# Accuracy of Measurement Problems

## Topic Overview

During your GCSE Maths Exam, you may be asked to give an answer in a simplified form. This guide will introduce you to several different methods which will make it easier to accurately simplify your answers.

This is often referred to as '**truncation**'. Truncation is the term given for limiting the number of digits right of the decimal point, by discarding the least significant ones.

For example, to truncate the number 8.9853409583490584 to 4 decimal digits, you are being asked to regard solely the 4 digits to the right of the decimal point: 8.9853. Truncation is equivalent to rounding towards zero.

## Key Concepts

In the new linear GCSE Maths paper, you will be work out several problems involving measurements and accuracy. According to the Edexcel Revision Checklist for the linear GCSE Maths paper, you will be required to:

- Approximate to specified or appropriate degrees of accuracy including a given power of 10, number of decimal places and significant figures

Listed below are a series of summaries and worked examples to help you solidify your knowledge about the accuracy of measurements.

## Worked Examples

**1 - Rounding numbers**

Rounding is often used when it is not necessary to give the exact figure. For example, if you wanted to know the attendance figure at an event, approximately 74,000 is acceptable rather than knowing the exact figure of 74,281.

In these cases, it is possible to 'round off' large numbers to the nearest thousand, hundred, ten, or whole number; depending on the size of the figure in question. When rounding, the following rule always applies:

If the next digit is **5 or more**, you 'round up' to the nearest figure. If the digit is **4 or less**, you '**round down**' and the figure stays the same.

*Example*

(a) - Round **56893** to the nearest thousand

*Solution*

(a) - In order to approximately round a number, you must break down the figure in question.

For example, in this question you are being asked to round to the nearest thousand. To do this, you must look at the digits in the thousands ; this is 6.

This means that 56893 lies between 56000 and 57000. The digit to the right of the thousands is 8. This means that 56893 is closer to 57000 than 56000.

Therefore, you can round 56893 to the nearest thousand, which is 57,000. 56893 to the nearest thousand = 57,000

*Example*

(b) - Round 56893 to the nearest ten

*Solution*

(b) - Once again, you need to look at the digits in question. When rounding the number 56893 to the nearest ten, it lies between 56890 and 56900.

The specific digit in question is 3, which is closer to 56890. Therefore, you must 'round down' to 56890.

56893 to the nearest ten = 56890

**2 - Decimal places**

In some cases, it is more accurate to round to the nearest decimal place or 'dp' rather than the nearest whole number.

When rounding to the nearest decimal place or decimal, the same rules apply as those for rounding up whole figures: if the decimal place in question is 5 or more, we 'round up' and if the decimal place is 4 or less, the digit stays as it is.

*Example*

(a) - Write 3.62859768 to 2 decimal places (dp)

*Solution*

(a) - As you are being asked to round to 2 decimal places, you need to look at the second decimal digit. In this case it is 2.

This means that the number lies between 3.62 and 3.63. Now you must look at the digit which follows it. This digit is 8. Therefore you must 'round up' to 3.63.

As such, 3.62859768 rounded to 2 decimal places = 3.63

*Example*

(b) - Round off the number 5.9634545 to 1 decimal place (1dp)

*Solution*

(b) - You have been asked to give your answer to one decimal place. To do this, look at the digit which is one place after the decimal point: 9.

This means that, to one decimal place, your answer lies between 5.9 and 6.0. The digit directly after the 9 is 6. Therefore you must round up.

5.9634545 to 1 decimal place (1dp) = 6.0

You may also be asked to round to a specific decimal place in a calculator based question.

*Example*

(c) - On a calculator, work out √47, giving your answer correct to one decimal place (1dp)

*Solution*

(c) - Using your calculator, you can work out that √47 = 6.8556546004...
You have been asked to give your answer to one decimal place. To do this, look at the digit which is one place after the decimal point: 8.

This means that, to one decimal place, your answer lies between 6.8 and 6.9. The digit directly after the 8 is 5. According to the rule mentioned previously, if this digit is 5 or more you must round up.

Therefore, √47 to one decimal place = 6.9

**3 - Significant figures**

Another method of giving an approximated answer is to round off using **significant figures**.

The term **'significant**' means 'having meaning', so you are only being asked to write down the numbers which are most important to the question you are being asked.

For example, when regarding larger numbers such as 57548, the 5 is the most significant figure because it tells us that the number is 5 hundred thousand and something.

If you were asked to give your answer to 2 significant figures, 7 would be the next most significant digit because it tells you that the number is 57 thousand and something, and so on.

Similarly, when regarding smaller numbers such as **0.0000073842**, the **7** is the most significant figure because it tells us that the number is **7 millionths and something**.

If you were asked to give your answer to 2 significant figures, 3 would be the next most significant digit because it would be 73 thousandths and something, and so on.

Once again, the same rules for rounding up are applied to significant figures: if the next digit is **5 or more**, you **round up** and if the next number is **4 or less**, you **do not round up**.

*Example*

(a) - Write the number 4624978 correct to 1 significant figure

*Solution*

(a) -

4 is the first significant figure. Therefore the number lies between 40,000 and 50,000. The digit which follows is 6, so you must round up.

Therefore, 4624978 correct to 1 significant figure = 50,000

*Example*

(b) - Write the number 0.00268 correct to 1 significant figure

*Solution*

(b) -

2 is the first significant figure. Therefore the number lies between 0.002 and 0.003. The digit which follows is 4 so you round down.

Therefore, 0.00268 correct to 1 significant figure = 0.002

*Example*

(c) - Write the number 0.0000067852 correct to 2 significant figures

*Solution*

(c) - In this example, the 2 most significant figures are 6 and 7. The digit following the 7 is 8 so you must round up.

Therefore 0.0000067852 correct to 2 significant figures = 0.0000068

**4 - Upper and Lower Bounds**

Upper and lower bounds are used to demonstrate what the range of values of a particular object may be.

*Example*

(a) - A fence is 10m long to the nearest m. What is the range of possible lengths it could be?

*Solution*

(a) - In this example, the smallest possible length of the piece of the fence in order for it to be rounded up to 10m would be 9.5 m.

Similarly, the highest possible length for the piece of wood to be for it to be rounded down to 10m must be less than 10.5m. This is because if the wood was 10.5m long or higher, it would then be rounded up to 11m.

Therefore, in this example, 10.5m is the upper limit for the length of wood. This value is known as the 'upper bound'.

Similarly, the lower limit for the length of wood is 9.5m. This value is known as the 'lower bound'.

*Example*

(b) - A pencil is 16.5cm long measured to the nearest tenth of a cm. What are the upper and lower bounds?

*Solution*

(b) - Anything above 16.45cm will round up to 16.5cm to the nearest tenth of a cm

Anything less than 16.55cm will round down to 16.5cm

Therefore the upper bound of the pencil is 16.55cm and the lower bound is 16.45cm.

## Exam Tips

- Remember that if a value is 5 or more, you round up
- If a value is 4 or less, you round down
- Upper bound refers to the highest possible value an object can be measured for it to be rounded down to a specific value
- Lower bound refers to the lowest possible value an object can be measured for it to be rounded up to a specific value

## Topic Summary

Once you understand the basic principles of rounding and estimating, it is a highly useful mathematical process which you can apply throughout your GCSE Maths exam and beyond. As long as you correctly place the decimal point, remember the rules of rounding up, and double check your working, you will be able to round and estimate mathematical values with ease!

## Related Topics

- Arithmetic with Fractions
- Indices
- Problem Solving with Decimals and Percentages
- Arithmetic with Positive and Negative Integers
- Standard Form
- Rational and Irrational Numbers
- Combinations