Lesson 6: Upper and Lower Bounds for Calculations

Objectives

At the end of the lesson, students should be able to:

• Give appropriate upper and lower bounds for data to a specified accuracy.
• Use upper and lower bounds for calculations.

When rounded values are used for calculations, we can find the upper and lower bounds for the results of the calculations.

Addition and multiplication follow the same rule;

To find the upper bound of the product (or sum) of any two numbers, multiply (or add) the upper bounds of the two numbers.

To find the lower bound of the product (or sum) of any two numbers, multiply (or add) the lower bounds of the two numbers.

Example 1

Calculate the upper and lower bounds for the following calculation, given that each number is given to the nearest whole number.

61 × 43

Solution

61 lies in the range  60.5 ≤ x<61.5

43 lies in the range 42.5 ≤ x< 43.5

The lower bound of the calculation is obtained by multiplying together the two lower bounds. Therefore the minimum product is 60.5 × 42.5 = 2571.25

The upper bound of the calculation is obtained by multiplying together the two upper bounds. Therefore the maximum product is 61.5 × 43.5 = 2675.25

Subtraction and division follow the same rule;

To find the upper bound of x/y,  divide the upper bound of x (numerator) by the lower bound of x (denominator). To find the lower bound of  x/y, divide the lower bound of  x (numerator) by the upper bound of y (denominator).

To find the upper bound of x – y  , subtract the lower bound of y from the upper bound of x. To find the lower bound of x – y, subtract the upper bound of y from the lower bound of x.

Example 2

Solution

Finding the upper and lower bounds of the numbers involve give;

180 lies in the range 175 ≤ x < 185

7.3 lies in the range 7.25 ≤ x < 7.35

4.5 lies in the range 4.45 ≤ x < 4.55

First find the upper and lower bounds of 7.3 – 4.5

The lower bound of the calculation is obtained by subtracting the upper bound of 4.5 from the lower bound of 7.3.

7.25 – 4.55 = 2.7

Therefore lower bound of (7.3 – 4.5) is 2.7.

The upper bound of the calculation is obtained by subtracting the lower bound of 4.5 from the upper bound of 7.3.

7.35 – 4.45 = 2.9

Therefore upper bound of (7.3 – 4.5) is 2.9.

If y represents 180 then the inequality becomes:175< x < 185

If x represents (7.3 – 4.5) then the inequality becomes: 2.7≤ x < 2.9

Example 3