Problem 56 - Prealgebra

Question

Without pencil and paper or a calculator. The quotient \(1,000 \div(-337)\) is closest to which of the following numbers? a. 663 b. \(-3\) c. \(-30\) d. \(-663\)

Step-by-Step Solution

Answer

The quotient is closest to option b, -3.

Step 1: Identity Positive Division Approximation

We want to approximate the division of 1000 by 337, but first ignore the negative sign. Estimate around which known multiple of 337 fits 1000. Since 337 is approximately close to 300, divide 1000 by 300 to check how close we get. This gives:\[ \frac{1000}{300} \approx 3.33 \]

Step 2: Adjust for the Sign

The original division was not just by 337 but by -337, thus our division answer would be negative. This changes the approximate positive result of 3.33 to a negative, so we have:\[ \frac{1000}{-337} \approx -3.33 \]

Step 3: Compare with Choices

Look at the options given and find which is closest to our approximation of -3.33. Among the choices, -3 is the closest to -3.33.

Key Concepts

Division of Integers

When dividing whole numbers, also known as integers, you are splitting the numerator (the number you are dividing) by the denominator (the number by which you divide) into equal parts. In essence, division is the process of determining how many times one number is contained within another.

• For example, in the equation \(1000 \div 337\), 1000 is the numerator and 337 is the denominator.
• Division can be thought of as repeated subtraction or finding a missing factor from multiplication.

Dividing integers can sometimes result in whole numbers (when the division is perfect with no remainder) or in fractions or decimals (when there is a remainder). Despite what form the quotient takes, the process of division remains the same.
Determining an exact division result without tools like paper or a calculator can seem challenging. However, estimating can simplify the process, providing a close approximation that helps solve problems quickly, especially when the calculation's precision is less crucial.

Approximating Division Results

Approximating in division involves estimating the result when an exact calculation is challenging. This becomes especially handy for large numbers or when avoiding complex arithmetic.
When estimating:

• Round one or both of the numbers to simpler, nearby numbers.
• This simplifies the arithmetic, making mental calculations more feasible.

In the example of dividing 1000 by 337, you can simplify 337 to 300 for easier calculations because 300 is a common and straightforward number to divide by.
This approximation gave us \(\frac{1000}{300} \approx 3.33\).
Thus, by approximating, we can more quickly identify a range for the quotient, even if we can't calculate it precisely at that moment. Taking a moment to simplify the math makes division far less daunting.

Negative Numbers in Division

Dealing with negative numbers in division requires understanding how negative and positive numbers interact. Remember the general rule: the quotient of a positive and a negative number is always negative.
Simply:

• A positive divided by a positive results in a positive.
• A negative divided by a negative results in a positive.
• A positive divided by a negative or vice versa results in a negative.

In our problem, since we are dividing by \(-337\), the quotient becomes negative, adjusting our earlier estimation from \(3.33\) to \(-3.33\).
Understanding how to handle negative signs is key to accurately interpreting division results, ensuring that your final answer is correctly signed according to mathematical rules.