Problem 18 - Fundamentals of Mathematics

Question

Find each value. \((-51) \div(-7)\)

Step-by-Step Solution

Answer

The result is 7.29.

Step 1: Identify the Operation

Recognize that the expression involves division: \((-51) \div (-7)\). This means we need to divide -51 by -7.

Step 2: Consider Sign Rule

Recall the rule for dividing integers: when dividing two numbers with the same sign, the quotient is positive. Both -51 and -7 are negative, so the result will be positive.

Step 3: Divide the Absolute Values

Calculate the division of the absolute values of the numbers: \(51 \div 7 = 7.285\). Note that in some cases, the question may require rounding or exact division; however, for this division, the repeating decimal can be rounded to two decimal places as 7.29.

Step 4: Apply the Sign

Since the rule from Step 2 establishes that the quotient of two negative numbers is positive, the result is positive. Therefore, \((-51) \div (-7) = 7.285\) when rounded is 7.29.

Key Concepts

Sign Rules

When working with integer division, understanding sign rules is crucial. The sign of the quotient in a division problem depends on the signs of the numbers being divided. Here are the basic rules for integer division sign rules:

• If both numbers have the same sign (both positive or both negative), the quotient is positive.
• If the numbers have different signs (one is positive and the other is negative), the quotient is negative.

In the example \((-51) \div (-7)\), both numbers are negative. According to our rules, this means the quotient will be positive.
Understanding these rules helps simplify calculations and reduce errors when dealing with integers.

Absolute Values

Absolute values refer to the magnitude of a number without considering its sign. When dividing integers, it is often helpful to first calculate the division of their absolute values, which provides a clearer view of the division itself.
The absolute value is denoted by two vertical bars; for instance, the absolute value of -51 is written as \(|-51|\) and equals 51.

• In integer division, compute the absolute values of each number before calculation. \(|-51| = 51\) and \(|-7| = 7\).
• Perform the division: \(51 \div 7 = 7.285\), which is the value before rounding.

By focusing on absolute values, we remove the complexity of signs until the end of our calculations, ensuring easier and more straightforward math operations.

Quotient

The quotient is the result of a division problem. In \((-51) \div (-7)\), after applying the sign rules and computing based on absolute values, we find the quotient.When dividing integers:

• Perform the division of the absolute values. Here, it results in 7.285.
• Determine the sign using the sign rules (in this case, positive).
• If required, round to the desired number of decimal places.

Ultimately, for the example given, the rounded quotient is 7.29. This final value answers the division problem based on our earlier computations and rules.