Upper and Lower Bounds for the TMUA

Finding Upper and Lower Bounds

When a number is rounded to a specified level of accuracy, it represents a range of possible values. The limits of this range are known as the upper and lower bounds. These are often referred to as the Greatest Lower Bound (GLB) and the Least Upper Bound (LUB).

The Greatest Lower Bound is defined as the smallest number that would round up to the given value $x$. The Least Upper Bound is defined as the smallest number that would round to a value larger than $x$.

For example, if a measurement is recorded as $3.84$ correct to 2 decimal places:

1. The lower bound is $3.835$. This is the smallest value that rounds to $3.84$.
2. The upper bound is $3.845$. This is the boundary at which a value would instead round to $3.85$.

Calculating with Bounds: Addition and Multiplication

When performing calculations involving only addition or multiplication, you find the bounds by using the same bounds of the quantities involved. To find the maximum possible result, use the upper bounds: to find the minimum possible result, use the lower bounds.

For addition: $\text{LUB}(A + B) = \text{LUB}(A) + \text{LUB}(B)$ $\text{GLB}(A + B) = \text{GLB}(A) + \text{GLB}(B)$

For multiplication: $\text{LUB}(A \times B) = \text{LUB}(A) \times \text{LUB}(B)$ $\text{GLB}(A \times B) = \text{GLB}(A) \times \text{GLB}(B)$

Calculating with Bounds: Subtraction and Division

Calculations involving subtraction and division require a different approach. To find the largest possible result (LUB), you must start with the largest possible value and divide by or subtract the smallest possible value. Conversely, to find the smallest result (GLB), start with the smallest value and divide by or subtract the largest.

For division: $\text{LUB}\left(\frac{A}{B}\right) = \frac{\text{LUB}(A)}{\text{GLB}(B)}$

$\text{GLB}\left(\frac{A}{B}\right) = \frac{\text{GLB}(A)}{\text{LUB}(B)}$

For subtraction: $\text{LUB}(A - B) = \text{LUB}(A) - \text{GLB}(B)$

$\text{GLB}(A - B) = \text{GLB}(A) - \text{LUB}(B)$

Worked Examples in Context

Example 1: Standard Measurement

A quantity is recorded as 236 grams correct to the nearest gram. Find the upper and lower bounds.

The number above 236 to the nearest gram is 237, and the number below it is 235. The boundary between 235 and 236 is $235.5$ (the lower bound). The boundary between 236 and 237 is $236.5$ (the upper bound). Alternatively, the bounds are $236 \pm 0.5$ grams.

Example 2: Perimeter and Area of a Rectangle

A rectangle has a length of $23.6$ cm and a width of $14.7$ cm, both correct to one decimal place. Find the upper bounds of the perimeter and the area.

The upper bound of the length is $23.65$ cm and the upper bound of the width is $14.75$ cm.

$\text{LUB of Perimeter} = 2 \times 23.65 + 2 \times 14.75 = 76.8$ cm. $\text{LUB of Area} = 23.65 \times 14.75 = 348.8375$ cm$^2$.

Example 3: Compound Measures (Density)

The volume of a piece of wood is $270$ cm$^3$ correct to the nearest $10$ cm$^3$. The mass is $540$ g correct to the nearest $10$ g. Calculate the upper and lower bounds for the density in g/cm$^3$ correct to 2 decimal places.

The bounds for the volume are 265 and 275. The bounds for the mass are 535 and 545. Using the rules for division:

$\text{LUB of Density} = \frac{275}{535} = 0.51$ g/cm$^3$. $\text{GLB of Density} = \frac{265}{545} = 0.49$ g/cm$^3$.

Example 4: Complex Formulae

Given $a = 6$, $b = 3$, and $c = 2$ correct to the nearest whole number, find the bounds for the expression $b^2 - 4ac$.

The bounds are: $a \in [5.5, 6.5)$, $b \in [2.5, 3.5)$, and $c \in [1.5, 2.5)$.

To find the upper bound of $b^2 - 4ac$, we need the largest possible value of the first term and the smallest possible value of the second term: $\text{Upper Bound} = 6.5^2 - (4 \times 2.5 \times 1.5) = 27.25$.

To find the lower bound, we use the smallest possible value of the first term and the largest possible value of the second term: $\text{Lower Bound} = 5.5^2 - (4 \times 3.5 \times 2.5) = -4.75$.