Question

Find the quotient. $$60 \div(-10)$$

Short answer

The quotient of 60 divided by -10 is -6.

Step-by-step solution

Write the Division Expression

The given division expression is \(60 \div(-10)\). Here, 60 is the dividend and -10 is the divisor.

Perform the Division

To get the quotient, simply divide the dividend (60) by the divisor (-10).

Write Down the Result

The result of the division values is -6, which is the quotient.

Key concepts

Negative Divisor

Understanding the concept of a negative divisor is crucial when performing division with integers. In the expression, such as in our problem, \(60 \div (-10)\), the number being divided by, which is -10, is known as the divisor. When the divisor is negative, it indicates that the division will result in a number with the opposite sign of the dividend. This rule is quintessential in the realm of integer arithmetic; it dictates that when we divide a positive dividend by a negative divisor, the quotient will be negative.

Such arithmetic reflects the consistent patterns found within the number line, where negative and positive values have a mirrored relationship. The exercise demonstrates that a dividend (the number being divided, which is 60 in this case) when divided by a negative divisor results in a negative quotient.

Dividend

In any division task, the number that you start with, which is divided by another number, is called the dividend. It represents the total or whole quantity that is being divided into equal parts. In the context of the given exercise, 60 represents our dividend. Understanding the role of the dividend is vital, as its magnitude and sign will significantly affect the result of the division - the quotient.

It's important to comprehend that the dividend acts as the numerator in a fraction, or what you might think of as 'the whole pie' before cutting it into slices. Students should remember that the magnitude of the dividend relative to the divisor affects whether the quotient is greater than or less than 1, an aspect that plays a fundamental role in proportion and ratio problems as well.

Quotient

The quotient is the result of dividing the dividend by the divisor. It essentially tells us how many times the divisor can be subtracted from the dividend until zero is reached or how many times a divisor fits into the dividend. In the given exercise, after dividing the dividend, 60, by the negative divisor, -10, we arrive at a quotient of -6.

This quotient conveys important information, particularly when dealing with negative numbers. It is essential to note that positive divided by negative, or vice versa, always results in a negative quotient. In contrast, dividing numbers with the same sign (both positive or both negative) will produce a positive quotient. Mastering the concept of quotients in integer arithmetic forms a solid foundation for advancing in algebra and other mathematical fields.

Integer Arithmetic

The term integer arithmetic encompasses the operations of addition, subtraction, multiplication, and division using whole numbers which can be positive, negative, or zero. When working with integer arithmetic, there are several key rules and properties students must grasp - one being the division of integers with different signs.

When an integer is divided by another, we can apply the concepts discussed briefly in the previous sections. Remembering when signs negate each other or reinforce each other is paramount. For instance, as illustrated in our example with a negative divisor and a positive dividend, the resulting quotient becomes negative. Conversely, had both the dividend and divisor been negative, the quotient would have been positive. Integer arithmetic can often feel counterintuitive because it operates in a sphere without the constructs of physical quantities and allows for abstractions like negative slices of pies, reinforcing the need to firmly embrace these rules in mathematical thinking.