Bounds (AQA GCSE Maths)

What are bounds?

• Bounds are the smallest (lower bound, LB) and largest (upper bound, UB) numbers that a rounded number can lie between
  • It simply means how low or high the number could have been before it was rounded
• The bounds for a number, , can be written as 
  • Note that the lower bound is included in the range of values could have taken but the upper bound is not

How do we find bounds when a number has been rounded?

• The basic rule is “Half Up, Half Down”
  • UPPER BOUND – To find the upper bound add on half the degree of accuracy
  • LOWER BOUND – To find the lower bound take off half the degree of accuracy
  • ERROR INTERVAL: LB ≤ x < UB
• Note that it is tempting to think that the Upper Bound should end in a 9, or 99, etc. but if you look at the Error Interval – LB ≤ x < UB – it does NOT INCLUDE the Upper Bound so all is well
  •  the upper bound is the cut off point for the greatest value that the number could have been rounded from but will not actually round to the number itself
• For example, the error interval for the number 1230, rounded to 3 significant figures will be 1225 ≤ x < 1235
  • The degree of accuracy is 10 (rounding to 3 s.f. here requires rounding to the nearest ten)
  • Half the degree of accuracy is 5

How do we find bounds when a number has been truncated?

• Remember that truncating a number means to round it down
  
  • This means that the UPPER BOUND is found by adding 1 to the digit in the place value that the number was truncated to
  • The LOWER BOUND is always the number the value was truncated to
• For example, the error interval for the number 1230, truncated to 3 significant figures will be 1230 ≤ x < 1240

How do I find the bounds of a calculation?

• If you are adding two numbers together: T =a + b
  • The upper bound of T can be found by adding together the upper bound of a and the upper bound of b
  • The lower bound of T can be found by adding together the lower bound of a and the lower bound of b
• If you are subtracting a number from another number: T =a - b
  • The upper bound of T can be found by using the upper bound of a and subtracting the lower bound of b
  • The lower bound of T can be found by using the lower bound of a and subtracting the upper bound of b
• If you are multiplying two numbers together: T =a × b
  • The upper bound of T can be found by multiplying together the upper bound of a and the upper bound of b
  • The lower bound of T can be found by multiplying together the lower bound of a and the lower bound of b
• If you are dividing a number by another number: T =a ÷ b
  • The upper bound of T can be found by using the upper bound of a and dividing it by the lower bound of b
  • The lower bound of T can be found by using the lower bound of a and dividing it by the upper bound of b

How can bounds help with calculations?

• You can use bounds to calculate the level of accuracy of a calculation
• This can be used to decide how to round your answer
  • e.g. If the lower bound of an value is 8.33217... and the upper bound is 8.33198...
  • The true value is between 8.33217... and 8.33198...
  • Both bounds round to 8.332 to 4sf
  • To 5sf they differ (first is 8.3322 and second is 8.3320)
  • Therefore you know the answer is definitely rounds to 8.332 to 4 significant figures