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Estimation

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Adding, multiplying, and performing similar mathematical operations in one's head can be difficult tasks, even for the most skilled mathematics students. By estimating, however, basic operations are easier to calculate mentally. This can make daily calculation tasks, from figuring tips to monthly budgets, quickly attainable and understandable.

How to Estimate

Although the core of estimation is rounding, place value (for example, rounding to the nearest hundreds) makes estimating flexible and useful. For instance, calculating 2.4 + 13.7 − 10.8 + 8 − 124.2 − 32 to equal −142.9 in one's head may be a daunting task. But if the equation is estimated by 10s, that is, if each number is rounded to the nearest 10, the problem becomes 0 + 10 − 10 + 10 −; 120 − 30, and it is easier to calculate its value at −140. Estimating to the 1s makes the equation 2 + 14 − 11 + 8 − 124 − 32 = −143, which is more accurate but more difficult to calculate mentally. Note that the smaller the place value used, the closer the estimation is to the actual sum.

Estimation by Tens

Multiplication and division can be estimated with any place value, but estimating by 10s is usually the quickest method. For example, the product 8 × 1,294 = 10,352 can be estimated by 10s as 10 × 1,290 = 12,900, which is calculated with little effort. Division is similar in that estimating by 10s allows for the quickest calculation, even with decimals. For instance, 1,232.322 ÷ 12.2 = 101.01 is quicker to estimate by 10s as 1,230.0 ÷ 10.0 = 123.0.

Regardless of the ease of estimating by 10s, there is a greater degree of inaccuracy as compared to estimating by 1s. However, this estimation method need not be abandoned in order to gain accuracy; instead, it can be used to obtain estimations that are more accurate, as the following example illustrates.

Suppose a couple on a date enjoys a dinner that costs $24.32. The customary tip is 15 percent, but the couple does not have a calculator, tip table, or pencil to help figure the amount that should be added to the bill. Using the estimating-by-10s method, they figure that 15 percent of $10 is __BODY__.50; if the bill is around $20, then the tip doubles to $3. However, a $3 tip is not enough because they have not included tip for the $4.32 remaining on the bill. Yet if __BODY__.50 is the tip for $10, then __BODY__.75 would be an appropriate tip for $5, which is near enough to $4.32. A total estimated tip of $3.75 is close (in fact, an overestimation) to 15 percent of $24.32, which is $3.65 (rounded to the nearest cent).

Conservative Estimation

As seen in several of the examples, estimations tend to be more (an overestimation) or less (an underestimation) than the actual calculation. Whether this is important depends upon the situation. For example, overestimating the distance for a proposed trip may be a good idea, especially in figuring how much gas money will be needed.

This property of rounding and estimation is the foundation of conservative estimation found in financial planning. When constructing a monthly budget, financial planners will purposely underestimate income and overestimate expenses, usually by hundreds. Although an accurate budget seems ideal, this estimating technique creates a "cushion" for unexpected changes, such as a higher water bill or fewer hours worked. Furthermore, financial planners will round down (regardless of rounding rules) for underestimation and round up for overestimation.

The following table represents a sample budget for an individual. The first column includes amounts expected to pay; the second is a conservative estimate of the next month's budget; the third is a list of the actual amounts incurred; and the fourth is the difference between actual and budgeted amounts. Note that negative numbers, or amounts that take away from income, are written in parentheses.

The table shows that the individual earned less than expected and in some cases spent more than expected. Nevertheless, because the budget is conservative, there is a surplus (money left over) at the end of the month.

ESTIMATING A MONTHLY BUDGET
	Expected Amount	Budget	Actual Amount	Difference
Income	$3,040	$3,000	$2,995	($5)
Tax	(578)	(600)	(579)	21
Rent	(575)	(600)	(575)	25
Utilities	(40)	(100)	(62)	38
Food	(175)	(200)	(254)	(54)
Insurance	(175)	(200)	(175)	25
Medical	(45)	(100)	(97)	3
Car Payments	(245)	(300)	(245)	55
Gas	(85)	(100)	(133)	(33)
Student Loans	(325)	(400)	(325)	75
Savings	(300)	(300)	(300)	0
Fun Money	(49)	(100)	(175)	(75)
Surplus (Deficit)	$28	__BODY__	$75	$75

Estimation by Average

Counting the number of words on a page can be a tedious task. Therefore, writers often estimate the total by averaging the number of words on the first few lines and then multiplying that average by the number of lines on the page.

Another application of estimation by average is the classic game of guessing how many jellybeans are in a jar. The trick is to average the number of beans on the top and bottom layers and then to multiply that average by the number of layers in the jar. Because it is customary to declare a winner who guessed the closest but not over the actual count, it is best to estimate conservatively.

Estimation is a powerful skill that can be applied to tasks from proofing arithmetic to winning a counting game. However, the use of estimation is not always appropriate to the task. For example, estimating distance and direction of space debris and ships is unwise, since even the smallest decimal difference can mean life or death. In addition, technology makes it possible to add and multiply large groups of numbers faster than it may take to estimate the total. Nevertheless, estimation is an important tool in managing the everyday mathematics of life.

SEE ALSO FINANCIAL PLANNER; ROUNDING.

Michael Ota

Bibliography

Pappas, Theoni. The Joy of Mathematics: Discovering Mathematics All Around You. San Carlos, CA: World Wide Publishing, 1989.