Basic Terms and Terminology Relating to Applying Estimation Strategies and Rounding Rules for Real World Problems

• Front end estimation: Estimating the answer to an algebraic, mathematical or arithmetic calculation by simplifying numbers.

The Front End Estimation Strategy

Although precise answers to algebraic, mathematical and arithmetic calculations is necessary, it is highly helpful to take an educated guess about the correct answer to a calculation particularly when the numerical values of the numbers are lengthy and complex.

For example, when you are asked to add or subtract very large and cumbersome numbers, it is highly helpful to take an educated guess about the correct answer to a calculation to avoid large and avoidable miscalculations and careless errors.

A highly useful estimation strategy called front end estimation. Front end estimation entails estimating the answer to an algebraic, mathematical or arithmetic calculation by simplifying numbers. When numbers are simplified and front end estimation is used, it will not give you the PRECISE answer to the calculation, but it will give you an approximate answer. You should, then, be able to compare your precise answer and calculation to the approximate answer that you got using front end estimation.

When you get the approximate answer to a calculation with front end estimation, you can then compare your precise mathematical calculation to this estimate and see whether or not your calculation is close to and in the same "ball park" front end estimation. If your precise answer to a calculation is in the same "ball park" as the front end estimation, you can be relatively sure that your precise calculation is perhaps accurate, but only if you carefully perform and check each calculation for any careless errors and mistakes. When, however, your "ball park" front end estimation is out of the range of with your precise calculation, something is wrong. When this happens, recheck your front end estimation and your precise calculation until these two are within the same "ball park".

The front end estimation strategy involves the changing of numbers into zeros with the exception of those in the "front end" which are the beginning numbers starting on the left of the numbers. Below are examples of front end estimations and how these estimations compare to precise calculations.

Example 1:

64 + 61

Front End Estimation:

Round off the 64 and the 61 leaving the front ends or the front, left most numbers, that is the 6 s, in place when the rounding off uses a number that is less than five, which the 4 and the 1 are. If, instead, the numbers were 67 and 69, for example, the 6 and 7 at the front end would be rounded up one number because both 7 and 9 are more than five.

64 + 61 can be approximated with 60 + 60 = 120

Precise Calculation:

64 + 61 = 125

As you can see, the front end estimation is very close to the precise mathematic calculation.

Example 2:

69 + 69 can be approximated with 70 + 70 = 140 and

The precise calculation is 138

Example 3:

64 + 64 can be approximated with 60 + 60 = 120 and

The precise calculation is 128

Example 4:

60 + 61 can be approximated with 60 + 60 = 120 and

The precise calculation is 121

Front End Estimations With Addition With Larger Numbers

Example 1:

604 + 716

Front End Estimation:

Round off the 604 and the 716 leaving the front ends that are the 6 and the 7 in place when the rounding off uses a number that is less than five.

604 + 716 can be approximated with 600 + 700 = 1300

Precise Calculation:

604 + 716 = 1, 320

As you can see, the front end estimation is very close to the precise mathematic calculation.

Example 2:

680 + 779 can be approximated with 700 + 800 = 1500 and

The precise calculation is 1459

Example 3:

891 + 234 can be approximated with 900 + 200 = 1100 and

The precise calculation is 1125

Example 4:

545 + 894 can be approximated with 500 + 900 = 1400 and

The precise calculation is 1439

Front End Estimations With the Addition of Even Larger Numbers

Example 1:

Front End Estimation:

1,234 + 5,678

Round off the 1,234 and the 5,678 leaving the front ends of 1 in place and rounding up the 5 to 6 because the 2 in the first number is less than five and the 6 in the second number is 5 or more.

1,234 + 5,678 can be approximated with 1000 + 6000 = 7000

Precise Calculation: 6912

Example 2:

1,234 + 5,678

Again, as you can see, the front end estimation is very close to the precise mathematic calculation.

Example 3:

13,567 + 19,987

Rounded off as 10,000 + 20,000 = 30,000

13,567 + 19,987 = 33,554

Front End Estimations With Subtraction

The front end estimation procedure for subtraction, multiplication and division is the same procedure that was done with addition above.

Here are some examples:

Example 1:

238 - 142

Rounded off as 200 - 100 = 100

238 - 142 = 96

Example 2:

398 - 289

Rounded off as 400 - 300 = 100

398 - 289 = 109

Example 3:

2,898 - 2,389

Rounded off as 3000 - 2000 = 1000

2,898 - 2,389 = 509

Example 4:

28,898 - 2,389

Rounded off as 30,000 - 2000 = 28,000

28,898 - 2,389 = 26,509

Front End Estimations With Multiplication

As you will see, the same rules apply to all arithmetic calculations including multiplication and division, when front end estimations are done.

Example 1:

Front End Estimation:

68 x 22

Rounded off as 70 x 20 = 1400

Precise Calculation: 1,496

Example 2:

689 x 2289

Rounded off as 700 x 2000 = 1,400,000

Precise Calculation: 1,577,121

Example 3:

68 x 987

Rounded off as 70 x 1000 = 70,000

Precise Calculation: 67,116

Example 4:

62 x 255

Rounded off as 60 x 300 = 18,000

Precise Calculation: 15,810

Front End Estimations With Division

Example 1:

Front End Estimation:

87 �- 22

Rounded off as 90 �- 20 = 4.5

Precise Calculation:

87 �- 22 = 3.9

Example 2:

879 �- 240

Rounded off as 900 �- 200 = 4.5

879 �- 240 = 3.6

Example 3:

18,700 �- 2,254

Rounded off as 20,000 �- 2,000 = 10

18,700 �- 2,254 = 8.3

Measurement System Conversions to The Nearest Estimation or Approximation

There are many occasions and situations where mathematics is not as precise as we often think it should. An example of this lack of precision is obvious when we consider our different measurements systems, including the household measurement system, the apothecary measurement system and the metric measurement system. There are few precise conversion factors from one measurement system to another; most are approximations. These approximations, however, when accurate, are acceptable to use according to the standards of measurement.

Although measurement system equivalents are often shown with an = sign, they should be more accurately shown with the ≈ sign which means about and approximately, rather than equal.

Examples of approximate system measurement conversions include:

1 fluid dram ≈ 4 to 5 milliliters

1 teaspoon ≈ 60 drops

1 kg ≈ 2.2 pounds

Rounding Rules for Real World Problems

Rounding numbers make them easier to work with to get some "ball park" guestimates, as discussed in the section above that detailed front end estimation; and the rounding of numbers is also used to calculate mathematical and arithmetic calculations to a shorter, more meaningful number, as shown below.

Rational numbers, including fractions, ratios, whole numbers, zero, positive numbers, negative numbers, and decimal numbers can be rounded off. Many mathematical and arithmetic calculations on TEAS will instruct you to round numbers off to the nearest place such as "to the nearest tens", "to the nearest hundreds", "to the nearest thousands place" or "to the nearest whole number", etc.. The TEAS may also instruct you to round off to the nearest tenths, hundredths, or thousandths place as well.

When rounding off numbers, you must examine the number AFTER the desired rounding off spot. If this number is 5 or more, you will round off the desired rounding off number by increasing it by one (1). If, however, the number AFTER the desired rounding off spot is LESS than 5, you would not change the number before it.

Some numbers must be rounded off or they will fill book after book. For example, the decimals 0.3 and 0.6 go to infinity. In this mathematical sense, infinity is never ending. The symbol for infinity is ∞.

0.03 will go to 0.033333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333 and even more. It will go to infinity.

Similarly, 0.6 will go to

0.66666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666 and even more. It will go to infinity (∞).

• 0333 rounded to the nearest hundredth would be 0.33 because the 3 in the thousandth place is less than 5
• 33333 rounded to the nearest thousandth would be 0.333 because the 3 in the ten thousandth place is less than 5

Similarly, 0.6 will go to

0.66666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666 and even more. It will go to infinity (∞).

• 0666 rounded to the nearest hundredth would be 0.07 because the 6 in the thousandth place is 5 or more.
• 66666 rounded to the nearest thousandth would be 0.667 because the 6 in the ten thousandth place is 5 or more.

The Positions of Numbers and Their Meanings: Whole Numbers: Ones, Tens, Hundreds, Thousands, Etc Positional notations are used in whole numbers, decimal numbers and combination whole numbers with decimal places.

These positional notations occur on both sides of a decimal point. The numbers to the left of the decimal point are whole numbers and the numbers on the right side of the decimal point are decimal numbers.

As you can see in the table above, the number positions to the left of the decimal place, in the correct sequential order are:

• Ones
• Tens
• Hundreds
• Thousands
• Ten thousands
• Hundred thousands
• Millions

Here are some examples:

Example 1:

Number: 67

Positional notations:

• 7 is in the ones place
• 6 is in the tens place

Example 2:

Number: 967

Positional notations:

• 7 is in the ones place
• 6 is in the tens place
• 9 is in the hundreds place

Example 3:

Number: 9672

Positional notations:

• 2 is in the ones place
• 7 is in the tens place
• 6 is in the hundreds place
• 9 is in the thousands place

Example 4:

Number: 19,672

Positional notations:

• 2 is in the ones place
• 7 is in the tens place
• 6 is in the hundreds place
• 9 is in the thousands place
• 1 is in the ten thousands place

Example 5:

Number: 129,672

Positional notations:

• 2 is in the ones place
• 7 is in the tens place
• 6 is in the hundreds place
• 9 is in the thousands place
• 2 is in the ten thousands place
• 1 is in the hundred thousands place

Example 6:

Number: 8,129,672

Positional notations:

• 2 is in the ones place
• 7 is in the tens place
• 6 is in the hundreds place
• 9 is in the thousands place
• 2 is in the ten thousands place
• 1 is in the hundred thousands place
• 8 is in the millions place

When rounding off whole numbers, you must examine the number AFTER the desired rounding off spot. If this number is 5 or more, you will round off the desired rounding off number by increasing it by one (1). If, however, the number AFTER the desired rounding off spot is LESS than 5, you would not change the number before it.

Here are some examples of rounding off to the nearest tens place with whole numbers:

• 276 is rounded off to 280 because the 6 in the ones place is 5 or more
• 1,456 is rounded off to 1460 because the 6 in the ones place is 5 or more
• 22,333 is rounded off to 22,330 because the 3 in the tens place is less than 5
• 567,987 is rounded off to 567,990 because the 7 in the tens place is 5 or more

Hare some examples of rounding off to the nearest hundreds place with whole numbers:

• 246 is rounded off to 200 because the 4 in the tens place is less than 5
• 1,457 is rounded off to 1,500 because the 5 in the tens place is 5 or more
• 55,525 is rounded off to 55,500 because the 2 in the tens place is less than 5
• 666,009 is rounded off to 666,000 because the 0 in the tens place is less than 5

Here are some examples of rounding off to the nearest thousands place with whole numbers:

• 1,457 is rounded off to 1,000 because the 4 in the hundreds place is less than 5
• 55,525 is rounded off to 56,000 because the 5 in the hundreds place is 5 or more
• 666,009 is rounded off to 666,000 because the 0 in the hundreds place is less than 5
• 7,987,987 is rounded off to 7,988,000 because the 9 in the hundreds place is 5 or more

Here are some examples of rounding off to the nearest ten thousands place with whole numbers:

• 666,009 is rounded off to 670,000 because the 6 in the thousands place is 5 or more
• 7,981,987 is rounded off to 7,980,000 because the 1 in the thousands place is less than 5
• 2,666,009 is rounded off to 2,670,000 because the 6 in the thousands place is 5 or more
• 23, 666,009 is rounded off to 23,670,000 because the 6 in the thousands place is 5 or more

Here are some examples of rounding off to the nearest hundred thousands place with whole numbers:

• 1,666,009 is rounded off to 1,700,000 because the 6 in the ten thousands place is 5 or more
• 7,981,987 is rounded off to 8,000,000 because the 8 in the ten thousands place is 5 or more
• 2,666,009 is rounded off to 2,7000,000 because the 6 in the ten thousands place is 5 or more
• 23, 666,009 is rounded off to 23,700,000 because the 6 in the ten thousands place is 5 or more

Here are some examples of rounding off to the nearest millions place with whole numbers:

• 31,666,009 is rounded off to 32,000,000 because the 6 in the hundred thousands place is 5 or more
• 27,981,987 is rounded off to 28,000,000 because the 9 in the hundred thousands place is less than 5
• 92,666,009 is rounded off to 93,000,000 because the 6 in the hundred thousands place is 5 or more
• 823,266,009 is rounded off to 823,000,000 because the 2 in the hundreds thousands place is less than 5

The Positions of Numbers and Their Meanings: Decimal Places As previously discussed, positional notations are used in whole numbers, decimal numbers and combination whole numbers with decimal places. Positional notations are used in whole numbers, decimal numbers and combination whole numbers with decimal places. These positional notations occur on both sides of a decimal point. The numbers to the left of the decimal point are whole numbers and the numbers on the right side of the decimal point are decimal numbers.

As you can see in the table above, the decimal number positions to the right of the decimal place, in the correct sequential order are:

• Tenths
• Hundredths
• Thousandths
• Ten thousandths
• Hundred thousandths
• Millionths

Here are some examples:

Number: 0.67

Positional notations:

• 6 is in the tenths place
• 7 is in the hundredths place

Number: 0.967

Positional notations:

• 9 is in the tenths place
• 6 is in the hundredths place
• 7 is in the thousandths place

Number: 0.9672

Positional notations:

• 9 is in the tenths place
• 6 is in the hundredths place
• 7 is in the thousandths place
• 2 is in the ten thousandths place

Number: 0.96728

Positional notations:

• 9 is in the tenths place
• 6 is in the hundredths place
• 7 is in the thousandths place
• 2 is in the ten thousandths place
• 8 is in the hundred thousandths place

Number: 0.129672

Positional notations:

• 1 is in the tenths place
• 2 is in the hundredths place
• 9 is in the thousandths place
• 6 is in the ten thousandths place
• 7 is in the hundred thousandths place
• 2 is in the millionth place

When you have to round off to the nearest hundredth, you must look at the next number, or thousandths, and determine if it is less than 5, equal to 5 or more than 5. In the examples below, the tenth place will be highlighted with green, the hundredth place will be highlighted in pink, the thousandths place will be highlighted in orange and the ten thousandths place will be highlighted in turquoise so you can readily see and distinguish among these decimal places.

The number 45.7589 is 45 and 7589 ten thousandths.

This number rounded off to the nearest thousandth is 45.759 because the number in the ten thousandth place is 5 or more. You round off a number to the nearest thousandth by looking at the number in the next place, which is the ten thousandth place. If this number is five or more, you would increase the 8 in the thousandth place to 9. If the number in the ten thousandth place had been less than 5, this number would be rounded to the nearest thousandth as 45.758; the 8 would not be increased by one when the number in the ten thousandth place is less than 5.

Now, here are some numbers rounded off to the nearest tenth. Remember, if the hundredth place, or 2nd number after the decimal, is 5 or more, the tenth place is increased by 1; and if the 2nd number after the decimal is less than 5, the number in the tenth place remains the same.

Here are some numbers rounded off to the nearest tenth. Please note the pink number in the hundredth place (2 numbers after the decimal) is the one that determines whether or not the number in the tenth place (the first number after the decimal) moves up 1 or remains the same. If the pink number in the hundredth place is 5 or more, the number in the tenth place is increased by 1; and if the number in the hundredth place is less than 5, the number in the tenths place remains the same.

• 3.44 = 3.4
• 4.01 = 4.0 which is simply written as 4 without a decimal at all
• 78.09 = 78.1
• 8.67 = 8.7
• 22.09 = 22.1
• 99.22 = 99.2
• 987.53 = 987.5

Here are some numbers rounded off to the nearest hundredth. Please note the orange number in the thousandth place (3 numbers after the decimal) is the one that determines whether or not the number in the hundredths place (the second number after the decimal) moves up 1 or remains the same. If it is 5 or more, the number in the hundredths place is increased by 1; and if the number in the thousandth place is less than 5, the number in the hundredths place remains the same.

• 3.447 = 3.45
• 4.011 = 4.01
• 78.099= 78.1
• 8.675 = 8.68
• 22.092 = 22.09
• 99.229 = 99.23
• 987.532 = 987.53

Here are some numbers rounded off to the nearest thousandth. Please note the turquoise number in the ten thousandth place (4 numbers after the decimal) is the one that determines whether or not the number in the thousandths place (the third number after the decimal) moves up 1 or remains the same. If it is 5 or more, the number in the thousandths place is increased by 1; and if the number in the ten thousandth place is less than 5, the number in the thousandths place remains the same.

• 3.4478 = 3.448
• 4.0111 = 4.011
• 78.0999= 78.1
• 8.6754 = 8.675
• 22.0923 = 22.092
• 99.2299 = 99.23
• 987.5328 = 987.533

The Positions of Numbers and Their Meanings: Whole Numbers With Decimal Places As previously discussed, positional notations are used in whole numbers, decimal numbers and combination whole numbers with decimal places. Positional notations are used in whole numbers, decimal numbers and combination whole numbers with decimal places. These positional notations occur on both sides of a decimal point. The numbers to the left of the decimal point are whole numbers and the numbers on the right side of the decimal point are decimal numbers.

There are many occasions when a whole number with decimal places has to be rounded off. In order to perform this rounding off accurately, it is highly important that you clearly differentiate between ten and tenths, hundreds and hundredths, thousands and thousandths, ten thousands and ten thousandths, hundred thousands and hundred thousandths, millions and millionths. For each of these pairs, the first place, for example hundreds and thousands, refer to the positional locations to the left of the decimal; and the second place in these pairs, like the hundredths and thousandths, indicate positional places to the right of the decimal.

For example, if you are asked to round off 67.91 to the nearest tens, you would round this number of to 70 because the 7 in the ones place is equal to or more than 5 and, if, you are asked to round 67.91 to the nearest ones, you would look at the tenths place to the right of the decimal point. Because this tenths place is 9 and 9 is equal to or more than 5, you would round up the 7 in the ones place to 8. The answer is now 68 rounded up to the nearest ones.

Here are some other examples:

Example 1:

Round off this number: 345.987 to the nearest hundred. You will leave the 3 in the hundreds place as 3 because the number 4 in the tens place is less than 5.

Example 2:

Round off this number: 3,653.987 to the nearest thousand. You will increase the 3 in the thousand place to 4 because the number 6 in the hundreds place is equal to or more than 5.

Example 3:

Round off this number: 345.987 to the nearest hundred. You will leave the 3 in the thousand place because the number 4 in the hundreds place is less than 5.