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1.2 H) Truncating – Upper & Lower Bounds
1.2 H) Truncating – Upper & Lower Bounds
In this section we are going to look at upper & lower bound questions with respect to numbers that have been truncated. Before we go through some of these questions, let’s just remind ourselves of how to truncate numbers. There are two steps involved in truncating numbers:
Truncating numbers is essentially just rounding down. Age and time on your phone are examples of numbers that are truncated. There is more information on truncating numbers in the truncation section (click here to be taken to the truncation section).
We are now going to have a few examples where we find the upper and lower bound for a number that has been truncated.
- Find the place value that we are truncating to and place a line to the right of this place value.
- We then replace all of the numbers to the right of the line and before the decimal place with a 0, and get rid of all of the numbers in the decimal positions on the right of the line.
Truncating numbers is essentially just rounding down. Age and time on your phone are examples of numbers that are truncated. There is more information on truncating numbers in the truncation section (click here to be taken to the truncation section).
We are now going to have a few examples where we find the upper and lower bound for a number that has been truncated.
Example 1
A number that has been truncated to 1 decimal place is 47.3. What is the upper and lower bound?
The lower bound for a number that has been truncated will always be the truncated number. For this question, the truncated number was 47.3, and this means that the lower bound for the truncated number is 47.3.
We find the upper bound for the truncated number by adding 1 onto the digit in the place value that the number has been truncated to. For this question, the number was truncated to one decimal place (the tenth) and the truncated number was 47.3. This means that we add 1 to the digit in the tenth place value for 47.3; we add 1 to the 3 in 47.3 to get 47.4. The upper bound is 47.4.
A number that has been truncated to 1 decimal place is 47.3. What is the upper and lower bound?
The lower bound for a number that has been truncated will always be the truncated number. For this question, the truncated number was 47.3, and this means that the lower bound for the truncated number is 47.3.
We find the upper bound for the truncated number by adding 1 onto the digit in the place value that the number has been truncated to. For this question, the number was truncated to one decimal place (the tenth) and the truncated number was 47.3. This means that we add 1 to the digit in the tenth place value for 47.3; we add 1 to the 3 in 47.3 to get 47.4. The upper bound is 47.4.
Questions like this can be extended by being asked to find the error interval, which shows the range of possible values a truncated number (or rounded number) can be. The error interval for truncated numbers (or rounded numbers) always follows the rule below (the number that has been truncated (or rounded) is x):
We now replace the lower bound (LB) with 47.3 and replace the upper bound (UB) with 47.4. The error interval for our truncated number (x) is:
The inequality sign on the left is a “less than or equal to” sign, and this is because 47.3 when truncated to 1 decimal place is 47.3.
The inequality sign on the right does not include “or equal to”, and this is because when 47.4 is truncated to one decimal place the outcome is 47.4.
If you are ever unsure whether you have the correct error interval for numbers that have been truncated (or rounded), you can choose a few values within your error interval and truncate (or round) them to see whether you obtain the truncated (or rounded) number. For example, I am going to choose 47.312 and 47.387. The process for truncating these numbers is shown below:
The inequality sign on the right does not include “or equal to”, and this is because when 47.4 is truncated to one decimal place the outcome is 47.4.
If you are ever unsure whether you have the correct error interval for numbers that have been truncated (or rounded), you can choose a few values within your error interval and truncate (or round) them to see whether you obtain the truncated (or rounded) number. For example, I am going to choose 47.312 and 47.387. The process for truncating these numbers is shown below:
Both of these numbers truncate to 47.3, which strongly implies that we have found the correct error interval.
Example 2
The number 34,700 has been truncated to 3 significant figures. Find the error interval for the number. Let the number be y.
The first step in answering this question is to find the place value that the number has been truncated to. We were told in the question that the number has been truncated to 3 significant figures and the third significant figure for the number 34,700 is the hundreds place value; the number has been truncated to the hundreds.
The lower bound for a truncated number is always the truncated number. This means that the lower bound is 34,700.
We find the upper bound for truncated numbers by adding 1 to the digit in the place value that we have truncated to. We truncated to the hundreds, which means that we add 1 to the digit in the hundreds place value for 34,700; we add 1 to the 7 in 34,700 to get 34,800. The upper bound is 34,800.
We now place the values for the lower bound (34,700) and upper bound (34,800) into the error interval formula. The error interval is:
The number 34,700 has been truncated to 3 significant figures. Find the error interval for the number. Let the number be y.
The first step in answering this question is to find the place value that the number has been truncated to. We were told in the question that the number has been truncated to 3 significant figures and the third significant figure for the number 34,700 is the hundreds place value; the number has been truncated to the hundreds.
The lower bound for a truncated number is always the truncated number. This means that the lower bound is 34,700.
We find the upper bound for truncated numbers by adding 1 to the digit in the place value that we have truncated to. We truncated to the hundreds, which means that we add 1 to the digit in the hundreds place value for 34,700; we add 1 to the 7 in 34,700 to get 34,800. The upper bound is 34,800.
We now place the values for the lower bound (34,700) and upper bound (34,800) into the error interval formula. The error interval is:
If we wanted to check whether we have obtained the correct error interval, we could choose a few values within the error interval and check whether they truncate to the 34,700. I am not going to do this, but feel free to choose a few values to truncate and check that we have the correct error interval.