Problem 85 - Prealgebra

Question

Estimating The quotient \(-253 \div 24\) is closer to which of the following? a. 5 b. \(-10\) c. 15 d. \(-20\)

Step-by-Step Solution

Answer

The quotient \(-253 \div 24\) is closer to option b, which is \(-10\).

Step 1: Identify the Division Sign

The problem given involves dividing a negative number, \(-253\), by a positive number, \(24\). This determines that our answer will eventually be a negative number.

Step 2: Approximate the Dividend

Notice that \(-253\) is close to \(-240\). This approximation is helpful because \(-240\) is easy to divide by \(24\).

Step 3: Calculate the Simple Quotient

Divide the approximated number by \(24\): \[-240 \div 24 = -10\]. This calculation shows that approximating \(-10\) could be a fitting choice.

Step 4: Evaluate Other Choices

Now consider the given choices: 5, -10, 15, and -20. Of these options, we notice that \(-10\) is appropriate as our division remains negative and is close to the approximated calculation \(-10\).

Key Concepts

Understanding Negative Numbers in Division

When dividing negative numbers, it is important to remember a few key rules that will help guide you to the right solution.
If a negative number is divided by a positive number, like in the exercise \(-253 \/ 24\), the result will always be negative. Conversely, if a positive number is divided by a negative number, the result will also be negative. If two negative numbers are divided, the result will be positive because the two negatives cancel each other out.

Recognizing these rules can simplify solving division problems with negative numbers and helps avoid common mistakes. Thus, knowing that the division result of \-253 \/ 24\ must be negative narrows down the options in the exercise.

The Art of Approximation in Division

Approximation is a powerful tool that provides ease, especially with complex divisions. In integer division, you might not get an exact answer, and that is where approximation comes in handy.
Consider the numbers in the exercise: \(-253 \/ 24\). An easier way to manage this problem is by approximating the dividend. Notice how \(-253\) is close to \(-240\).

Mathematically speaking, \(-240\) is more straightforward to divide by \24\, simplifying the calculation to get a quick estimate. Approximating helps manage numbers for mental calculation, especially when dealing with tricky figures. The ability to approximate effectively streamlines the division process, providing a quick path to a viable quotient.

Estimating Quotients for Enhanced Understanding

Estimation is an essential skill when dividing complex numbers, as it helps you get a rough idea of what the answer should be. To estimate quotients effectively, consider the following steps:

• First, simplify the numbers by rounding them. This makes the numbers easier to handle mentally.
• Next, think of simpler, rounded numbers that are easy to divide.
• Finally, calculate the quotient of the approximated numbers.

Using these methods simplifies finding a close result quickly, like determining that \(-10\) is a reasonable quotient for \(-253 \/ 24\).
This makes a challenging problem more approachable and ensures that the estimation remains within a logical range. By consistently practicing estimation, students can enhance their mathematical intuition and bolster confidence in handling more complex division tasks.