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Chapter 6: Problem 16

For Exercises \(1-20,\) estimate. Then find the actual quotient. $$-21.7 \div(0.001)$$

Short Answer

Expert verified
Answer: The estimated quotient is -2200, and the actual quotient is -21700.

Step by step solution

01

Evaluate the numbers

In this problem, we are given the numbers \(-21.7\) and \(0.001\). We need to find the quotient, which means dividing \(-21.7\) by \(0.001\).
02

Estimate the numbers

When estimating the numbers, we will round them to the nearest whole number. The number \(-21.7\) can be rounded to \(-22\), and the number \(0.001\) can be rounded to \(0\). However, we can't divide by zero, so let's round \(0.001\) to the nearest tenth instead: \(0.001 \approx 0.01\).
03

Perform the Estimated Division

Now that we've rounded the numbers, we can perform a simpler division: $$-22 \div 0.01$$ Let's divide \(-22\) by \(0.01\): $$\frac{-22}{0.01} = -2200$$ So, the estimated quotient is \(-2200\).
04

Perform the Actual Division

Now let's find the actual quotient by dividing \(-21.7\) by \(0.001\): $$\frac{-21.7}{0.001} = -21700$$ The actual quotient is \(-21700\). In summary, the estimated quotient is \(-2200\), and the actual quotient is \(-21700\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quotient Calculation
Understanding quotient calculation is fundamental in mathematics. A quotient is the result you get when you divide one number by another. In this exercise, our task was to divide \(-21.7\) by \(0.001\), aiming to find both an estimated and actual quotient.
To perform a division and get a quotient:
  • Identify the dividend, which is the number being divided. Here, it is \(-21.7\).
  • Identify the divisor, which is the number you divide by. In our exercise, this is \(0.001\).
  • Use division: Divide the dividend by the divisor.
It's important to remember that dividing by very small numbers (like \(0.001\)) is like multiplying by a large number, hence the actual quotient being \(-21700\). Division transforms our problem into exploring relationships between numbers, which helps solve proportions and balance equations.
Estimating Quotients
Often, exact answers are not necessary, and estimates suffice. Estimating quotients helps simplify complex problems by making numbers easier to work with. The key is to round numbers to make calculations manageable.
For instance:
  • First, round \(-21.7\) to a more convenient whole number, \(-22\).
  • Next, since dividing by zero is undefined, avoid rounding \(0.001\) to \(0\). Instead, opt for something close like \(0.01\).
  • Then, divide \(-22\) by \(0.01\) to easily get an estimate: \(-2200\).
Estimating can be incredibly helpful for quickly checking work, especially when verifying more complex division problems or during tests where time is limited.
Negative Numbers Division
Dividing negative numbers follows specific sign rules that govern multiplication and division in mathematics.
Here are some key points:
  • When you divide a negative number by a positive one, the quotient is negative. In this exercise, \(-21.7\) divided by \(0.001\) results in \(-21700\).
  • The opposite holds true as well, meaning that dividing a positive number by a negative one also results in a negative quotient.
  • Conversely, dividing two negative numbers results in a positive quotient. This is due to the negative signs canceling each other out.
Understanding these principles helps in solving division problems across various mathematical contexts. Remember to always track your signs carefully to avoid mistakes.

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