# Upper and lower bounds

**What are upper and lower bounds?**

Anything that can be measured has a margin of error. The magnitude of error depends on how precisely it is measured and the accuracy of the measuring tool.

Upper bound and lower bounds helps to provide measurement a degree of accuracy.

**E1.10A: Give appropriate upper and lower bounds for data given to a specified accuracy.**

**Upper bound and lower bound**

Numbers can be written to a distinct degree of accuracy, for example, $$4.5$$, $$5.50$$, and $$5.500$$, even though the numbers seem alike, they are not. The reason is these numbers are written to distinct degrees of accuracy.

Now, $$5.5$$ is written around one decimal place, and hence some number from $$5.45$$ up to $$5.55$$, but excluding $$5.55$$, would be approximated to $$5.5$$. On a number line, this could be presented as follows:

In the process of inequality, when $$x$$ serves the number, it can be formulated as $$5.45\leq x<5.55$$. Here, $$5.45$$ is the lower bound to $$5.5$$, while $$5.55$$ is the upper bound. If it is in touch with two decimal places then numbers from $$5.495$$ and $$5.505$$ are the lower and upper bound to $$5.50$$.

**Worked examples**

**Example 1:** A journey of a route is predicted $$150\;\text{miles}$$ to the nearest mile. What is the upper and lower bound?

**Step 1: Formulate the upper bound and the lower bound of the distance of the journey.**

$$Lower bound = 149.5\;\text{miles}$$

$$Upper bound = 150.5\;\text{miles}$$

**Step 2: Represent this on the number line.**

**Step 3: Assume that the distance is $$d\;\text{miles}$$ and present this scope as an inequality.**

$$149.5\leq x<150.5$$

**Example 2:** The height of a hill is calculated as $$480\;\text{m}$$ to the nearest $$10\;\text{m}$$. What is the upper and lower bound?

**Step 1: Formulate the upper bound and the lower bound of the height of the hill.**

$$\text{Lower bound} = 475\;\text{m}$$

$$\text{Upper bound} = 485\;\text{m}$$

**Step 2: Represent this on the number line.**

**Step 3: Assume that height of the hill is $$h\;\text{m}$$ and present this as an inequality.**

$$475\leq x<485$$

**E1.10B: Obtain appropriate upper and lower bounds to solutions of simple problems given data to a specified accuracy.**

When data is given to a specified accuracy, the margin of error is half of it. Presume that accuracy is given to the nearest $$1$$ decimal point; hence, the margin of error is $$0.05$$. If accuracy is given to the nearest whole number, the margin of error will become $$0.5$$.

**Worked examples**

**Example 1: **Calculate the upper bound and lower bound for the perimeter of a rectangle of length $$5.75\;\text{m}$$ and width $$3.00\;\text{m}$$, where the measurements are accurate to $$2$$ decimal points.

**Step 1: Formulate the upper bound and the lower bound of the length and the width separately.**

The lower bound of length and width are $$5.645\;\text{m}$$ and $$2.995\;\text{m}$$, respectively.

The upper bound of length and width are $$5.755\;\text{m}$$ and $$3.005\;\text{m}$$, respectively.

**Step 2: Calculate the lower bound of the perimeter, substituting the lower bound value of length and width.**

Perimeter of the rectangle is $$2\times (l+w)$$.

$$ \text{Lower bound of perimeter} =2\times(5.645+2.995)=17.28 \;\text{m}$$

**Step 3: Calculate the upper bound of the perimeter substituting the upper bound value of length and width.**

Perimeter of the rectangle is $$2\times (l+w)$$.

$$\text{Lower bound of perimeter} =2\times (5.755+3.005)=17.52\; \;\text{m}$$

**Example 2:** Calculate the upper bound and lower bound of the area of a square whose edge is $$8.5\;\text{m}$$ correct to the nearest $$1$$ decimal point.

**Step 1: Formulate the upper bound and lower bound of the given edge of the square.**

$$\text{Lower bound} =8.45\;\text{m}$$

$$\text{Upper bound} =8.55\;\text{m}$$

**Step 2: Calculate the lower bound of the area, substituting the lower bound value of the edge.**

The area of a square is $$edge^2$$.

Lower bound of area is $$8.45^2=71.40\; \text{m}^2$$.

**Step 3: Calculate the upper bound of the area, substituting the upper bound value of the edge**

The area of square is $$(\text{side})^2$$.

Upper bound of area is $$8.55^2 = 73.10\;\text{m}^2$$.