Algebra and Bounds
Neil Trivedi
Teacher
Algebra and Bounds
In the previous note, Error Intervals, we looked at how to find error intervals for rounded or truncated values and how we can use these in bound calculations for real life scenarios.
Similarly, in this note, we will use bounds in algebraic formulae. The main idea is to decide which combination of bounds gives the smallest or greatest possible value.
For example, if
and we would like to make as large as possible given that and have been rounded.
Here, we need to decide if we want and to be as large or as small as possible.
Let’s think about this in words first:
If we want to be then we must first start with a number. Then we must decide if we want to divide this number by a or number to leave us with as large of a number as possible.
The way to do this would be to take a number and divide it with as of a number as possible.
Therefore,
which in terms of our original equation is
and now, in terms of upper bounds and lower bounds
The following is true when trying to find the largest possible value given the four operations and that and are positive:
If then
If then
If then
If then
For all the lower bound calculations, swap with everywhere.
Example 1:
Given the formula,
If correct to the nearest unit, to the nearest tenth and to decimal places, find the upper and lower bounds of correct to significant figures.
Step 1: Find the lower and upper bounds of each value.
1) Since to the nearest unit, the error here is which when divided by gives
Therefore,
the lower bound of is
the upper bound of is
2) Since to the nearest tenth, the error here is which when divided by gives
Therefore,
the lower bound of is
the upper bound of is
3) Since to two decimal places, the error here is (the second decimal’s place value) which when divided by gives
Therefore,
the lower bound of is
the upper bound of is
Step 2: Find the upper bound of by considering whether you want and to be as large or as small as possible.
Thinking about this in terms of and first helps us plan better.
First, let’s think about the overall fraction before we focus in on the numerator and denominator individually.
However, the denominator is a subtraction, so we now need to consider what would make that subtraction as small as possible.
So, we can now rewrite our fraction:
and now in terms of and
Substituting our values:
Step 3: Find the lower bound of by considering whether you want and to be as large or as small as possible.
Like with the upper bound, thinking about this in terms of and first helps us plan better.
First, let’s think about the overall fraction before we focus in on the numerator and denominator individually.
The denominator is a subtraction, so we now need to consider what would make that subtraction as big as possible.
So, we can now rewrite our fraction:
and now in terms of and
Substituting our values:
Therefore, the upper and lower bound of are and respectively, to significant figures.
Note: Once you’ve worked out how to find one bound (upper bound in step 2 in this case), you can just swap all the and terms (so, the terms replace the terms and vice versa) to find the other bound.
Finding a Value to a Suitable Degree of Accuracy
Sometimes, the question asks us to find a value to a suitable degree of accuracy. This means we need to find the lower and upper bounds of the answer, then look at how accurately they agree.
Example 2:
Given the formula,
where correct to decimal places and to the nearest tenth. By considering bounds, work out the value of to a suitable degree of accuracy. Give a reason for your answer.
Step 1: Find the lower and upper bounds of and
1) Since correct to decimal places, the error here is (the third decimal’s place value) which when divided by gives
Therefore,
the lower bound of is
the upper bound of is
2) Since to the nearest tenth, the error here is which when divided by gives
Therefore,
the lower bound of is
the upper bound of is
Step 2: Find the upper and lower bound of by considering whether you want and to be as large or as small as possible.
Thinking about this in terms of and first helps us plan better.
Even though we have a square root in the fraction, they are just a distraction as there are only single terms in the numerator and denominator.
For the upper bound:
Now, for the lower bound:
The final step is to find the maximum degree of accuracy that both the upper bound and lower bound agree to.
To decimal place, both the upper and lower bounds round to
Note: While both bounds agree to decimal place, our aim is to be as accurate as possible, so we need to check if the bounds would agree to a greater degree of accuracy before giving our answer.
To decimal places, both the upper and lower bounds round to
To decimal places, the upper bound rounds to while the lower bound rounds to and so, they do not agree with each other.
Therefore, correct to decimal places since both the upper and lower bound agree to that degree of accuracy.
Note: You can also compare the degrees of accuracy using significant figures. In this example they would both agree to significant figures.
Challenging Questions