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When we talk about division, we are looking at one of the four primary operations in mathematics, along with addition, subtraction, and multiplication. Division is used to determine how many times a number, known as the divisor, is contained within another number, called the dividend. For example, in the expression \( \frac{49}{7} \), 49 is the dividend, and 7 is the divisor. The result of this division, known as the quotient, tells us how many times 7 can be included in 49.

• Dividend: The number being divided (49).
• Divisor: The number you divide by (7).
• Quotient: The result of the division (in this case, 7).

To solve \( \frac{49}{7} \), you divide 49 by 7. Therefore, you find out how many whole sets of 7 you can extract from 49, which is 7 in this scenario. This is a straightforward division because 49 is evenly divisible by 7.

When dealing with negative numbers in division, it's critical to understand how a negative sign affects the quotient. A negative sign in front of a division expression, like in \(-\frac{49}{7}\), changes the sign of the result. Here's how it works:

• If both the dividend and divisor are positive, the quotient is positive.
• If the dividend is negative and the divisor is positive, like \(-\frac{49}{7}\), the quotient is negative.
• If the dividend is positive and the divisor is negative, the quotient remains negative.
• If both the dividend and divisor are negative, the quotient is positive.

So for \(-\frac{49}{7}\), although dividing 49 by 7 gives 7, the negative sign makes the final answer \(-7\). This rule helps maintain the consistent logic in mathematical operations involving negative numbers.

Solving mathematical problems step-by-step is a powerful strategy to understand not only the solution but the process as well. Let's break down the steps used in solving the problem \(-\frac{49}{7}\).
First, identify the operation involved. Here, it's division and we observe a negative sign in the expression.

• Identify: Recognize that this is a division operation with a negative sign.

Next, execute the division. Divide 49 by 7 to see how many sets of 7 fit into 49, which results in 7.

• Calculate: 49 divided by 7 equals 7.

Finally, apply the negative sign learned in the second step.

• Apply Negative Sign: The calculation finalizes as -7 due to the negative sign preceding the division.

By going through each step systematically, you can confidently arrive at the result \(-7\), and this approach can be useful for tackling more complex problems with ease.