When numbers are given to a certain degree of accuracy, they represent a range of possible values. When performing calculations, the lower bound and upper bound of the result can be found by using the bounds of the original values.

 

Before calculating bounds, always identify the lower and upper bounds of each number involved.

 

Addition

 

For addition, the smallest possible result comes from adding the lower bounds. The largest possible result comes from adding the upper bounds.

 

Suppose a length is given as \( 12 \) cm to the nearest centimetre and another length is given as \( 5 \) cm to the nearest centimetre.

 

Their bounds are:

$$
11.5 \le x < 12.5
$$

$$
4.5 \le y < 5.5
$$

 

Lower bound of the sum:

$$
11.5 + 4.5 = 16
$$

 

Upper bound of the sum:

$$
12.5 + 5.5 = 18
$$

 

So the result lies between:

$$
16 \le x + y < 18
$$

 

Subtraction

 

For subtraction, the smallest possible result comes from subtracting the upper bound of the second value from the lower bound of the first. The largest possible result comes from subtracting the lower bound of the second value from the upper bound of the first.

 

Using the same bounds:

 

Lower bound of the difference:

$$
11.5 - 5.5 = 6
$$

 

Upper bound of the difference:

$$
12.5 - 4.5 = 8
$$

 

So the result lies between:

$$
6 \le x - y < 8
$$

 

Multiplication

 

For multiplication, the smallest possible result comes from multiplying the lower bounds. The largest possible result comes from multiplying the upper bounds, provided all values are positive.

 

Lower bound of the product:

$$
11.5 \times 4.5 = 51.75
$$

 

Upper bound of the product:

$$
12.5 \times 5.5 = 68.75
$$

 

So the result lies between:

$$
51.75 \le xy < 68.75
$$

 

Division

 

For division, the smallest possible result comes from dividing the lower bound of the numerator by the upper bound of the denominator. The largest possible result comes from dividing the upper bound of the numerator by the lower bound of the denominator.

 

Lower bound of the quotient:

$$
11.5 \div 5.5 = 2.0909
$$

 

Upper bound of the quotient:

$$
12.5 \div 4.5 = 2.777\dot{7}
$$

 

So the result lies between:

$$
2.0909 \le \frac{x}{y} < 2.777\dot{7}
$$

 

When calculating bounds, always use the extreme values that make the result as small or as large as possible. This ensures the final bounds fully capture all possible values of the calculation.