MTE3101 Knowing Numbers

Topic 3 Estimation of Quantities

3.1 Synopsis

This topic covers the skill of rounding off real numbers including whole numbers, fractions,

and decimals. A definition of standard forms, square roots and surds is also given.

Guidelines to enable you to round off numbers and find better estimates are provided to

refresh your memory.

3.2 Learning Outcome

Estimate quantities by rounding off real numbers including whole numbers, fractions

and decimals.

3.3 Conceptual Framework

3.4 Estimation of quantities

Estimating is an important mathematical skill and is a very handy tool for everyday life.

As such, we need to teach children to estimate amounts of money, lengths of time,

distances, and many other physical quantities. Various techniques can be used to

estimate quantities using certain guidelines. The process of estimation can take on

several forms, ranging from merely making a guess to finding an estimate or

approximate answer to a computation, often involving the use of mental mathematics.

Estimation of quantities

Rounding off numbers:Whole Numbers

Rounding off numbersFractions Decimals

Standard Forms Square roots and surds

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MTE3101 Knowing Numbers

Rounding is used to replace complicated numbers with simpler numbers. It is the best-

known and most useful computational estimation technique taught in schools. Unlike

other computational estimation techniques, rounding is often applied to an answer as

well as to individual numbers before a computation is performed.

3.5 Rounding off numbers

We often use rounding when estimating. Rounding gives an approximation answer

There are several rounding techniques that can be used to obtain an estimate. Each

technique involves rounding to a particular place. For example, when rounding, you are

finding the closest multiple of ten (or one hundred, or other place value) to your number.

Rounding off is a kind of estimating. When rounding off numbers, we can either round

up or round down. There are certain rules to be followed when rounding numbers, be it

whole numbers, fractions or decimals.

3.5.1 Rounding off whole numbers

Whole numbers can be rounded to the tens place, hundreds place, thousands place,

and so on. On the other hand, decimal numbers can be rounded to the nearest tenth,

hundredth, thousandth, or other decimal place.

Rounding is used to make a number easier to work with mentally. Rounded numbers are

only approximate and an exact answer generally cannot be obtained using rounded

numbers. There are two important reasons for estimation: (1) to form a reasonable opinion,

or (2) to check the reasonableness of an answer.

On a number line, you can see how rounding a number approximates its value.

The most common rounding technique used in schools is the round a 5 up method. This

method always rounds up numbers ending with a 5. One disadvantage of this method is

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MTE3101 Knowing Numbers

that estimates obtained when several 5’s are involved tend to be on the higher side. For

example, applying the “round a 5 up” to the nearest ten estimate for finding an estimate to

the sum of 35 + 45 + 55 + 65 + 75 gives a result that is 25 more than the exact sum of

275, since 40 + 50 + 60 + 70 + 80 = 300

Let us look at the example of rounding to powers of ten below:

Example:

The population of England is about 60 million. The populations, to the nearest hundred

thousand, of the five largest cities are:

London 6.4 million

Birmingham 1.0 million

Liverpool 0.5 million

Sheffield 0.4 million

Leeds 0.4 million

(a) What is the greatest population London could have?

(b) The tenth largest city is Coventry with a population of 0.3 million. Estimate the

percentage of the population of England who live in the ten largest cities.

Solution:

(a) 6.4 million is 6 400 000 and has been rounded to the nearest hundred thousand.

The greatest number that would be rounded to 6 400 000 is 6 449 999. If it was

one more, 6 450 000, it would be rounded to 6.5 million.

(b) The sixth to the ninth largest cities must have populations between 0.3 million

and 0.4 million. We estimate that the mean population of these population of

these cities is 0.35 million.

Total millions in ten largest cities = 6.4 + 1 + 0.5 + 0.4 + 0.4 + 4 x 0.35 + 0.3

= 10.4 million.

Percentage in ten largest cities = x 100%

= 17 % to the nearest percent.

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10.4 60

MTE3101 Knowing Numbers

3.5.2 Rounding off fractions and decimals

As mentioned earlier, decimal numbers can be rounded to the nearest tenth, hundredth,

thousandth, or other decimal places. There are rules to follow when rounding off numbers

to a particular number of decimal places.

Follow the steps below to round to a given number of decimal places.

(i) Keep the number of digits asked for after the decimal point.

(ii) Before rounding off, look at the next decimal place to the right.

(iii) If this digit is 5 or greater, increase the last digit you are keeping by 1 (round up).

Otherwise leave it as it is (round down).

(iv) Discard the unwanted digits away.

What number will you get when you round 3.417824 to 2 decimal places?

Yes, you are right! The answer is 3.42 .

How would you round a given decimal number to the nearest whole number? Can we use

the same method as above? Do the same rules apply when you round decimal numbers to

the nearest whole number?

The rules are the same as above. In other words, when rounding a decimal number to the

nearest whole number, we are actually rounding the number to 0 decimal place.

Let’s look at the example below.

Example:

6.5489 rounded to the nearest whole number is 7.

Try doing the following questions.

Round the following numbers to the nearest whole number.

0.985

325.092

45.7

¼

⅞

(Note: When rounding off fractions, convert the fractions to decimal form before rounding

to

a particular decimal place.)

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MTE3101 Knowing Numbers

When estimating quantities, we often asked ourselves if our estimate was correct or

otherwise. An important feature of estimation of quantities pertains to how accurate is the

estimate obtained. When rounding off numbers, the degree of accuracy can vary.

Sometimes an exact answer is not always needed and an estimate is sufficient. Other

times an exact answer is needed and an estimate is not sufficient.

Useful Tip: Never round the number until the final answer is found so that a more

accurate answer can be obtained.

Example:

If we are finding the answer to , do not round the answer of

before multiplying by 4.8. Perform the stated operations first before rounding off the final

product. That is, the answer should be 22.03 instead of 22.08 to 2 decimal places.

The degree of accuracy of a rounded number depends on the situation and the user’s

requirements. For example:

An estimate is good enough when calculating the amount of drinks needed for a

party.

An exact answer is needed when calculating the amount of dose to be injected to a

patient.

Now, put on your thinking cap!

Decide the degree of accuracy needed for the following situations using the choices

provided below. Give reasons for your answer.

(i) A veterinarian uses the mass of a cat to calculate how much medicine to inject.

(ii) A builder calculates the amount of wood he needs to buy to build a shed.

(iii) A chef calculates how much flour and sugar are needed for a cake.

(iv) A reporter calculates the number of people who have attended an Art exhibition

last week.

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9.64

2.1

X 4.8 9.64

2.1

A. As accurate as possible.B. A rough estimate will do.C. An estimate is fine, but it must be reasonably accurate.

MTE3101 Knowing Numbers

Estimation is an important part of problem solving and involves using common sense to

think about the reasonableness of the answers you obtained. According to Webster’s New

World Dictionary, to ‘estimate’ means ‘to form an opinion or a judgment about’ or to

‘calculate approximately’. Thus, developing estimation skills is an important aspect in the

mathematics classroom as they can be applied in our daily lives. Some guidelines for

estimating are given below.

Guidelines for estimating

Look for ‘nice’ numbers that enable you to do the calculation mentally.

Example: 200 ÷ 5.8 ≈ 200 ÷ 5 rather than 200 ÷ 6

Example:

( ≈ means ‘is approximately equal to’ )

Look for numbers that will cancel.

Example:

When multiplying or dividing never approximate a number to 0.

Use 0.1, 0.01 or 0.001, etc.

Example:

105.6 x 0.014 should not be approximated as 100 x 0. It is better to use 100 x 0.01 or

100 x 1 , which gives an estimate of 1 .

100

When multiplying two numbers, try to round one up and one down.

When dividing two numbers, try to round both numbers up or both numbers

down.

Example: It is better to estimate 4.5 x 3.5 as 5 x 3 or 4 x 4 rathan than 5 x 4 since

4.5 x 3.5 = 15.75 so 5 x 3 = 15 or 4 x 4 = 16 both give a closer estimate than

5 x 4=20.

Example: It is better to estimate as rather than . since

and =

( Note: There is often more than one possible estimate for an answer. )

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72.6 x 347.05 0.86 ≈ (100 x 350) ÷ 1

12 x 500 4≈

=1500

12.46 x 486.21 3.78

83.2 8.5

80 8

81 9

83.2 8.5

80 8= 9.79 (2 d.p.) and = 10 gives a closer estimate than

81 9 = 9.

MTE3101 Knowing Numbers

3.6 Standard form

The Standard form is a way of writing down very large or very small numbers in a more tidy

and convenient manner. For example, 10³ = 1000, so 4 × 10³ = 4000 . Therefore, 4000

can be written as 4 × 10³. Much larger numbers than the above example can also be

written down easily using standard form.

Small numbers can also be written in standard form. However, instead of the index being

positive (e.g. In the example above, the index was 3), it will be negative.

The rules when writing a number in standard form is that first you write down a number

between 1 and 10, followed by ‘ X 10’ (that is, times to the powers of ten).

Scientists often work with very large numbers such as distances from the Earth to other

planets measured in light years, the Earth’s mass measured in kilograms, etc. The

scientific notation for a nonzero number is that number written as a power of 10 times

another number x, such that x is between 1 and 10, including 1; that is , 1 ≤ x < 10 . In

short, the scientific notation is a method of writing numbers in terms of a decimal number

between 1 and 10 multiplied by a power of 10. For example, 10,592 is written as 1.0592 ×

104 in the scientific notation. Scientific notation has a number of useful properties and is

favoured by scientists, mathematicians and engineers, who work with such numbers.

Some examples of measurement in scientific notation are given below:

The speed of light in SI units is 2.99792458×108 m/s .

An electron's mass is about

0.000 000 000 000 000 000 000 000 000 000 910 938 22 kg. In scientific notation,

this is written as 9.109 382 2×10-31 kg.

The Earth's mass is about 5,973,600,000,000,000,000,000,000 kg. In scientific

notation, this is written as 5.9736×1024 kg.

The Earth's circumference is approximately 40,000,000 m. In scientific notation, this

is written as 4×107 m.

You would have noticed that the definition of scientific notation is similar to that of the

standard form. Hence, the scientific notation is also known as the Standard form. It is also

known as the exponential notation since it involves powers of ten. In short, it is a way of

writing numbers that accommodates values too large or too small to be conveniently written

in standard decimal notation.

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MTE3101 Knowing Numbers

The chart below gives some other examples of numbers expressed in scientific notation or

standard form.

Ordinary decimal notation Scientific notation 300 3×102

4,000 4×103

5,720,000,000 5.72×109

−0.000 000 006 1 −6.1×10−9

In conclusion, it is important to note that:

Numbers in standard form can be rounded to a particular decimal place for a neater presentation.

Now, let’s look at some examples.

Example 1

Write 81 900 000 000 000 in standard form:

81 900 000 000 000 = 8.19 × 1013

Note:The answer involves 1013 because the decimal point has been moved 13 places to

the left to get the number to be 8.19

Example 2

Write 0.000 001 2 in standard form:

0.000 001 2 = 1.2 × 10-6

The answer involves 10-6 because the decimal point has been moved 6 places to the right

to get the number to be 1.2

Now, do the following:

Write in standard form. Round your answer to 2 decimal places.

7 891 124

0.000 005 437

- 124 809

Well done!

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MTE3101 Knowing Numbers

Fun with calculators!

A calculator can also be used easily to help you to write very large or very small numbers in

standard form. On a calculator, you usually enter a number in standard form as follows:

Type in the first number (the one between 1 and 10). Press EXP . Type in the power to

which the 10 is risen. Explore on your own. Have fun!

Manipulation in Standard Form

This is best explained with an example:

Example

If the values of p and q are 8 × 105 and 5 × 10-2 respectively, calculate (i) p x q ; (ii) p ÷ q.

Give your answers in standard form.

Solution

(i) Multiply the two first bits of the numbers together and the two second bits together:

8 × 5 × 105 × 10-2 = 40 × 103

(Remember : 105 × 10-2 = 103)

The question asks for the answer to be in standard form, but this is not in standard form

because the first part (that is, 40) should bewritten as a number between 1 and 10, that

is, it should be written as 4 × 104 or 4.0 x 104 .

(ii) This time, divide the two first bits of the standard forms. Divide the two second bits.

(8 ÷ 5) × (105 ÷ 10-2) = 1.6 × 107

Now try these:

Compute and give your answers in standard form. Round your answers to 2

decimal places.

67 X 1289

8942 ÷ 0.127

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MTE3101 Knowing Numbers

3.7 Square roots and surds

The square root of a number is that special value that when multiplied by itself gives the

original number. For example, the square root of 9 is 3, because when 3 is multiplied

by itself you get 9. A better understanding of what a square root is can be derived by

using the following analogy.

Note: When you see "root" think

"I know the tree, but what is the root that produced it?"

In this case the tree is "9", and the root is "3".

The symbol below is used to denote the square root.

The Square Root Symbol

This is the special symbol that means "square root", it is sort of like a tick, and actually started hundreds of years ago as a dot with a flick upwards.

It is called the radical, and always makes mathematics look important!

For example, you can use it like this: . This is read as "the square root of 9

equals 3" .

Square Roots of Other Numbers

It is easy to work out the square root of a perfect square, but it is really hard to work out

other square roots. For example, what is the square root of 10?

Solution:

Since 3 × 3 = 9 and 4 × 4 = 16, we can make an estimate by guessing that the answer

should be between 3 and 4. Working further to arrive at the answer can be a slow and

laborious process. For example,

Let's try 3.5: 3.5 × 3.5 = 12.25

Let's try 3.2: 3.2 × 3.2 = 10.24

Let's try 3.1: 3.1 × 3.1 = 9.61

From the above, we should be able to guess that the answer lies between 3.1 and 3.2 . A

good and reasonable estimate will be approximately 3.15 .

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MTE3101 Knowing Numbers

Checking out the answer with a calculator gives the following answer:

3.1622776601683793319988935444327

However, in this case, the digits just go on and on, without any pattern. Thus, even the

calculator's answer is only an approximation ! To make the answer simpler, it can be

rounded to a particular place, say 3 decimal places or more. Based on the above figure,

what is √10 in standard form rounded to 3 decimal places? Right! It is 3.162 .

The answers for square roots of numbers which are not perfect squares are usually

rounded off to either two decimal places or more.

Can you round the square root of ten to 2 decimal places or 1 decimal place?

What happens if we round it to the nearest whole number? The answer will be 3 which is

actually the square root of 9 and not 10. Thus, we will get an estimate which is way too low

and is therefore not accurate or exact. Thus, rounding to the nearest whole number always

turns out too low.

Surds are numbers that cannot be simplified to remove a square root (or cube root, etc.).

In other words, they are numbers that are left in 'square root form' (or 'cube root form' etc).

The reason we leave them as surds is because in decimal form they would go on forever

and thus, this is a very clumsy way of writing them.  We can also estimate the values of

surds.

For example, √2 (square root of 2) cannot be simplified further so it is a surd but √4

(square root of 4) can be simplified (it equals 2), thus, it is not a surd. The chart below

can give you a clearer picture of what surds are.

Have a look at these:

NumberSimplife

dAs a Decimal

Surd ornot?

√2 √2 1.4142135(etc) Surd

√3 √3 1.7320508(etc) Surd

√4 2 2 Not a surd

√(1/4) 1/2 0.5 Not a surd3√(11) 3√(11) 2.2239800(etc) Surd3√(27) 3 3 Not a surd5√(3) 5√(3) 1.2457309(etc) Surd

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MTE3101 Knowing Numbers

As you can see, the surds have a decimal which goes on forever without repeating, and

that makes them Irrational numbers.

Point of Interest:

Do you know the origin of the word "Surd" ?

Around 820 A.D., the Persian mathematician, al-Khwarizmi, from whom we got the name

"Algorithm", called irrational numbers "'inaudible". This was later translated to the Latin

surdus ("deaf" or "mute"). As a matter of fact, "Surd" used to be another name for

"Irrational", but it is now used for a root that is irrational.

Parting shot:

If it is a root and is irrational, then it is a surd. Remember: NOT ALL roots are surds!

Now, go through the following example.

Example:

Estimate the value of √29.

Solution

Since √29 is between √25 = 5 and √36 = 6 so the value of √29 is between 5 and 6.

Try this:

Give an estimate for the surds given below? Check your answers with a calculator

and round off to 2 decimal places.

√37

√230

√0.0078

√0.01 569

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MTE3101 Knowing Numbers

Things to do:

1. Refer to your Resource Materials and read the notes on “Place value, Ordering and

Rounding” in Tipler, M. J. et al. (2003). New national framework mathematics. ( pp. 26 –

33 and 67–72.)

Complete the tasks on pp. 29 and pp. 67

Do related exercises from Exercise 6 on pp. 69 – 72.

2. Search for more exercises on the estimation of quantities from other sources such as

internet or books. Do the exercises obtained.

Congratulations! You have now come to the end of this module.

Happy Studying and Good Luck!

References

Tipler, M. J. et al. (2003). New national framework mathematics. United Kingdom: Nelson

Thornes Limited.

Websites on:

1. Rounding numbers:

http://www.enchantedlearning.com/math/rounding/

2. Estimating and rounding decimals:

http://www.math.com/school/subject1/lessons/S1U1L3GL.html#

GCSE Maths/www.mathsrevision.net/gcse/pages324e.html?page=43 ?

3. Square roots and surds:

http://www.mathsisfun.com/square-root.html

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MTE3101 Knowing Numbers

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