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Have you ever been faced with making a quick estimate of a numerical quantity or determining the order of magnitude of a numerical quantity? If you were faced with determining the result of 78,359 divided by 415 without paper and pencil or a calclator, you might be hard pressed to come up with an answer. The use of compatible numbers makes the calculation easier but it must be recognized that the answer derived is only an approximation.
Compatible numbers are those whole, or rounded off, numbers that allow the mental execution of a mathematical operation between the numbers easier. They are used primarily for estimating the approximate result of multiplying or dividing numbers containing many different digits.
Given two numbers containing several different digits, it is possible to round them off to numbers that are more easy to operate on. As an illustration, if you were asked to obtain the result of 2807 ÷ 42 you would either divide it out long hand or head straight for the calculator. On the other hand, upon examination, you can readily see that 2807 is approximately 2800 and 42 is approximately 40. Dividing these two rounded off numbers yields 70 which is a reasonably fair "order of magnitude" estimation of the exact answer of 66.83. Alternatively, dividing 28 by 4 gives you 7 and replacing the difference in the deleted zeros brings you back to 70. The rounded off numbers are referred to as compatible numbers.
How would you estimate 78,359 ÷ 415 using compatible numbers?
Rounding off the 78,359 we get 78,000 and rounding off the 415 we get 400. Dividing these two numbers gives us 195 compared to the exact answer of 188.8.
The use of compatible numbers is clearly justified when one is seeking what we call an "order of magnitude" answer. For instance, consider a farmer giving some early thought as to how much feed he should order for his livestock for the upcoming winter. In other words, if a farmer were trying to make a rough estimate of how many pounds of feed to order for his livestock, he is concerned with whether he will be needing 1, 10, 100, 1000 or 10,000 tons, so to speak. This is typically referred to as the order of magnitude. Lets say that at the time he was interested in making the estimate, he remembered that over the previous winter he used approximately 23 pounds per day for 218 animals over a 90 day period. Assuming he had no paper, pencil or calculator available, he elects to use the equivalent compatible numbers to make his estimate. He rounds off the 23 to 20, the 218 to 220 and the 90 to 100. These compatible numbers are multiplied together to yield a result of 20x220 = 4400 and 4400x100 = 440,000 pounds. Divided by 2000 pounds to the ton, gives him his reasonably close order of magnitude answer of 220 tons compared to the exact answer of 225.63 tons. (The numbers are used only for illustrative purposes and bear no resemblance to a real world situation.)
Would it be 75?
Since it's easier to work with numbers that end in 0, I think 80 is the most compatible number.
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