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Accuracy and Bounds: A Key Skill for IGCSE Mathematics Success


- How to find the upper and lower bounds of a number given to a certain accuracy? - How to use bounds to solve simple problems? H2: Finding bounds for numbers rounded to the nearest 10, 100 or 1000 - What is the rule for finding the upper and lower bounds of a number rounded to the nearest 10, 100 or 1000? - Examples and exercises with solutions H2: Finding bounds for numbers rounded to a certain number of decimal places - What is the rule for finding the upper and lower bounds of a number rounded to a certain number of decimal places? - Examples and exercises with solutions H2: Finding bounds for numbers rounded to a certain number of significant figures - What is the rule for finding the upper and lower bounds of a number rounded to a certain number of significant figures? - Examples and exercises with solutions H2: Using bounds to solve simple problems - How to find the upper and lower bounds of an expression involving addition, subtraction, multiplication or division? - How to use interval notation to represent the range of possible values for an expression? - Examples and exercises with solutions H1: Conclusion - Summary of the main points and key takeaways - Call to action and further resources Table 2: Article with HTML formatting Introduction




In this article, we are going to learn about limits of accuracy and how to use them in mathematics. Limits of accuracy are also known as bounds or errors. They help us to estimate how accurate a measurement or calculation is, and how much it could vary from the true value.




IGCSE Mathematics Limits of accuracy exercises Example.pdf 1



When we measure or calculate something, we often round the result to make it easier to work with. For example, if we measure the length of a pencil as 12.7 cm, we might round it to 13 cm. However, by doing this, we lose some information about the exact value. How do we know how close 13 cm is to the true length of the pencil?


This is where limits of accuracy come in. They tell us the smallest and largest possible values that would round to the given value. For example, if we round a number to the nearest 10, then the smallest possible value is 5 less than that number, and the largest possible value is 5 more than that number. These are called the lower bound and the upper bound respectively.


In this article, we will learn how to find the upper and lower bounds of a number given to a certain accuracy, such as the nearest 10, 100 or 1000, or a certain number of decimal places or significant figures. We will also learn how to use bounds to solve simple problems involving addition, subtraction, multiplication or division.


Finding bounds for numbers rounded to the nearest 10, 100 or 1000




Let's start with an easy case. Suppose we have a number that has been rounded to the nearest 10, such as 60. What are the smallest and largest possible values that would round to 60?


The rule is simple: add or subtract half of the rounding unit. In this case, the rounding unit is 10, so half of it is 5. Therefore, we add 5 to get the upper bound, and subtract 5 to get the lower bound.


The upper bound is 60 + 5 = 65.


The lower bound is 60 - 5 = 55.


This means that any number between 55 and 65 would round to 60 when rounded to the nearest 10. We can write this using inequality symbols as follows:


55 ≤ x < 65


We can also use interval notation to represent this range of values:


[55,65)


The square bracket means that 55 is included in the interval, while the round bracket means that 65 is not included. This is because 65 would round up to 70, not 60.


Here are some more examples of finding bounds for numbers rounded to the nearest 10:


  • 40: lower bound = 40 - 5 = 35, upper bound = 40 + 5 = 45, range = [35,45)



  • 90: lower bound = 90 - 5 = 85, upper bound = 90 + 5 = 95, range = [85,95)



  • 120: lower bound = 120 - 5 = 115, upper bound = 120 + 5 = 125, range = [115,125)



The same rule applies for numbers rounded to the nearest 100 or 1000. We just need to change the rounding unit and half of it accordingly. For example, if we have a number rounded to the nearest 100, such as 500, then the rounding unit is 100 and half of it is 50. Therefore, we add or subtract 50 to get the bounds.


The upper bound is 500 + 50 = 550.


The lower bound is 500 - 50 = 450.


This means that any number between 450 and 550 would round to 500 when rounded to the nearest 100. We can write this as:


450 ≤ x < 550


Or using interval notation:


[450,550)


Here are some more examples of finding bounds for numbers rounded to the nearest 100:


  • 300: lower bound = 300 - 50 = 250, upper bound = 300 + 50 = 350, range = [250,350)



  • 800: lower bound = 800 - 50 = 750, upper bound = 800 + 50 = 850, range = [750,850)



50 = 1450, range = [1350,1450)


Similarly, if we have a number rounded to the nearest 1000, such as 6000, then the rounding unit is 1000 and half of it is 500. Therefore, we add or subtract 500 to get the bounds.


The upper bound is 6000 + 500 = 6500.


The lower bound is 6000 - 500 = 5500.


This means that any number between 5500 and 6500 would round to 6000 when rounded to the nearest 1000. We can write this as:


5500 ≤ x < 6500


Or using interval notation:


[5500,6500)


Here are some more examples of finding bounds for numbers rounded to the nearest 1000:


  • 2000: lower bound = 2000 - 500 = 1500, upper bound = 2000 + 500 = 2500, range = [1500,2500)



  • 9000: lower bound = 9000 - 500 = 8500, upper bound = 9000 + 500 = 9500, range = [8500,9500)



  • 15000: lower bound = 15000 - 500 = 14500, upper bound = 15000 + 500 = 15500, range = [14500,15500)



Now that we have learned how to find bounds for numbers rounded to the nearest 10, 100 or 1000, let's try some exercises to practice our skills.


Exercises




Find the upper and lower bounds and the range of possible values for each of the following numbers rounded to the nearest 10, 100 or 1000.


  • 70



  • 400



  • 7000



  • 230



  • 16000



Solutions




  • 70: lower bound = 70 - 5 = 65, upper bound = 70 + 5 = 75, range = [65,75)



  • 400: lower bound = 400 - 50 = 350, upper bound = 400 + 50 = 450, range = [350,450)



  • 7000: lower bound = 7000 - 500 = 6500, upper bound = 7000 + 500 = 7500, range = [6500,7500)



  • 230: lower bound = 230 - 5 = 225 , upper bound = 230 + 5 = 235 , range = [ 225 , 235 )



  • 16000: lower bound = 16000 - 500 = 15500 , upper bound = 16000 + 500 = 16500 , range = [ 15500 , 16500 )



Finding bounds for numbers rounded to a certain number of decimal places




Sometimes we have numbers that have been rounded to a certain number of decimal places. For example, we might have a number like 3.14 that has been rounded to two decimal places. How do we find the upper and lower bounds for such numbers?


The rule is similar to the previous case. We add or subtract half of the last decimal place. In this case, the last decimal place is the hundredths place, so half of it is $$1 \over 200$$ or $$5 \over 1000$$ or $$50 \over 10000$$ or $$500 \over 100000$$ or $$500000 \over 100000000$$ or $$5 \times 10^ - n$$ where n is any positive integer. Therefore, we add or subtract this amount to get the bounds.


The upper bound is 3.14 + $$5 \times 10^ - n$$.


The lower bound is 3.14 - $$5 \times 10^ - n$$.


This means that any number between 3.14 - $$5 \times 10^ - n$$ and 3.14 + $$5 \times 10^ - n$$ would round to 3.14 when rounded to two decimal places. We can write this using inequality symbols as follows:


3.14 - $$5 \times 10^ - n$$ ≤ x < 3.14 + $$5 \times 10^ - n$$


We can also use interval notation to represent this range of values:


[3.14 - $$5 \times 10^ - n$$, 3.14 + $$5 \times 10^ - n$$)


Here are some more examples of finding bounds for numbers rounded to a certain number of decimal places:



  • 4.6 (rounded to one decimal place): lower bound = 4.6 - $$5 \times 10^ - 2$$ = 4.55, upper bound = 4.6 + $$5 \times 10^ - 2$$ = 4.65, range = [4.55,4.65)



  • 2.35 (rounded to two decimal places): lower bound = 2.35 - $$5 \times 10^ - 3$$ = 2.345, upper bound = 2.35 + $$5 \times 10^ - 3$$ = 2.355, range = [2.345,2.355)



  • 0.123 (rounded to three decimal places): lower bound = 0.123 - $$5 \times 10^ - 4$$ = 0.1225, upper bound = 0.123 + $$5 \times 10^ - 4$$ = 0.1235, range = [0.1225,0.1235)



Now that we have learned how to find bounds for numbers rounded to a certain number of decimal places, let's try some exercises to practice our skills.


Exercises




Find the upper and lower bounds and the range of possible values for each of the following numbers rounded to a certain number of decimal places.


  • 1.8 (rounded to one decimal place)



  • 3.141 (rounded to three decimal places)



  • 0.007 (rounded to three decimal places)



  • 9.87 (rounded to two decimal places)



  • 0.0012 (rounded to four decimal places)



Solutions




  • 1.8 (rounded to one decimal place): lower bound = 1.8 - $$5 \times 10^ - 2$$ = 1.75, upper bound = 1.8 + $$5 \times 10^ - 2$$ = 1.85, range = [1.75,1.85)



.1415)


  • 0.007 (rounded to three decimal places): lower bound = 0.007 - $$5 \times 10^ - 4$$ = 0.0065, upper bound = 0.007 + $$5 \times 10^ - 4$$ = 0.0075, range = [0.0065,0.0075)



  • 9.87 (rounded to two decimal places): lower bound = 9.87 - $$5 \times 10^ - 3$$ = 9.865, upper bound = 9.87 + $$5 \times 10^ - 3$$ = 9.875, range = [9.865,9.875)



  • 0.0012 (rounded to four decimal places): lower bound = 0.0012 - $$5 \times 10^ - 5$$ = 0.00115, upper bound = 0.0012 + $$5 \times 10^ - 5$$ = 0.00125, range = [0.00115,0.00125)



Finding bounds for numbers rounded to a certain number of significant figures




Another way of rounding numbers is to use significant figures. Significant figures are the digits that carry meaning in a number and indicate its precision. For example, the number 1234 has four significant figures, while the number 0.0123 has three significant figures.


When we round a number to a certain number of significant figures, we keep only the first few digits that are significant and drop the rest. For example, if we round 1234 to two significant figures, we keep the first two digits (12) and drop the rest (34), so we get 1200.


How do we find the upper and lower bounds for numbers rounded to a certain number of significant figures? The rule is similar to the previous cases, but we need to be careful about the place value of the last significant figure.


For example, if we have a number like 1200 that has been rounded to two significant figures, then the last significant figure is in the hundreds place, so half of it is $$50$$ or $$500 \over 10$$ or $$5000 \over 100$$ or $$50000 \over 1000$$ or $$500000 \over 10000$$ or $$5 \times 10^n - 1$$ where n is any positive integer. Therefore, we add or subtract this amount to get the bounds.


The upper bound is 1200 + $$5 \times 10^n - 1$$.


The lower bound is 1200 - $$5 \times 10^n - 1$$.


This means that any number between 1200 - $$5 \times 10^n - 1$$ and 1200 + $$5 \times 10^n - 1$$ would round to 1200 when rounded to two significant figures. We can write this using inequality symbols as follows:


1200 - $$5 \times 10^n - 1$$ ≤ x < 1200 + $$5 \times 10^n - 1$$


We can also use interval notation to represent this range of values:


[1200 - $$5 \times 10^n - 1$$, 1200 + $$5 \times 10^n - 1$$)


Here are some more examples of finding bounds for numbers rounded to a certain number of significant figures:



0}}$$ = 2.5, upper bound = 3 + $$5 \times 10^0$$ = 3.5, range = [2.5,3.5)


  • 0.04 (rounded to one significant figure): lower bound = 0.04 - $$5 \times 10^-2$$ = 0.035, upper bound = 0.04 + $$5 \times 10^-2$$ = 0.045, range = [0.035,0.045)



  • 24 (rounded to two significant figures): lower bound = 24 - $$5 \times 10^-1$$ = 23.5, upper bound = 24 + $$5 \times 10^-1$$ = 24.5, range = [23.5,24.5)



  • 0.0067 (rounded to two significant figures): lower bound = 0.0067 - $$5 \times 10^-4$$ = 0.00665, upper bound = 0.0067 + $$5 \times 10^-4$$ = 0.00675, range = [0.00665,0.00675)



  • 356 (rounded to three significant figures): lower bound = 356 - $$5 \times 10^-2$$ = 355.95, upper bound = 356 + $$5 \times 10^-2$$ = 356.05, range = [355.95,356.05)



  • 0.00123 (rounded to three significant figures): lower bound = 0.00123 - $$5 \times 10^-5$$ = 0.001225, upper bound = 0.00123 + $$5 \times 10^-5$$ = 0.001235, range = [0.001225,0.001235)



Now that we have learned how to find bounds for numbers rounded to a certain number of significant figures, let's try some exercises to practice our skills.


Exercises




Find the upper and lower bounds and the range of possible values for each of the following numbers rounded to a certain number of significant figures.


  • 9 (rounded to one significant figure)



  • 0.02 (rounded to one significant figure)



  • 67 (rounded to two significant figures)



  • 0.0089 (rounded to two significant figures)



  • 123 (rounded to three significant figures)



  • 0.000456 (rounded to three significant figures)



Solutions




0}}$$ = 8.5, upper bound = 9 + $$5 \times 10^0$$ = 9.5, range = [8.5,9.5)


  • 0.02 (rounded to one significant figure): lower bound = 0.02 - $$5 \times 10^-2$$ = 0.015, upper bound = 0.02 + $$5 \times 10^-2$$ = 0.025, range = [0.015,0.025)



  • 67 (rounded to two significant figures): lower bound = 67 - $$5 \times 10^-1$$ = 66.5, upper bound = 67 + $$5 \times 10^-1$$ = 67.5, range = [66.5,67.5)



  • 0.0089 (rounded to two significant figures): lower bound = 0.0089 - $$5 \times 10^-4$$ = 0.00885, upper bound = 0.0089 + $$5 \times 10^-4$$ = 0.00895, range = [0.00885,0.00895)



  • 123 (rounded to three significant figures): lower bound = 123 - $$5 \times 10^-2$$ = 122.95, upper bound = 123 + $$5 \times 10^-2$$ = 123.05, range = [122.95,123.05)



  • 0.000456 (rounded to three significant figures): lower bound = 0.000456 - $$5 \times 10^-6$$ = 0.0004555, upper bound = 0.000456 + $$5 \times 10^-6$$ = 0.0004565, range = [0.0004555,0.0004565)



Using bounds to solve simple problems




Now that we know how to find the upper and lower bounds of numbers given to a certain accuracy, we can use them to solve simple problems involving addition, subtraction, multiplication or division.


For example, suppose we have a rectangle with a length of 12 cm and a width of 8 cm, both rounded to the nearest centimeter. What is the area of the rectangle?


We can't just multiply 12 by 8 and get 96 cm, because that would be using the rounded values and not the exact values. The exact values could be anywhere between the upper and lower bounds of 12 and 8.


To find the upper and lower bounds of the area, we need to use the upper and lower bounds of the length and width.


The upper bound of the length is 12 + $$5 \times 10^ - n$$ cm.


The lower bound of the length is 12 - $$5 \times 10^ - n$$ cm.


The upper bound of the width is 8 + $$5 \times 10^ - n$$ cm.


The lower bound of the width is 8 - $$5 \times 10^ - n$$ cm.


To find the upper bound of the area, we need to multiply the upper bounds of the length and width:


Upper bound of area = (12 + $$5 \times 10^ - n$$(8 + $$5 \times 10^ - n$$(cm


To find the lower bound of the area, we need to multiply the lower bounds of the length and width:


Lower bound of area = (12 - $$5 \times 10^ - n$$(8 - $$5 \times 10^ - n$$(cm


This means that the area of the rectangle could be anywhere between these two values:


(12 - $$5 \times 10^ - n$$(8 - $$5 \times 10^ - n$$( ≤ Area < (12 + $$5 \times 10^ - n$$(8 + $$5 \times 10^ - n$$(


We can also use interval notation to represent this range of values:


[(12 - $$5 \times 10^ - n$$(8 - $$5 \times 10^ - n$$(, (12 + $$5 \times 10^ - n$$(8 + $$5 \times 10^ - n$$()


Here are some more examples of using bounds to solve simple problems:



  • Suppose we have a circle with a radius of 5 cm, rounded to the nearest centimeter. What is the circumference of the circle?



Solution: The upper bound of the radius is 5 + $$5 \times 10^ - n$$ cm.


The lower bound of the radius is 5 - $$5 \times 10^ - n$$ cm.


The circumference of a circle is given by the formula C = 2πr, where r is the radius and π is approximately equal to 3.14.


To find the upper bound of the circumference, we need to multiply 2π by the upper bound of the radius:


Upper bound of circumference = 2π(5 + $$5 \times 10^ - n$$(cm


To find the lower bound of the circumference, we need to multiply 2π by the lower bound of the radius:


Lower bound of circumference = 2π(5 - $$5 \times 10^ - n$$(cm


This means that the circumference of the circle could be anywhere between these two values:


2π(5 - $$5 \times 10^ - n$$( ≤ Circumference < 2π(5 + $$5 \times 10^ - n$$(


We can also use interval notation to represent this range of values:


  • [2π(5 - $$5 \times 10^ - n$$(, 2π(5 + $$5 \times 10^ - n$$()



  • Suppose we have two numbers, 3.6 and 4.7, both rounded to one decimal place. What is the difference between them?



Solution: The upper bound of 3.6 is 3.6 + $$5 \times 10^-1$$ = 3.65.


The lower bound of 3.6 is 3.6 - $$5 \times 10^-1$$ = 3.55.


The upper bound of 4.7 is 4.7 + $$5 \times 10^-1$$ = 4.75.


The lower bound of 4.7 is 4.7 - $$5 \times 10^-1$$ = 4.65.


The difference between two numbers is given by subtracting them.


To find the upper bound of the difference, we need to subtract the lower bound of 3.6 from the upper bound of 4.7:


Upper bound of difference = 4.75 - 3.55 = 1.2


To find the lower bound of the difference, we need to subtract the upper bound of 3.6 from the lower bound of 4.7:


Lower bound of difference = 4.65 - 3.65 = 1


This means that the difference between the two numbers could be anywhere between these two values:


1 ≤ Difference < 1.2


We can also use interval notation to represent this range of values:


  • [1,1.2)



  • Suppose we have two numbers, 2.3 and 1.4, both rounded to one decimal place. What is the product of them?



Solution: The upper bound of 2.3 is 2.3 + $$5 \times 10^-1$$ = 2.35.


The lower bound of 2.3 is 2.3 - $$5 \times 10^-1$$ = 2.25.


The upper bound of 1.4 is 1.4 + $$5 \times 10^-1$$ = 1.45.


The lower bound of 1.4 is 1.4 - $$5 \times 10^-1$$ = 1.35.


The product of two numbers is given by multiplying them.


To find the upper bound of the product, we need to multiply the upper bounds of 2.3 and 1.4:


Upper bound of product = 2.35 × 1.45 = 3.4075


To find the lower bound of the product, we need to multiply the lower bounds of 2.3 and 1.4:


Lower bound of product = 2.25 × 1.35 = 3.0375


This means that the product of the two numbers could be anywhere between these two values:


3.0375 ≤ Product < 3.4075


We can also use interval notation to represent this range of values:


  • [3.0375,3.4075)



Conclusion




In this article, we have learned about limits of accuracy and how to use them in mathematics. We have learn


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