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Upper and Lower Bounds

Upper and Lower Bounds

When combining upper and lower bounds, you need to work out how the measurements relate to one another. Some examples are given below. To work out which example applies in which situation, think of practical examples and what effect you would expect to see.

Example LB = Lower Bound UB = Upper Bound
Addition Measuring two lengths of wood joined end to end Shortest length: LB + LB Longest Length: UB + UB
Subtraction Fitting a shelf into an alcove and measuring the gap Smallest gap: LB (alcove) - UB (shelf) Largest gap: UB (alcove) - LB (shelf)
Multiplication Area of a tennis court Smallest Area: LB (length) x LB (width) Largest Area: UB (length) x UB (width)
Division Loading a lorry with pallets by weight Smallest number of pallets: LB (lorry) ÷ UB (pallets) this is the maximum number of pallets for a safe load Largest number of pallets: UB (lorry) ÷ LB (pallets)

Example 1

A shelf is being fitted into an alcove. The shelf is 2.3m long, with an error interval of 5cm; and the alcove is 2.4m long, with an error interval of 5cm.

What is the biggest gap that there might be when fitting the shelf to the alcove?

The gap is given by the smallest shelf size and the largest alcove size.

Shelf lower bound: 2.3 - 0.05 ÷ 2 = 2.275

Alcove Upper Bound = 2.4 + 0.05 ÷ 2 = 2.425

Difference = 2.425 - 2.275 = 0.15m

Answer: 0.15m

Example 2

Crates are being loaded for lifting by a crane. The maximum weight the crane can lift is 1000kg to the nearest 10kg. Each crate weighs 85kg, to the nearest kg.

What is the maximum number of crates that can be lifted?

Higher bound for each crate = 85 + 0.5 = 85.5kg.

The crane can take 1000 - 10 = 990Kg

990 ÷ 85.5 = 11.57

Rounding 11.57 is not appropriate as this could take it over the safety limit: in this instance the answer must be rounded down.

Answer: 11 crates