Problem 19 - Algebra 1

Question

Find the quotient. $$49 \div(-7)$$

Step-by-Step Solution

Answer

The quotient of \(49 \div(-7)\) is -7.

Step 1: Identify the Dividend and the Divisor

The dividend, which is the number to be divided, is 49 and the divisor (the number by which the dividend is to be divided) is -7.

Step 2: Division Process

Next, perform the division as normal, ignoring the negative sign at first. Divide 49 by 7 to get a quotient of 7.

Step 3: Account for the Negative Sign

Since the divisor is negative, the result has to be negative as well. Hence, the final answer will be -7.

Key Concepts

Dividend and Divisor

Understanding the roles of the dividend and divisor is essential when learning the basics of division. In any division equation, the number that is being divided is called the dividend, and the number that divides the dividend is known as the divisor. For instance, in the equation \( 49 \div (-7) \), 49 is the dividend, and -7 is the divisor.

It's important to distinguish between these two because it allows us to set up the problem correctly. Regardless of whether the numbers are positive or negative, the division process follows the same rules where the dividend is divided by the divisor to yield the quotient.

Division Process

The division process can be seen as a method to determine how many times the divisor can be subtracted from the dividend until the remainder is less than the divisor. When dividing, one usually starts by setting up the problem, ensuring the dividend is first, and the divisor is second. In case of 49 divided by -7, ignore the signs at first to focus on the numbers:

\( 49 \div 7 = 7 \).

By following this procedure step by step, we achieve an intermediate quotient, and this stage forms the basis for accurately determining the final outcome. Especially in educational contexts, mastering this process lays a vital foundation for understanding more complex mathematical operations.

Negative Number Division

Negative number division adds an extra rule to the normal division process. The key to this concept lies in how the signs of the dividend and divisor affect the sign of the quotient. There's a simple rule to remember:

If either the dividend or the divisor is negative, but not both, the quotient will be negative. Conversely, if both are negative or both are positive, the quotient will be positive. For our problem, \( 49 \div (-7) \), since the divisor is negative and the dividend is positive, the quotient is negative. Hence the solution, -7.

This rule helps maintain consistency across arithmetic operations and is an important one for students to memorize as they encounter algebra and more advanced mathematics. Understanding this also helps in avoiding common mistakes when dealing with negative numbers.