Problem 36 - Algebra A Combined Function

Question

Divide. \(-60 \div 5\)

Step-by-Step Solution

Answer

-12

Step 1: Understand the Division

Division is the process of splitting a number into equal parts. In the expression \(-60 \div 5\), \(-60\) is the dividend, and \(5\) is the divisor.

Step 2: Determine the Sign of the Quotient

A negative number divided by a positive number yields a negative result. Therefore, \(-60 \div 5\) will have a negative quotient.

Step 3: Calculate the Absolute Values

Ignore the signs and divide the absolute values. Perform the operation \(60 \div 5\). This represents splitting 60 into 5 equal parts, which equals 12.

Step 4: Apply the Sign to the Quotient

As determined in Step 2, the final result will be negative. Thus, apply the negative sign to the quotient obtained, giving us \(-12\).

Key Concepts

Negative Numbers

Negative numbers are numbers that are less than zero. They are usually represented with a minus sign \(-\). These numbers are crucial in mathematics and can greatly influence the outcome of operations like division. For instance, when dividing negative numbers, different rules apply depending on the sign of the other number involved. Here's a quick guide:

• Dividing two negative numbers results in a positive quotient.
• Dividing a negative number by a positive number results in a negative quotient.
• Dividing a positive number by a negative number also results in a negative quotient.

Understanding these rules can help in correctly solving division problems involving negative numbers. Each rule revolves around the principle that negatives cancel each other out when paired or remain negative when only one factor is negative.

Quotient

The quotient is the result of a division problem. It's important to know that the quotient can be positive or negative, depending on the signs of the numbers involved. In the problem \(-60 \div 5\), we have a negative number divided by a positive one.From what we learned about negative numbers, that means our quotient will be negative. Computing this correctly involves determining the sign first. Then, focus on the division of their absolute values. This ensures the accuracy of both the value and sign of the quotient. In simple terms:

• Sign matters! Determine the expected sign first.
• Divide as usual, disregarding signs initially.
• Apply the sign to your result.

These steps help in efficiently calculating the quotient in any division involving negative numbers.

Absolute Values

When dealing with division, especially with negative numbers, absolute values play a crucial role. The absolute value of a number refers to its distance from zero on the number line, without considering direction. For instance, the absolute value of both \(-60\) and \(60\) is 60, denoted as \(|-60| = 60\).In calculations, particularly division, we often ignore the signs at first and focus on the absolute values. This helps simplify the arithmetic while applying the correct rules for signs later. For example, with \(-60 \div 5\), ignoring the signs gives us \(60 \div 5 = 12\). Once this part is done, the sign from the division rule is applied.

• Use absolute values to simplify division calculations.
• Decide on the sign of the result separately.

This method keeps calculations straightforward, allowing focus on obtaining the correct mathematical result.

Divisor

A divisor is the number you divide by in a division operation. It plays a significant role in determining the outcome. In the division \(-60 \div 5\), 5 is the divisor. A divisor can be any number except zero because division by zero is undefined.The effect of the divisor's sign is critical. If the divisor is positive and the dividend is negative, the quotient will be negative, as seen here. Always ensure that your divisor is non-zero, and when calculating, first determine how it will affect the sign of the quotient.

• A positive divisor with a negative dividend results in a negative quotient.
• A negative divisor with a positive dividend equally results in a negative quotient.

These basic rules ensure accuracy and comprehension in operations involving division. Keeping these in mind will guide you through the process efficiently.