Problem 18 - Prealgebra

Question

Complete the following tables. $$\begin{array}{|c|c|c|} \hline\begin{array}{c}\text { First } \\\\\text { Number } \\\a\end{array} & \begin{array}{c}\text { Second } \\\\\text { Number } \\\b\end{array} & \begin{array}{c}\text { The Quotient } \\ \text { of } a \text { and } b\\\ \frac{a}{b} \end{array} \\ \hline 24 & -4 \\\\\hline 24 & -3 \\\\\hline 24 & -2 \\ \hline 24 & -1 \\\\\hline\end{array}$$

Step-by-Step Solution

Answer

Quotients: -6, -8, -12, -24.

Step 1: Understanding the Table Structure

The table provided asks us to complete the last column, which represents the quotient of the 'First Number' (denoted as \(a\)) and the 'Second Number' (denoted as \(b\)). The expression for the quotient is given as \(\frac{a}{b}\). We need to calculate this for each pair of \(a\) and \(b\).

Step 2: Calculating First Quotient

For the first row of the table, \(a = 24\) and \(b = -4\). Calculate the quotient: \(\frac{24}{-4} = -6\). This value is the entry for the third column in the first row.

Step 3: Calculating Second Quotient

In the second row, \(a = 24\) and \(b = -3\). The quotient is \(\frac{24}{-3} = -8\). Enter this value in the corresponding cell.

Step 4: Calculating Third Quotient

For the third row, \(a = 24\) and \(b = -2\). Compute \(\frac{24}{-2} = -12\). Add this result to the table.

Step 5: Calculating Fourth Quotient

For the last row, \(a = 24\) and \(b = -1\). The quotient is \(\frac{24}{-1} = -24\). Place this in the last cell.

Key Concepts

Quotient Calculation

Quotient calculation is a basic arithmetic operation where one number is divided by another. It is crucial in prealgebra as it introduces students to division concepts in mathematics. When calculating the quotient of two numbers, you determine how many times the divisor fits into the dividend. In other words, you are calculating how many times the second number, called the divisor, can be subtracted from the first number, the dividend. For example:

• If you have a dividend of 24 and a divisor of 6, the quotient is: \(\frac{24}{6} = 4\).
• This means you can subtract 6 from 24 exactly 4 times before reaching zero.

Understanding this fundamental process is the first step towards mastering division and further mathematical studies.

Negative Numbers

Negative numbers are essential in mathematics, representing values less than zero. They can be a little tricky when it comes to operations like division, making it essential to understand how to handle them. When calculating quotients involving negative numbers, it's necessary to remember:

• If you divide a positive number by a negative number, the quotient will be negative.
• Similarly, if you divide a negative number by a positive number, the quotient will also be negative.
• For instance, \(\frac{24}{-4} = -6\), because a positive 24 divided by a negative 4 gives a negative result.

This rule results from the properties of dividing numbers with different signs. It is an important concept for prealgebra students to understand and apply confidently in problem-solving.

Fractions in Mathematics

Fractions are another pivotal component of prealgebra, often used in quotient calculations. A fraction represents a part of a whole and is expressed as \(\frac{a}{b}\), where \(a\) is the numerator and \(b\) the denominator. In division problems, the quotient is essentially a fraction. Understanding fractions involves knowing that:

• The numerator is divided by the denominator.
• If the numerator and the denominator have different signs, the fraction is negative, like \(\frac{24}{-3} = -8\).
• Reducing fractions to their simplest form helps make calculations easier.

Fractions provide a powerful way to express non-integer results in division, allowing students to grasp the full range of division outcomes in mathematics.