] because dividing two negative numbers yields a positive result.

Step 3: Compute the Quotient for the Second Row

For the second row, the numbers given are \( a = -100 \) and \( b = 5 \). The quotient is \( \frac{-100}{5} \). We calculate this as follows:\[ \frac{-100}{5} = -20 \] because dividing a negative number by a positive number yields a negative result.

Step 4: Compute the Quotient for the Third Row

For the third row, the numbers given are \( a = 100 \) and \( b = -5 \). The quotient is \( \frac{100}{-5} \). We calculate this as follows:\[ \frac{100}{-5} = -20 \] because dividing a positive number by a negative number yields a negative result.

Step 5: Compute the Quotient for the Fourth Row

For the fourth row, the numbers given are \( a = 100 \) and \( b = 5 \). The quotient is \( \frac{100}{5} \). We calculate this as follows:\[ \frac{100}{5} = 20 \] because dividing a positive number by a positive number yields a positive result.

Step 6: Fill in the Completed Table

Based on our calculations, the completed table is as follows:\[ \begin{array}{|c|c|c|} \hline \text{First Number } a & \text{Second Number } b & \text{The Quotient of } a \text{ and } b = \frac{a}{b} \ \hline -100 & -5 & 20 \ \hline -100 & 5 & -20 \ \hline 100 & -5 & -20 \ \hline 100 & 5 & 20 \ \hline \end{array} \]

Key Concepts

Understanding Quotients

In mathematics, a quotient is the result obtained when one number is divided by another. It represents how many times the divisor fits into the dividend. For example, if you divide 10 by 2, the quotient is 5 because 2 fits into 10 exactly five times. This concept is fundamental in prealgebra and forms the basis for understanding division.

When calculating quotients, remember these key points:

• If the dividend and divisor are both positive, the quotient will be positive.
• If both numbers are negative, the quotient will also be positive. This is because two negatives make a positive.
• However, if the dividend and divisor have opposite signs (one is positive, the other is negative), the quotient will be negative.

Working with Negative Numbers in Division

Working with negative numbers can be a bit tricky at first, but with practice, it becomes intuitive. A negative number is any number that is less than zero, and these numbers are crucial in various fields including finance and physics.

When it comes to division, the rules for signs are pretty straightforward:

• Dividing two numbers that have the same sign (both negative or both positive) will always give a positive quotient. For example, \(-100 \/ -5 = 20\) and \(100 \/ 5 = 20\).
• If the numbers have different signs, the quotient will be negative. For instance, \(-100 \/ 5 = -20\) and \(100 \/ -5 = -20\).

These rules help maintain consistency and predictability in arithmetic operations. It’s important to understand these principles to avoid common mistakes during calculations.

To get comfortable with negative numbers, practice dividing different combinations of positive and negative numbers to see how the sign of the quotient changes.

Calculating Division Step-by-Step

To effectively calculate division, especially when dealing with negative numbers, following a step-by-step approach is beneficial. Let's break it down:
1. **Identify the Numbers**: Note the dividend (the number being divided) and the divisor (the number you are dividing by). For example, \(a = -100\) and \(b = 5\).2. **Determine the Sign**: Based on the rules for dividing negative numbers:

• If the numbers have the same sign, the quotient is positive.
• If they have different signs, the quotient is negative.

3. **Perform the Division Ignoring the Signs**: Calculate as if both numbers are positive. With \(-100 \/ 5\), disregard the sign and divide 100 by 5 to get 20.4. **Apply the Determined Sign**: After calculating, apply the sign determined in step 2. Hence, \(-100 \/ 5 = -20\).
By following these steps, you’ll master calculating division, even when negative numbers are involved. These guiding principles ensure accuracy and help avoid simple errors when performing mathematical operations.