- Bounds are the smallest
**(lower bound, LB)**and largest**(upper bound, UB)**numbers that a**rounded number**can lie between- It simply means how low or high the number could have been before it was rounded

- The bounds for a number,
, can be written as
- Note that the lower bound is included in the range of values

- Note that the lower bound is included in the range of values

- The basic rule is “Half Up, Half Down”
- UPPER BOUND – To find the upper bound add on half the degree of accuracy
- LOWER BOUND – To find the lower bound take off half the degree of accuracy
- ERROR INTERVAL: LB ≤
**x**< UB

- Note that it is tempting to think that the Upper Bound should end in a 9, or 99, etc. but if you look at the Error Interval – LB ≤ x < UB – it does NOT INCLUDE the Upper Bound so all is well
- the upper bound is the cut off point for the greatest value that the number could have been rounded from but will not actually round to the number itself

- For example, the error interval for the number 1230, rounded to 3 significant figures will be 1225 ≤
*x*< 1235- The degree of accuracy is 10 (rounding to 3 s.f. here requires rounding to the nearest ten)
- Half the degree of accuracy is 5

- Remember that
**truncating**a number means to round it down

- This means that the UPPER BOUND is found by adding 1 to the digit in the place value that the number was truncated to
- The LOWER BOUND is always the number the value was truncated to

- For example, the error interval for the number 1230, truncated to 3 significant figures will be 1230 ≤
*x*< 1240

The length of a road, , is given as , correct to 1 decimal place.

Find the lower and upper bounds for

The degree of accuracy is 1 decimal place, or 0.1 km so the true value could be up to 0.05 km above or below this

Upper bound:

3.6 + 0.05 = 3.65 km

Lower bound:

3.6 – 0.05 = 3.55 km

**
Upper bound: 3.65 km
**

**
Lower bound: 3.55 km
**

We could also write this as an error interval of , although this is not asked for in this question

- If you are
**adding**two numbers together:*T*=*a*+*b*- The
**upper bound of**can be found by adding together the*T***upper bound of**and the*a***upper bound of***b* - The
**lower bound of**can be found by adding together the*T***lower bound of**and the*a***lower bound of***b*

- The
- If you are
**subtracting**a number from another number:*T*=*a*–*b*- The
**upper bound of**can be found by using the*T***upper bound of**and subtracting the*a***lower bound of***b* - The
**lower bound of**can be found by using the*T***lower bound of**and subtracting the*a***upper bound of***b*

- The
- If you are
**multiplying**two numbers together:*T*=*a*×*b*- The
**upper bound of**can be found by multiplying together the*T***upper bound of**and the*a***upper bound of***b* - The
**lower bound of**can be found by multiplying together the*T***lower bound of**and the*a***lower bound of***b*

- The
- If you are
**dividing**a number by another number:*T*=*a*÷*b*- The
**upper bound of**can be found by using the*T***upper bound of**and dividing it by the*a***lower bound of***b* - The
**lower bound of**can be found by using the*T***lower bound of**and dividing it by the*a***upper bound of***b*

- The

- You can use bounds to calculate the
**level of accuracy**of a calculation - This can be used to decide how to
**round your answer**- e.g. If the lower bound of an value is 8.33217… and the upper bound is 8.33198…
- The true value is between 8.33217… and 8.33198…
- Both bounds round to 8.332 to 4sf
- To 5sf they differ (first is 8.3322 and second is 8.3320)
- Therefore you know the answer is definitely rounds to 8.332 to 4 significant figures

(a) A room measures 4 m by 7 m, where each measurement is made to the nearest metre

Find the upper and lower bounds for the area of the room

Find the bounds for each dimension, you could write these as error intervals, or just write down the upper and lower bounds

As they have been rounded to the nearest metre, the true values could be up to 0.5 m bigger or smaller

Calculating the lower bound of the area, using the two smallest measurements

3.5 × 6.5 =

**
Lower Bound = 22.75 m ^{2}
**

Calculating the upper bound of the area, using the two largest measurements

4.5 × 7.5 =

**Upper Bound = 33.75 m ^{2}
**

(b) David is trying to work out how many slabs he needs to buy in order to lay a garden path.

Slabs are 50 cm long, measured to the nearest 10 cm.

The length of the path is 6 m, measured to the nearest 10 cm.

Find the maximum number of slabs David will need to buy.

Find the bounds for each measurement, you could write these as error intervals, or just write down the upper and lower bounds

As they have been rounded to the nearest 10 cm, the true values could be up to 5 cm bigger or smaller

We have a mixture of centimetres and metres, so it is useful to change them both to metres for later calculations

Length of the slabs: or in metres:

Length of the path:

The maximum number of slabs needed will be when the path is as long as possible (6.05 m), and the slabs are as short as possible (0.45 m)

Max number of slabs =

Assuming we can only purchase whole slabs

**
The maximum number of slabs to be bought is 14
**

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