YOU ARE LEARNING:
Upper and Lower Bounds
Upper and Lower Bounds
When we round numbers, upper and lower bounds contain the range of exact values which the rounded number could be.
Sometimes we cannot be sure of the exact value of a number. Upper and lower bounds contain the range of exact values that a rounded number could correspond to.
For example, when a person's height is measured we usually give it to the nearest centimetre or nearest inch. This is close to, but not the same as, the person's actual height.
If a person's height is 176cm to the nearest centimetre, it could be as small as 175.5cm and have just been rounded up to 176cm, or it could be anything up to 176.499999.... and have just been rounded down to 176cm.
Your friend's height is 176cm. To 6 significant figures, what is the tallest they could be?
In practice, they could be anything up to 176.499999999 cm.
In this example, 175.5 is considered the lower bound and 176.5 is the upper bound. We do not use 176.49999...as it is harder to write down and only slightly smaller than 176.5
Mathematically, if x=176 to the nearest centimetre, then:
175.5≤x<176.5
x=51 to the nearest integer. What is the lower bound of x?
x=40 to the nearest ten. What is the lower bound of x?
Sometimes we want to know what the upper bound and lower bound are for a particular calculation.
If x=12.95 to 2 decimal places and y=23.4 to 1 decimal place, what are the upper and lower bounds for yx?
Find the largest possible answer for yx
As it is a division, we need to carry out the calculation with the upper bound of the numerator and lower bound of the denominator.
What is the upper bound of 12.95?
What is the lower bound of 23.4?
Calculate with upper bound (numerator) and lower bound (denominator)
23.3512.955=0.555. Therefore, the upper bound is 0.555
Find the smallest possible answer for yx
For the smallest answer, we need to use the lower bound of the numerator and the upper bound of the denominator.
What is the lower bound of x?
What is the upper bound of y?
Calculate the lowest possible answer
23.4512.945=0.552. Therefore, the lower bound is 0.552
Great work! 👍
0.552≤yx≤0.555
What is the lower bound of yx to 2 decimal places?
x=2.3 y=4.5
What is the lower bound of ba correct to 2 decimal places?
a=15.6 b=7.6