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Chapter 6: Problem 8

Estimate then find the actual sum. $$1964.86-213.5$$

Short Answer

Expert verified
Question: Estimate and calculate the sum when subtracting the second number from the first number based on the following numerical values: 1964.86 and 213.5. Answer: The estimated sum is 1751, and the actual sum is 1751.36.

Step by step solution

01

Estimate the sum by rounding off

Firstly, we round off both the numbers to the nearest whole number, which gives us: $$1965 - 214$$
02

Calculate the estimated sum

Now, subtract the smaller number (214) from the larger number (1965) to get the estimate: $$1965 - 214 = 1751$$ This means our estimate for the actual sum is 1751.
03

Calculate the actual sum

Now, let's find the actual sum. Subtract 213.5 from 1964.86: $$1964.86 - 213.5 = 1751.36$$ So, the actual sum is 1751.36. Comparing this to our estimate, we can see that our estimation was close: only the decimal part differs by 0.36.

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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 make calculations simpler by dealing with whole numbers instead of their more complicated decimal forms. When rounding, you look at the digits in the number to determine which whole number it is closest to. This makes mental math easier and faster.
For example, when rounding the numbers 1964.86 and 213.5 to the nearest whole number, we look at the digit in the tenths place. For 1964.86, the tenths digit is 8. Since 8 is greater than 5, we round up to 1965. Similarly, for 213.5, the tenths digit is 5, which indicates a rounding up as well, resulting in 214. Thus, rounding simplifies 1964.86 to 1965 and 213.5 to 214.
Estimating Calculations
Estimating calculations allows us to predict the result of a mathematical operation without the need for exact computations. It is especially useful when you just need a rough idea of what the answer might be. In our example, after rounding both numbers, we estimated the subtraction.
The rounded numbers were 1965 and 214. Estimating simply involves subtracting the smaller rounded number from the larger rounded number: 1965 minus 214 equals 1751. This estimate provides a clear picture of what the final answer should approximate, facilitating a quick check against actual results.
Decimal Subtraction
Decimal subtraction is similar to whole number subtraction but requires careful alignment of the decimal points. The decimal point of each number should be directly above the other in a column arrangement. This ensures that each digit is subtracted from its respective place value.
Consider 1964.86 subtracted by 213.5. First, align the decimal points:
1964.86
- 213.50
-----
Now, subtract starting from the rightmost digit. Borrowing may be needed, just as in basic subtraction, but keep in mind the decimal point remains steady and properly positioned. This will result in an accurate decimal difference.
Calculating Differences
Calculating differences, especially with decimals, helps identify the gap or distance between two numbers. After both estimating and computing decimal subtraction, you can compare the estimated value with the actual difference.
For our example, you compute the difference between 1964.86 and 213.5 and obtain 1751.36. Comparing this to your estimate of 1751, you focus on the decimal component. The actual calculation shows that the decimal difference is 0.36, showing that our estimation was quite accurate despite rounding. Calculating differences helps confirm the reliability of estimates and points out minor inaccuracies due to decimal involvement.

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