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Upper and Lower Bounds
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Upper and Lower Bounds

lesson introduction

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 176176cm. To 6 significant figures, what is the tallest they could be?

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In practice, they could be anything up to 176.499999999176.499999999 cm.

In this example, 175.5175.5 is considered the lower bound and 176.5176.5 is the upper bound. We do not use 176.49999...176.49999...as it is harder to write down and only slightly smaller than 176.5176.5

Mathematically, if x=176x = 176 to the nearest centimetre, then:

175.5x<176.5 175.5 \le x <176.5

x=51x = 51 to the nearest integer. What is the lower bound of xx?

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x=40x = 40 to the nearest ten. What is the lower bound of xx?

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Sometimes we want to know what the upper bound and lower bound are for a particular calculation.

If x=12.95x = 12.95 to 2 decimal places and y=23.4y = 23.4 to 1 decimal place, what are the upper and lower bounds for xy\dfrac{x}{y}?

1

Find the largest possible answer for xy\dfrac{x}{y}

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.

2

What is the upper bound of 12.9512.95?

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3

What is the lower bound of 23.423.4?

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4

Calculate with upper bound (numerator) and lower bound (denominator)

12.95523.35=0.555\dfrac{12.955}{23.35}=0.555. Therefore, the upper bound is 0.5550.555

5

Find the smallest possible answer for xy\dfrac{x}{y}

For the smallest answer, we need to use the lower bound of the numerator and the upper bound of the denominator.

6

What is the lower bound of xx?

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7

What is the upper bound of yy?

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8

Calculate the lowest possible answer

12.94523.45=0.552\dfrac{12.945}{23.45}=0.552. Therefore, the lower bound is 0.5520.552

9

Great work! 👍

0.552xy0.5550.552\leq\dfrac{x}{y}\leq0.555

What is the lower bound of xy\dfrac{x}{y} to 2 decimal places?

x=2.3   y=4.5x = 2.3\space\space\space y = 4.5

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What is the lower bound of ab\dfrac{a}{b} correct to 2 decimal places?

a=15.6   b=7.6a = 15.6\space\space\space b = 7.6

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