Problem 20 - Prealgebra

Question

Complete the following tables. $$\begin{array}{|c|c|c|} \hline\begin{array}{c}\text { First } \\\\\text { Number } \\\a\end{array} & \begin{array}{c}\text { Second } \\\\\text { Number } \\\b\end{array} & \begin{array}{c}\text { The Quotient } \\ \text { of } a \text { and } b\\\ \frac{a}{b} \end{array} \\ \hline -24 & -2 \\\\\hline -24 & -4 \\\\\hline -24 & -6 \\ \hline -24 & -8 \\\\\hline\end{array}$$

Step-by-Step Solution

Answer

1st: 12, 2nd: 6, 3rd: 4, 4th: 3.

Step 1: Understand the Task

We are given a table with two columns titled 'First Number' and 'Second Number', and we need to fill in the third column, 'The Quotient of a and b', which represents the division result of the 'First Number' by the 'Second Number'.

Step 2: Analyze the Table

Identify the given values in the table. The 'First Number' is consistently \(-24\) for each row, while the 'Second Number' varies (\(-2\), \(-4\), \(-6\), and \(-8\)). We will calculate \(\frac{a}{b}\) for each row.

Step 3: Calculate the Quotients

Determine the quotient for each row:- For the first row, \(-24\) divided by \(-2\) is \(\frac{-24}{-2} = 12\).- For the second row, \(-24\) divided by \(-4\) is \(\frac{-24}{-4} = 6\).- For the third row, \(-24\) divided by \(-6\) is \(\frac{-24}{-6} = 4\).- For the fourth row, \(-24\) divided by \(-8\) is \(\frac{-24}{-8} = 3\).

Step 4: Fill in the Table

Insert the calculated results into the quotient section of each row:1. First row: \(12\)2. Second row: \(6\)3. Third row: \(4\)4. Fourth row: \(3\)

Key Concepts

Table Completion

When dealing with table completion in math, we are often asked to fill in missing values based on the information given in other parts of the table.
In our exercise, you're given a table with three key components:

• First Number (constant at -24)
• Second Number (varies across rows)
• The Quotient of the division of these two numbers

As you complete the table, your task is to perform specific calculations that will allow you to fill in any missing data. In this case, you'll solve for the quotient.
To do this, identify the values, substitute them into your equation, and compute to find a solution. This requires your understanding of integer division and the relationship between the divisors and the dividend.

Integer Division

Integer division is a crucial concept in prealgebra, especially when dealing with whole numbers that might be negative.
Here, integer indicates that you are dividing whole numbers, which can either be positive or negative, as in our example. When dividing integers like \(-24\) by any of the second numbers (-2, -4, -6, -8), you should remember:

• Division of two negative integers results in a positive quotient.
• Always handle the negative signs before performing the arithmetic division of absolute values.

For instance, in the exercise, dividing \(-24\) by \(-2\) gives a quotient of \(12\). This shows how negative divided by negative yields a positive number. Understanding this will guide you in accurately completing the table with the right answers.

Quotients Calculation

The calculation of quotients is where you find the result of dividing one number by another.
This involves applying the process of division and involves understanding how the quotient relates to division, which is particularly vital in filling out tables or when checking table calculations. Here is how it works:

• To find your quotient, identify your 'First Number' (dividend) and 'Second Number' (divisor).
• Apply the formula \(\frac{a}{b}\) where \(a\) is the dividend and \(b\) is the divisor.
• Solve for the quotient, ensuring correct arithmetic operation, especially with attention to signs.

In this specific exercise, you performed calculations like \(\frac{-24}{-4} = 6\). The accuracy in these calculations ensures that you fill the table correctly with precise mathematical solutions. Practicing these calculations will enhance your comfort with similar exercises in prealgebra.