Problem 55 - Prealgebra

Question

Without pencil and paper or a calculator. The quotient \(-121 \div 27\) is closest to which of the following numbers? a. \(-150\) b. \(-100\) c. \(-4\) d. 6

Step-by-Step Solution

Answer

The quotient is closest to -4.

Step 1: Estimate the Approximate Quotient

We know that dividing a negative by a positive number will yield a negative result. So, \(-121 \div 27\) will be negative. Let's estimate the quotient. We can approximate \(121\) as roughly \(120\), and \(120 \div 30 = 4\). Therefore, the closest integer quotient of \(-121 \div 27\) should be close to \(-4\).

Step 2: Verify Against Given Options

Examine the effect of rounding when dividing. With our estimate of \(-4\), compare this to the choices: a. \(-150\) is far too negative,b. \(-100\) is also too negative, c. \(-4\) matches our estimate, d. 6 is positive, which is not possible. Thus, \(-4\) is the closest to our estimate of the actual quotient.

Key Concepts

Estimation Method

The estimation method is a powerful tool to quickly find approximate values without exact calculations. This approach simplifies complex problems by rounding numbers to make calculations easier.
For instance, when you need to estimate \(-121 \div 27\), you might round \(-121\) to \(-120\) and \(27\) to \(30\). This way, the division becomes easier as \(120 \div 30 = 4\). Since the original problem involves a negative number, we know the approximate result will be negative.
Thus, by understanding how to break down the division, you gain insight into arriving at the closest approximate result without detailed calculations. Estimating in this way helps make complex numerical problems manageable.

Division of Negative Numbers

When you divide a negative number by a positive number, the result is always negative. This is a fundamental concept in mathematics and essential to understand for accurate problem-solving.
In our example, dividing \(-121\) by \(27\), we anticipate a negative quotient because the sign of the numbers involved dictates the result's sign.

• Negative divided by positive = Negative
• Positive divided by negative = Negative
• Negative divided by negative = Positive

Understanding these rules helps you predict the outcome, making it easier to quickly eliminate answer options that don't fit the expected sign, such as a positive result in a situation involving division by a positive and a negative number.

Approximate Quotient

The approximate quotient finds its place when exact calculations aren't necessary, but an estimated answer is sufficient.
In the exercise, we approached the problem by rephrasing the numbers involved for efficiency. By rounding \(-121\) to \(-120\) and \(27\) to \(30\), we concluded the approximate quotient is around \(-4\). This estimation aligned with the given options allowing us to identify \(-4\) as the nearest correct choice.
This method excels in multiple-choice scenarios where examining a range of outcomes can quickly guide you to the nearest correct answer, easing the process in classroom settings, exams, or quick problem-solving activities.