Estimation
An estimateAn answer to a problem that is close to the exact number, but not necessarily exact. is an answer to a problem that is close to the solution, but not necessarily exact. Estimating can come in handy in a variety of situations, such as buying a computer. You may have to purchase numerous devices: a computer tower and keyboard for `$1,295`, a monitor for `$679`, the printer for `$486`, the warranty for `$196`, and software for `$374`. Estimating can help you know about how much you’ll spend without actually adding those numbers exactly.
Estimation usually requires roundingFinding a number that’s close to a given number, but is easier to think about.. When you round a number, you find a new number that’s close to the original one. A rounded number uses zeros for some of the place values. If you round to the nearest ten, you will have a zero in the ones place. If you round to the nearest hundred, you will have zeros in the ones and tens places. Because these place values are zero, adding or subtracting is easier, so you can find an estimate to an exact answer quickly.
It is often helpful to estimate answers before calculating them. Then if your answer is not close to your estimate, you know something in your problem-solving process is wrong.
Suppose you must add a series of numbers. You can round each addend to the nearest hundred to estimate the sum.
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Example |
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Problem
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Estimate the sum `1,472 + 398 + 772 + 164` by rounding each number to the nearest hundred. |
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First, round each number to the nearest hundred.
Then, add the rounded numbers together.
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Answer The estimate is `2,900`. |
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In the example above, the exact sum is `2,806`. Note how close this is to the estimate, which is `94` greater.
In the example below, notice that rounding to the nearest ten produces a far more accurate estimate than rounding to the nearest hundred. In general, rounding to the lesser place value is more accurate, but it takes more steps.
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Example |
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Problem
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Estimate the sum `1,472 + 398 + 772 + 164` by first rounding each number to the nearest ten. |
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First, round each number to the nearest ten.
Next, add the ones and then the tens. Here, the sum of `7`, `7`, and `6` is `20`. Regroup.
Now, add the hundreds. The sum of the digits in the hundreds place is `18`. Regroup.
Finally, add the thousands. The sum in the thousands place is `2`.
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Answer The estimate is `2,800`. |
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Note that the estimate is `2,800`, which is only `6` less than the actual sum of `2,806`.
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In three months, a freelance graphic artist earns `$1,290` for illustrating comic books, `$2,612` for designing logos, and `$4,175` for designing web sites. Estimate how much she earned in total by first rounding each number to the nearest hundred.
A) `$8,200`
B) `$7,900`
C) `$8,000`
D) `$8,100`
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You can also estimate when you subtract, as in the example below. Because you round, you do not need to subtract in the tens or hundreds places.
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Example |
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Problem
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Estimate the difference of `5,876` and `4,792` by first rounding each number to the nearest hundred. |
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First, round each number to the nearest hundred.
Subtract. No regrouping is needed since each number in the minuend is greater than or equal to the corresponding number in the subtrahend.
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Answer |
The estimate is `1,100`. |
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The estimate is `1,100`, which is `16` greater than the actual difference of `1,084`.
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Estimate the difference of `474,128` and `262,767` by rounding to the nearest thousand.
A) `212,000`
B) `211,000`
C) `737,000`
D) `447,700`
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Estimating is handy when you want to be sure you have enough money to buy several things.
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Example |
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Problem
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When buying a new computer, you find that the computer tower and keyboard cost `$1,295`, the monitor costs `$679`, the printer costs `$486`, the `2`-year warranty costs `$196`, and a software package costs `$374`. Estimate the total cost by first rounding each number to the nearest hundred. |
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First, round each number to the nearest hundred.
Add.
After adding all of the rounded values, the estimated answer is `$3,100`.
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Answer The total cost is approximately `$3,100`. |
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Estimating can also be useful when calculating the total distance one travels over several trips.
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Example |
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Problem
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James travels `3,247` meters to the park, then travels `582` meters to the store. He then travels `1,634` meters back to his house. Find the total distance traveled by first rounding each number to the nearest ten. |
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First, round each number to the nearest ten.
Adding the numbers in the tens place gives `16`, so you need to regroup.
Adding the numbers in the hundreds place gives `14`, so regroup.
Adding the numbers in the thousands place gives `5`.
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Answer The total distance traveled was approximately `5,460` meters. |
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In the example above, the final estimate is `5,460` meters, which is `3` less than the actual sum of `5,463` meters.
Estimating is also effective when you are trying to find the difference between two numbers. Problems dealing with mountains like the example below may be important to a meteorologist, a pilot, or someone who is creating a map of a given region. As in other problems, estimating beforehand can help you find an answer that is close to the exact value, preventing potential errors in your calculations.
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Example |
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Problem
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One mountain is `10,496` feet high and another mountain is `7,421` feet high. Find the difference in height by first rounding each number to the nearest `100`. |
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First, round each number to the nearest hundred.
Then, align the numbers and subtract.
The final estimate is `3,100`, which is `25` greater than the actual value of `3,075`.
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Answer The estimated difference in height between the two mountains is `3,100` feet. |
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A space shuttle traveling at `17,581` miles per hour decreases its speed by `7,412` miles per hour. Estimate the speed of the space shuttle after it has slowed down by rounding each number to the nearest hundred.
A) `10,100` miles per hour
B) `10,200` miles per hour
C) `25,000` miles per hour
D) `25,100` miles per hour
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Estimation is very useful when an exact answer is not required. You can use estimation for problems related to travel, finances, and data analysis. Estimating is often done before adding or subtracting by rounding to numbers that are easier to think about. Following the rules of rounding is essential to the practice of accurate estimation.