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Rounding to the nearest 10, 100 and 1000 Decimal places Significant figures Truncation Inequalities on a number lineThis topic is relevant for:

Here we will learn about **error intervals** including how to find error intervals for numbers that have been rounded and for truncated numbers.

There are also error intervals worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

**Error intervals** are the limits of accuracy when a number has been rounded or truncated. They are the range of possible values that a number could have been before it was rounded or truncated.

To do this we think about what the smallest and biggest numbers that would round, or be truncated, to a value for a given degree of accuracy.

E.g.

In order to find the error interval of a rounded or truncated number:

**Identify the place value of the degree of accuracy stated – this will be the interval size required for the error interval.**

2

**If rounded:**

Divide this place value by 2, and then add and subtract this amount to the given value to give the maximum and minimum values for your error interval

**If truncated:**

Add the place value to the given value. This will be the maximum value of your error interval, the given value

will be your minimum.

3 **Write your error interval as an inequality in the form Min ≤ x < Max**

For the case of rounded numbers, the maximum and minimum are referred to as the **upper bound** and **lower bound** of the number.

Get your free error intervals worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREEGet your free error intervals worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREEThe height of a house,

Find the error interval of the height of the house.

**Identify the place value of the degree of accuracy stated – this will be the interval size required for the error interval.**

The place value of the degree of accuracy is

2 **If rounded: Divide this place value by 2, and then add and subtract this amount to the given value to give the maximum and minimum values for your error interval.**

\[1\div 2=0.5\]

Maximum

Minimum

3 **Write your error** **interval as an inequality in the form M**

The mass of a stone,

The place value of the degree of accuracy is

**If rounded: Divide this place value by 2, and then add and subtract this amount to the given value to give the maximum and minimum values for your error interval.**

\[100\div 2=50\]

Maximum

Minimum

**Write your error interval as an inequality in the form ****in ≤ x < Max**

The length of a pencil,

The place value of the degree of accuracy is

\[0.1\div 2=0.05\]

Maximum

Minimum

**Write your error interval as an inequality in the form ****in ≤ x < Max**

The attendance,

The second significant figure is the

The place value of the degree of accuracy is

\[1000\div 2=500\]

Maximum

Minimum

**Write your error interval as an inequality in the form ****in ≤ x < Max**

When asked her age, Imogen said that she was 13 years old. Imogen truncated her age to the integer value. Find the error interval for Imogen’s age,

The place value of the degree of accuracy is

\[13 + 1 = 14\]

Maximum

Minimum

**Write your error interval as an inequality in the form ****in ≤ x < Max**

Sam calculated the area of a circle and truncated the answer to 2 decimal places. His answer was ^{2}

The place value of the degree of accuracy is

\[49.26 + 0.01 = 49.27\]

Maximum

Minimum

**Write your error interval as an inequality in the form ****in ≤ x < Max**

^{2} ≤ A < 49.27 cm^{2}

**Maximum value is incorrect**

It is very common for students to put an incorrect maximum value,

E.g

If a length has been rounded to 16 cm to the nearest centimetre, they may incorrectly write the error interval as

The maximum value is given using a strict inequality.

**Incorrect use of inequality notation**

It is important that the correct inequality symbols are used in the error interval. The range of values must be greater than or equal to the minimum, and strictly less than the maximum.

**The wrong methods are used for rounded or truncated numbers**

It is important to read the questions very carefully and look out for the word “truncated”. The exam questions will not always use the word “rounded” in a question.

E.g.

“The length

Error intervals are part of our series of lessons to support revision on rounding numbers, and upper and lower bounds. You may find it helpful to start with the main rounding numbers lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

1. The mass, m , of a rock is 24 kilograms to the nearest kilogram.

Complete the error interval for the mass of the rock.

….. ≤ m < ……

24 kg ≤ m < 25 kg

24 kg ≤ m < 24.5 kg

23.5 kg ≤ m < 24.5 kg

23.5 kg ≤ m < 24.49 kg

The mass is rounded to the nearest integer (the ones column). Divide this place value by 2 , and then add and subtract this amount to the given value to give the maximum and minimum values for your error interval.

2. Jonny estimates the number of steps, s , he has walked in one day to be 7000 to the nearest 1000 .

Complete the error interval for Jonny’s estimate.

….. ≤ s < ……

6000 ≤ s < 7500

6000 ≤ s < 7000

6500 ≤ s < 7500

6999 ≤ s < 7500

The number of steps is rounded to the nearest 1000 . Divide this place value by 2 , and then add and subtract this amount to the given value to give the maximum and minimum values for your error interval.

3. The circumference, C , of a circle is 63.5 cm , rounded to three significant figures.

Complete the error interval for the circumference of the circle.

….. ≤ C < ……

63.4 cm ≤ C < 63.5 cm

63.49 cm ≤ C < 63.59 cm

63.45 cm ≤ C < 63.55 cm

63 cm ≤ C < 64 cm

The circumference is rounded to three significant figures, which takes us up to the tenths. Divide this place value by 2 , and then add and subtract this amount to the given value to give the maximum and minimum values for your error interval.

4. The length, L , of a river is 5700 kilometres, truncated to two significant figures.

Find the error interval of the length of the river.

5600 km ≤ L < 5800 km

5650 km ≤ L < 5750 km

5700 km ≤ L < 5800 km

5650 km ≤ L < 5800 km

The length is truncated to two significant figures, which is in the hundreds column. Add this place value to the given value. This will be the maximum value of your error interval, the given value will be your minimum.

5. The age, A , of a fossil is 34,000,000 years to the nearest million.

Find the error interval for the age of the fossil.

33,000,000 years ≤ A < 34,000,000 years

33,500,000 years ≤ A < 34,500,000 years

33,550,000 years ≤ A < 34,550,000 years

34,000,000 years ≤ A < 35,000,000 years

The age is rounded to the nearest 1,000,000 . Divide this place value by 2 , and then add and subtract this amount to the given value to give the maximum and minimum values for your error interval.

6. An average speed calculation, S , is 93.2 km/h truncated to one decimal place.

Find the error interval for S .

93.25 km/h ≤ S < 93.35 km/h

93.15 km/h ≤ S < 93.35 km/h

93 km/h ≤ S < 94 km/h

93.2 km/h ≤ S < 93.3 km/h

The speed is truncated to the tenths. Add this place value to the given value. This will be the maximum value of your error interval, the given value will be your minimum.

1. Kevin truncates the number x to one decimal place. The result is 8.4 .

Write the error interval for x .

**(2 marks)**

Show answer

8.4 or 8.5

**(1)**

8.4 ≤ x < 8.5

**(1)**

2. Ana rounds the number y to 2 significant figures. The result is 7200 .

Write the error interval for y .

**(2 marks)**

Show answer

7150 or 7250

**(1)**

7150 ≤ y < 7250

**(1)**

3. Amir measures a door to be 782mm wide to the nearest mm .

Write the error interval for the width of the door w .

**(2 marks)**

Show answer

781.5 or 782.5

**(2)**

781.5 mm ≤ w < 782.5 mm

**(2)**

You have now learned how to:

- Apply and interpret limits of accuracy when rounding or truncating

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