Upper & Lower Bounds (DP IB Applications & Interpretation (AI)): Revision Note

Upper & Lower Bounds

What are bounds?

• Bounds are the smallest (lower bound, LB) and largest (upper bound, UB) numbers that a rounded number can lie between

  • The bounds for a rounded number, , can be written as 

• e.g. to 1 decimal place has the bounds

  • The lower bound is included in the inequality

    • as could have been (it rounds to )

  • The upper bound is not included in the inequality

    • as could not have been (it rounds to )

    • Despite this, we still call the 'upper bound'

How do we find bounds?

• To find bounds, the basic rule is “half up, half down”

  • To find the upper bound, add on half the degree of accuracy

  • To find the lower bound, take off half the degree of accuracy

How do we combine bounds?

• The following rules can be used when doing calculations with bounds:

  • When adding, UB = UB + UB and LB = LB + LB

  • When subtracting, UB = UB - LB and LB = LB – UB

    • i.e. start with the biggest and take off something small

    • or start with the smallest and take off something big

  • When multiplying, UB = UB × UB and LB = LB × LB

  • When dividing, UB = UB / LB and LB = LB / UB

    • This is because dividing by a smaller number makes the fraction bigger

    • and vice versa