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# 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 $176$cm. 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:

$175.5 \le x <176.5$

$x = 51$ 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 $\dfrac{x}{y}$?

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

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.

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)

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

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

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

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

Great work! 👍

$0.552\leq\dfrac{x}{y}\leq0.555$

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

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

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

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