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Chapter 1: Problem 20

Estimate each difference by rounding. Then find the actual difference. $$6,005,002-258,496$$

Short Answer

Expert verified
Answer: The estimated difference is 5,747,000, and the actual difference is 5,746,506.

Step by step solution

01

Estimate by rounding

To estimate the difference, round each number to the nearest thousand: $$6,005,002 \approx 6,005,000$$ $$258,496 \approx 258,000$$ Now that we have rounded the numbers, we can proceed to estimating the difference.
02

Estimate the difference

Now that the numbers have been rounded, we can estimate the difference: $$6,005,000 - 258,000 = 5,747,000$$ The estimated difference is 5,747,000.
03

Find the actual difference

Now, we will find the actual difference between the two numbers given: $$6,005,002 - 258,496 = 5,746,506$$ The actual difference is 5,746,506.

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Key Concepts

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

Rounding Numbers
Rounding numbers is a technique used to simplify complex calculations. By approximating numbers to the nearest ten, hundred, thousand, or more, you can make arithmetic much easier to handle visually and mentally. This approach doesn't provide the exact figure, but a version that is "close enough" for estimating purposes.

When rounding to the nearest thousand, look at the hundreds digit:
  • If the digit is 5 or greater, round up.
  • If the digit is less than 5, round down.
For example, in our exercise, we rounded
  • 6,005,002 to 6,005,000
  • 258,496 to 258,000
By rounding, we can quickly see a rough estimate of what the actual answer might be without precise calculation.
Arithmetic Difference
The arithmetic difference is the result you get when you subtract one number from another. Subtraction involves taking away the value of one number from another to find how they differ in magnitude or size. Understanding how to calculate this difference accurately is essential because it gives you the precise measure of how far apart two numbers are.

In the exercise given, the arithmetic difference is the result of:
  • Subtracting 258,496 from 6,005,002.
The actual difference, an exact calculation, yields 5,746,506. Calculating the arithmetic difference accurately is important for tasks requiring precise values, such as financial computations and scientific data analysis.
Estimating Differences
Estimating differences involves making a quick guess of the outcome of a subtraction problem. This method uses rounded numbers to simplify subtraction, making it faster and easier to mentally compute or provide an answer when exact precision is not needed.

In the exercise, rounding the numbers before subtracting allows you to estimate the difference between 6,005,002 and 258,496 more easily:
  • The estimate was found by calculating 6,005,000 minus 258,000, giving an estimated difference of 5,747,000.
Estimating differences is useful in situations where a fast, close answer suffices, like checking the reasonableness of results or making decisions based on approximated data. This approach helps to handle complex calculations more efficiently without delving into detailed computations.

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