How To Find Upper And Lower Bound

Information technology is very common to run into a customer and a seller bargaining on the price that should exist paid for an item. No affair how good the customer's negotiating skill is, the seller would not sell the item below a specific amount. You can telephone call that specific amount the lower jump. The customer has an amount in mind likewise and is non willing to pay in a higher place that. You tin phone call this amount the upper bound.

This aforementioned concept is applied in mathematics. There is a limit in which a measurement or value cannot become beyond and above. In this article, we will acquire about lower and upper-spring limits of accuracy, their definition, rules, and formulas, and run into examples of their applications.

Lower and Upper bounds definition

The lower jump (LB) refers to the everyman number that tin be rounded to become an estimated value.

The upper bound (UB) refers to the highest number that tin be rounded to get an estimated value.

Another term that you'll come across in this topic is mistake interval.

Error intervals show the range of numbers that are within the limits of accurateness. They are written in the form of inequalities.

The lower and upper bounds tin can besides be chosen the limits of accuracy.

Consider a number 50 rounded to the nearest 10.

Many numbers can be rounded to get 50, but the lowest is 45. This means that the lower bound is 45 because it is the lowest number that can be rounded to get 50.

The upper spring is 54 because it is the highest number that can be rounded to go l.

As explained earlier, the lower and upper bound tin exist found by just figuring out the lowest and highest number that can be rounded to get the estimated value, merely in that location is a simple process that yous can follow to achieve this. The steps are below.

ane. You should beginning know the degree of accuracy, DA.

The degree of accuracy is the measure to which a value is rounded.

2. Split the caste of accuracy by two,

.

3. Add what y'all got to the value to get the upper bound, and decrease to get the lower bound.

Rules and formulas for upper and lower bounds

You may come across questions involving formulas, and you will have to work with multiplication, sectionalisation, add-on, and subtraction. In cases like this, you lot have to follow some rules to get the correct answers.

For Add-on.

This unremarkably happens when nosotros have a value that undergoes an increment. We then accept an original value and its range of increase.

When you accept a question involving addition, exercise the following:

1. Find the upper and lower bounds of the original value, UB_value, and of its range of increase, UB_range.

2. Use the post-obit formulas to find the upper and lower bounds of the answer.

3. Considering the bounds, determine on a suitable degree of accurateness for your respond.

For Subtraction.

This usually happens when we have a value that undergoes a subtract. We and then have an original value and its range of decrease.

When y'all have a question involving subtraction, exercise the following.

one. Find the upper and lower bounds of the original value, UB _value , and of its range of increment, UB _range.

2. Utilize the following formulas to discover the upper and lower bounds of the answer.

3. Considering the bounds, decide on a suitable degree of accuracy for your answer.

For Multiplication.

This usually happens when we have quantities that involve the multiplication of other quantities, such as areas, volumes, and forces.

When you lot have a question involving multiplication, do the post-obit.

1. Observe the upper and lower premises of the numbers involved. Allow them be quantity 1, q1, and quantity 2, q2.

2. Use the post-obit formulas to notice the upper and lower bounds of the answer.

3. Considering the bounds, determine on a suitable degree of accuracy for your respond.

For Partitioning.

Similarly to the multiplication, this usually happens when nosotros have a quantity that involves the sectionalization of other quantities, such as velocity, and density.

When you have a question involving division, practise the following.

1. Find the upper and lower premises of the numbers involved. Let's denote them quantity 1, q1, and quantity 2, q2.

2. Apply the following formulas to find the upper and lower premises of the reply.

3. Considering the bounds, decide on a suitable degree of accuracy for your answer.

Upper and Lower bounds examples

Let'south take some examples.

Find the upper and lower jump of the number 40 rounded to the nearest x.

Solution.

There are lots of values that could be rounded to 40 to the nearest 10. It can be 37, 39, 42.5, 43, 44.9, 44.9999, and so on.

Merely the lowest number which will be the lower jump is 35 and the highest number is 44.4444, so nosotros will say the upper bound is 44.

Let's call the number that we get-go with, 40, . The mistake interval will be:

This means x tin can be equal to or more than than 35, but less than 44.

Allow's take another example, now following the steps we've mentioned earlier.

The length of an object y is 250 cm long, rounded to the nearest 10 cm. What is the mistake interval for y?

Solution.

To know the error interval, yous have to kickoff notice the upper and lower bound. Let's apply the steps nosotros mentioned earlier to get this.

Step i: First, we have to know the degree of accuracy, DA. From the question, the degree of accuracy is DA = x cm.

Step 2: The next step is to divide it by 2.

Step three: We will now subtract and add 5 to 250 to go the lower and upper bound.

The error interval volition be:

This ways that the length of the object can exist equal to or more than than 245 cm, but less than 255 cm.

Let's have an example involving addition.

The length of a rope x is 33.vii cm. The length is to be increased by 15.five cm. Because the bounds, what volition exist the new length of the rope?

Solution.

This is a case of add-on. Then, post-obit the steps for addition above, the first matter is to observe the upper and lower premises for the values involved.

Footstep 1: Let'south start with the original length of the rope.

The lowest number that can be rounded to 33.7 is 33.65, pregnant that 33.65 is the lower spring, LB _value.

The highest number is 33.74, simply we will use 33.75 which can be rounded down to 33.vii, UB _value.

And so, we can write the error interval every bit:

We volition do the same for 15.5 cm, let's denote it y.

The lowest number that tin be rounded to xv.v is 15.45 meaning that 15.45 is the lower bound, LB _range.

The highest number is 15.54, but we will use 15.55 which can exist rounded down to 15.5, UB _range.

Then, we can write the error interval as:

Stride 2: We volition employ the formulas for finding upper and lower bounds for addition.

We are to add both upper premises together.

The lower bound is:

Footstep iii: We at present have to decide what the new length will exist using the upper and lower bound nosotros simply calculated.

The question nosotros should be asking ourselves is to what degree of accuracy does the upper and lower bound round to the same number? That will exist the new length.

Well, we take 49.3 and 49.1 and they both circular to 49 at one decimal place. Therefore, the new length is 49 cm.

Let'due south take another example involving multiplication.

The length L of a rectangle is 5.74 cm and the latitude B is 3.iii cm. What is the upper bound of the expanse of the rectangle to 2 decimal places?

Solution.

Step 1: First thing is to get the error interval for the length and breadth of the rectangle.

The lowest number that can be rounded to the length of 5.74 is v.735 meaning that v.735 is the lower bound, LB_value.

The highest number is 5.744, simply we will employ five.745 which tin exist rounded down to v.74, UB_value.

So, we can write the fault interval as:

The lowest number that can exist rounded to the latitude of 3.3 is 3.25 meaning that 3.25 is the lower bound.

The highest number is three.34, merely we will utilise 3.35, so we can write the error interval every bit:

The area of a rectangle is:

Step 2: And then to get the upper bound, we will apply the upper jump formula for multiplication.

Step 3: The question says to go the respond in two decimal places. Therefore, the upper bound is:

Permit's have some other example involving division.

A man runs 14.8 km in four.25 hrs. Find the upper and lower bounds of the man's speed. Requite your respond in 2 decimal places.

Solution

We are asked to observe the speed, and the formula for finding speed is:

Step ane: Nosotros volition starting time notice the upper and lower bounds of the numbers involved.

The distance is 14.8 and the lowest number that can be rounded to 14.eight is 14.75 meaning that 14.75 is the lower bound, LB_d.

The highest number is 14.84, but we will use xiv.85 which can be rounded down to 14.viii, UB_d.

So, we tin write the mistake interval equally:

The speed is 4.25 and the lowest number that can be rounded to iv.25 is iv.245 meaning that 4.245 is the lower bound, LB_t.

The highest number is 4.254, just we volition apply four.255 (which tin be rounded down to 4.25), UB_t, then nosotros can write the error interval as:

Step 2: We are dealing with partitioning hither. So, we will utilise the division formula for calculating the upper and lower bound.

The lower spring of the man's speed is:

is the symbol for approximation.

Step iii: The answers for the upper and lower bound are approximated because nosotros are to requite our answer in 2 decimal places.

Therefore, the upper and lower leap for the man's speed are 3.50 km/hr and 0.47 km/hr respectively.

Let's take 1 more example.

The peak of a door is 93 cm to the nearest centimetre. Find the upper and lower premises of the height.

Solution.

The starting time stride is to determine the degree of accuracy. The degree of accuracy is to the nearest one cm.

Knowing that the side by side step is to split up by two.

To find the upper and lower bound, we volition add and subtract 0,5 from 93 cm.

The Upper jump is:

The Lower spring is:

Lower and Upper bound limits of accuracy - Primal takeaways

• The lower jump refers to the lowest number that tin can exist rounded to get an estimated value.
• The upper jump refers to the highest number that can be rounded to become an estimated value.
• Fault intervals show the range of numbers that are within the limits of accuracy. They are written in the grade of inequalities.
• The lower and upper premises tin also be called the limits of accurateness .

Source: https://www.studysmarter.co.uk/explanations/math/pure-maths/lower-and-upper-bounds/

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