The Significance of Upper and Lower Bounds in Measurements

When measuring with a ruler, we may record a string's length as 8.2 cm. However, due to limitations of the ruler, the actual measurement could be 8.23 cm, making 8.2 cm the limit of accuracy with a precision of 0.1 cm. In this article, we will explore the concept of limits of accuracy and how it affects measurements in different scenarios. Let's start by defining what limits of accuracy are exactly.

Understanding Limits of Accuracy

Limits of accuracy refer to the possible range of values a measurement may have based on its rounded value. For instance, if a room's length is recorded as 10 meters with a tolerance of 1 meter, the accuracy is limited to 1 meter. Similarly, if someone's weight is rounded to the nearest 5 kg at 65 kg, the degree of accuracy is 5 kg.

In situations where a measurement is recorded with decimals, such as 10.2 cm rounded to the nearest 0.1 cm or £110 rounded to the nearest £10, the degree of accuracy is determined by the smallest unit of measure.

For example, if we agree to meet at 10 am, it could mean any time between 9:50 am and 10:10 am. This range, also known as the lower and upper limits, defines the degree of accuracy for the given time. Any time outside this range would be considered early or late.

Calculating Upper and Lower Bounds

To determine the limits of accuracy, we use a simple formula to calculate the upper and lower bounds. First, let's define what these terms mean.

The lower bound is the smallest possible value the measurement could be when rounded, while the upper bound is the largest possible value. The formula for the upper bound is the degree of accuracy divided by two plus the actual measurement, while the lower bound is determined by subtracting the degree of accuracy divided by two from the measurement.

An Example of Finding Upper and Lower Bounds

Let's take a look at an example to better understand how to find upper and lower bounds.

If a bag of oranges is recorded as weighing 3 kg, rounded to the nearest kilogram, what could be the upper and lower bounds for the weight of the oranges?

The oranges could weigh 2.8 kg, 2.6 kg, or any weight less than 3 kg. To calculate the lower bound, we subtract half of the degree of accuracy, which is 1 kg, from 3 kg, giving us a lower bound of 2.5 kg. Calculating the upper bound is a bit more complicated. We know that 3 kg rounds up to 4 kg, but there is no single largest value that would round down to 3 kg. This can be seen with 3.5 kg, which rounds up to 4 kg. As a result, the upper bound is 3.5 kg, and the lower bound is 2.5 kg.

Illustrating Limits of Accuracy

Let's consider a common question about limits of accuracy that one may come across in a GCSE exam.

If the average mass of an apple is recorded as 175 g, rounded to the nearest gram, what could be the smallest and largest possible masses for the apple?

The accuracy is limited to 1 gram in this case. To calculate the lower bound, we subtract half of the degree of accuracy, which is 0.5 g, from 175 g, giving us a lower bound of 174.5 g. Similarly, we add 0.5 g to 175 g to get an upper bound of 175.5 g. Using <, we can represent the range of possible values as an error interval, indicating that the mass of the apple cannot be exactly 175.5 g but can be any value less than that.

The Importance of Limits of Accuracy and Error Intervals

An error interval refers to the potential range of values a measurement can have, expressed through an inequality. This concept can be better understood through examples.

For instance, imagine a car traveling at a speed of 70 mph, rounded to the nearest 10 mph.

What are the Maximum and Minimum Possible Speeds of a Car?

Solution: The degree of accuracy for this measurement is 10 mph. By dividing this value in half, we get 5 mph. Adding 5 mph to 70 mph, we get the upper bound of 75 mph. Subtracting 5 mph from 70 mph, we get the lower bound of 65 mph. If we define the speed of the car as x mph, the possible speeds fall within the range of x ≤ 75 and x ≥ 65.

Applying Limits of Accuracy: This concept is often used in GCSE-style questions, where the limits of accuracy play a key role in solving problems.

For instance, let's say Matthew measures the height and base of a triangle. He finds the height to be 16 cm rounded to the nearest centimeter and the base to be 20 cm rounded to the nearest 10 centimeters. What is the maximum possible area for the triangle?

Solution: We can use the formula for the area of a triangle: A = ½bh.

The maximum possible area can be achieved by using the highest possible values for both the base and height. The upper bound for the height is 16.5 cm, and the upper bound for the base is 25 cm. Thus, the maximum possible area is ½(25)(16.5) = 206.25 cm².

Now, let's consider Tony who went on a 40-mile bike ride rounded to the nearest 10 miles. He traveled at a constant speed of 15 mph rounded to the nearest mph. What is the minimum time it took him to complete the bike ride?

Solution: Rearranging the formula for distance, d=rt, we get t=d/r. In order to minimize the time, we need to divide the smallest distance by the largest speed. The minimum distance is 35 miles (since 40 has been rounded to the nearest 10) and the maximum speed is 15.5 mph. Thus, the minimum time is 2.26 hours.

Next, let's imagine a triangle ABC with side AB equal to 10 cm and angle ACB equal to 34 degrees (rounded to the nearest degree).

Solution: We can use trigonometry to determine the upper and lower bounds for the length of AB. If you are unfamiliar with trigonometry, we recommend reviewing it before proceeding.

To find the lower and upper bounds, we must use the trigonometric ratio tan. Recall that tan = opposite/adjacent. In this case, we are trying to find the opposite side, so we can say that tan 34 = AB/10.

For the lower bound, we need to find the values of AB and 34 that result in the smallest possible value for the opposite side. This is achieved when AB = 9.5 cm and 34 is the lower bound for angle ACB. Thus, the lower bound for the opposite side is 5.503 cm.

Similarly, for the upper bound, we need to find the values of AB and 34 that result in the largest possible value. This is achieved when AB = 10.5 cm and 34 is the upper bound for angle ACB. Thus, the upper bound for the opposite side is 6.402 cm.

Therefore, the lower bound for side AB is 5.503 cm and the upper bound is 6.402 cm.

Limits of Accuracy: Key Takeaways

The limit of accuracy of a measurement refers to the range of possible values a measurement can be when rounded. The lower bound is the smallest possible value, while the upper bound is the largest possible value. Inequalities are used to define the error interval of a measurement.

Frequently Asked Questions About Limits of Accuracy

• What is the formula for finding the limit of accuracy?
  The lower bound can be calculated by halving the degree of accuracy and then subtracting this from the measurement. Similarly, the upper bound can be calculated by halving the degree of accuracy and adding this to the measurement.
• Why do we use limits of accuracy?
  Limits of accuracy indicate that a value has been rounded and can potentially have a range of possible values.
• How do you calculate the limits of accuracy?
  To find the lower bound, half the degree of accuracy and then subtract this from the measurement. To find the upper bound, half the degree of accuracy and then add this from the measurement.
• What is limits of accuracy?
  Limits of accuracy refer to the degree of precision in a measurement. For example, if we round a measurement to the nearest 0.1 cm, we are measuring with a limit of accuracy of 0.1 cm.
• When do we apply limits of accuracy?
  Limits of accuracy are applied when a measurement is rounded, indicating that the actual value can fall within a range of possible values.Tips for Accurately Rounding Answers
• As students, it's common for us to round our solutions to whole numbers or decimal places when completing assignments or solving problems. However, it's crucial to consider the accuracy of our rounded answer. So how can we determine if our answer is precise enough? This is where finding the range of a rounded answer becomes essential.
• When a number is rounded, it is no longer the exact value. For instance, if we round 3.75 to the nearest whole number, we get 4. However, the true value could range from 3.5 to 4.49. This means that our rounded answer falls within a range of possible values. To ensure accuracy, we need to look at the smallest and largest possible values within this range.
• To determine the smallest possible value, we subtract half of the rounding increment from our rounded answer. For example, if we rounded to the nearest whole number, we would subtract 0.5. This gives us the closest value below our rounded answer. Similarly, to find the largest possible value, we add half of the increment to our rounded answer. This gives us the closest value above our rounded answer.
• Let's take an example. If our rounded answer is 28, and we rounded to the nearest ten, we need to subtract 5 to find the smallest possible value. This means that the true value could be as low as 23. On the other hand, we must add 5 to find the largest possible value, which could be as high as 33. By considering this range, we can determine if our rounded answer is accurate enough for the problem.
• In summary, when rounding an answer, it's crucial to consider the range of potential values to ensure accuracy. By finding the smallest and largest possible values, we can determine if our rounded answer is appropriate for the problem at hand. Remember to always double-check the range of your rounded answer to avoid any errors in your calculations. Keeping these tips in mind will help you accurately round your answers and achieve success in your assignments or problem-solving tasks.