Chapter 7: Estimation and approximation
7.1 Estimation
Estimation is the process of making a guess about the size or cost of something, without doing the actual measurement or calculation. Estimations are sometimes needed because we do not have all the facts, or because we are pressed for time. Estimation can also help you to see whether the answer to a problem makes sense.
estimation Estimation is the process of making a guess without doing the actual measurement or calculation.
Estimating distances and dimensions
A distance can be described as the length of the space between two points. An example of a distance is the length of the path between the door of the classroom and the door to the principal's office.
The dimensions of an object are the measurable lengths that we use to determine the size of the object. Examples of dimensions are the length, breadth (or width) and height of a box.
distance Distance is the length of the space between two points.
dimensions The dimensions of an object are the measurable lengths that we use to determine the size of the object, such as its length, breadth and height.
When we estimate distances and dimensions, it is important to use the correct unit of measurement. The SI unit for length is the metre. A length of one metre (1 m) is more or less equal to:
 the height of a 6yearold child
 a large step of an adult
 the length of a prayer rug.
Other units of measurement used often are multiples of ten of 1 m. The most common multiples are shown in the table below.
\begin{array}{cccc} \hline \textbf{Name} & \textbf{Abbreviation} & \textbf{Size} & \textbf{Example} \newline \hline \text{millimetre} & \text{mm} & 1\text{ mm}=\frac{1}{1,000}\text{ m}=0,001\text{ m} & \text{thickness of a plastic bank card } \newline \hline \text{centimetre} & \text{cm} & 1\text{ cm}=\frac{1}{100}\text{ m}=0,01\text{ m} & \text{width of a finger nail} \newline \hline \text{kilometre} & \text{km} & 1\text{ km}=1,000\text{ m} & \text{distance walked in 12 minutes} \newline \hline \end{array}SI unit An SI unit is a basic unit of measurement that forms part of the International System of Units.
Worked example 7.1: Evaluating an estimation of a dimension
Adebankole is 14 years old. He estimated his own height as 160 cm. Evaluate his estimation.

Step 1: Evaluate whether the unit of measurement is appropriate.
Look at your ruler to see how long 1 cm is. It is quite short compared to the height of a teenager.
It might be better to use metres, but then Adebankole would need to use a decimal point in his estimation. To avoid the decimal point, centimetres is an appropriate unit.

Step 2: Evaluate whether the value of the estimation makes sense.
The height of a 6yearold child is roughly 1 m. That is equal to 100 cm. So Adebankole's height is most probably more than 100 cm.
Most humans are not taller than 2 m, which is equal to 200 cm.
Adebankole's estimation is between 100 cm and 200 cm, so it makes sense.
Exercise 7.1: Estimate distances and dimensions in your classroom
Estimate the following distances and dimensions. Include an appropriate unit of measurement in each case. Then use a ruler, metre rule or tape measure to get the real measurements. Compare the real measurements to your estimations.
 Length of your pencil
 Width of your eraser
 Length of your Mathematics workbook
 Height of your thickest textbook
 Length of the classroom
 Width of the classroom
 Width of the blackboard
 Height of your teacher's table
 Width of the classroom door
 Distance from the doorknob of the Mathematics classroom to the doorknob of the classroom next to it
Exercise 7.2: Estimate everyday distances and dimensions
Estimate the following distances and dimensions by deciding which one of the options a) to d) is the most correct.
 Diameter of a 50 kobo coin
 5 mm
 24.5 mm
 80.5 mm
 120 mm
b) 24.5 mm

Length of a ₦10 note
 20 mm
 1.5 cm
 25 cm
 130 mm
d) 130 mm

Height of a standard door
 450 mm
 35 cm
 2.1 m
 12 m
c) 2.1 m

Maximum length of a standard football field
 120 m
 25 m
 850 cm
 950 m
a) 120 m

Shortest driving distance from Murtala Muhammed International Airport to Ibadan
 127 km
 1,270 m
 11,270 km
 100,270 mm
a) 127 km
Estimate capacity
Volume is the amount of space occupied by an object or the amount of space inside a container. The amount of liquid that a container can hold is called the capacity of the container.
capacity The amount of liquid that a container can hold is called the capacity of the container.
The SI unit for capacity is the litre. A capacity of one litre (1 L) is more or less equal to:
 six cups of tea
 a carton of milk
 three cans of soft drink.
Other units of measurement used often are multiples of ten of 1 L. The most common multiples are shown in the table below.
\begin{array}{cccc} \hline \textbf{Name} & \textbf{Abbreviation} & \textbf{Size} & \textbf{Example} \newline \hline \text{millilitre} & \text{ml} & 1\text{ ml}=\frac{1}{1,000}\text{ L}=0,001\text{ L} & \text{twenty drops of water } \newline \hline \text{centilitre} & \text{cl} & 1\text{ cl}=\frac{1}{100}\text{ L}=0,01\text{ L} & \text{two teaspoons of water} \newline \hline \text{kilolitre} & \text{kl} & 1\text{ kl}=1,000\text{ L} & \text{mediumsized inflatable swimming pool} \newline \hline \end{array}Worked example 7.2: Evaluating an estimation of capacity
Amaka is feeling sick. The doctor gives her a bottle of medicine. She estimates that the capacity of the bottle is 250 cl. Evaluate her estimation.

Step 1: Evaluate whether the unit of measurement is appropriate.
Think about how much liquid 1 cl is. It is about two teaspoons.
One teaspoon is about 5 ml. The instructions on a bottle of medicine normally tells you how many teaspoons to take at time. Therefore, both millilitres and centilitres are appropriate units.

Step 2: Evaluate whether the value of the estimation makes sense.
If there is 250 cl in the bottle, it means the bottle holds 500 teaspoons of medicine.
At two teaspoons twice a day, it would take days to finish the medicine. That is more than 4 months! It is not possible to get so much medicine from a normal medicine bottle.
Amaka's estimation does not make sense. An estimation of 25 cl, which is 250 ml, would be a better estimation.
Exercise 7.3: Estimate capacity of containers
You might use the containers listed below at school or at home. Estimate the capacity of the containers. Include an appropriate unit of measurement in each case. Then use measuring cylinders from the science laboratory, or a set of measuring spoons and cups used for baking, to get the real measurements. Compare the real measurements to your estimations.
 Tea cup
 Soft drink bottle
 Test tube
 Lunch box
 Water glass
Exercise 7.4: Estimate capacity of everyday containers
Estimate the capacity of the following containers by deciding which one of the options a) to d) is the most correct.
 Milk tin
 3 cl
 25 ml
 300 cl
 350 ml
d) 350 ml

Kerosene tin
 1.5 kl
 7 ml
 2 L
 5 cl
c) 2 L

Water bucket
 750 cl
 35 cl
 550 ml
 45 L
a) 750 cl

Petrol tank of a small car
 850 ml
 50 L
 950 L
 15 kl
b) 50 L

Petrol tanker (large truck that transports petrol)
 1,200 cl
 950 cl
 18 kl
 75 L
c) 18 kl
Estimate mass
The mass of an object is a measure of how much matter is in that object. It tells us how heavy or how light the object is.
mass The mass of an object is a measure of how much matter is in an object.
The SI unit for mass is the gram. A mass of one gram (1 g) is more or less equal to:
 a small paperclip
 one millilitre of pure water
 a quarter teaspoon of sugar.
Other units of measurement used often are multiples of ten of 1 g. The most common multiples are shown in the table below.
\begin{array}{cccc} \hline \textbf{Name} & \textbf{Abbreviation} & \textbf{Size} & \textbf{Example} \newline \hline \text{milligram} & \text{mg} & 1\text{ mg}=\frac{1}{1,000}\text{ g}=0,001\text{ g} & \text{small grain of sand } \newline \hline \text{kilogram} & \text{kg} & 1\text{ kg}=1,000\text{ g} & \text{one litre of water} \newline \hline \text{tonne} & \text{t} & 1\text{ t}=1,000,000\text{ g}=1,000\text{ kg} & \text{small car } \newline \hline \end{array}Worked example 7.3: Evaluating an estimation of mass
Danladi's mother sends his to the corner shop to buy a cup of rice. He carries it home in a small bag. He estimates that the mass of the rice is 50 g. Evaluate his estimation.

Step 1: Evaluate whether the unit of measurement is appropriate.
Think about how light 1 g is. It is about a quarter teaspoon of sugar. This is not a lot compared to a cup of rice.
It might be better to use kilograms, but then Danladi would need to use a decimal point in his estimation. To avoid the decimal point, gram is an appropriate unit.

Step 2: Evaluate whether the value of the estimation makes sense.
The mass of a teaspoon of sugar is about 4 g. So 50 g of sugar is = 12.5 teaspoons. Even though the same amount of sugar and rice have different masses, there is a big difference between 12.5 teaspoons and 1 cup.
Danladi's estimation does not make sense. An estimation of 150 g would be a better estimation.
Exercise 7.5: Estimate mass of objects
Estimate the mass of the following objects. Include an appropriate unit of measurement in each case. Then use an electronic balance from the science laboratory or a kitchen scale to get the real measurements. Compare the real measurements to your estimations.
 Your eraser
 One of your pens
 Your thinnest textbook
 Full box of chalk
 Volleyball
Exercise 7.6: Estimate mass of everyday objects
Estimate the mass of the following objects by deciding which one of the options a) to d) is the most correct.
 50 kobo coin
 8 mg
 5.5 g
 250 g
 1.5 kg
b) 5.5 g

Packet of sugar cubes
 475 g
 45 g
 550 mg
 45 kg
a) 475 g

Basketball
 950 mg
 45 g
 625 g
 12 kg
c) 625 g

A male teenager
 250 kg
 70 kg
 950 g
 0.5 t
b) 70 kg

An adult cow
 65 kg
 700 kg
 50 t
 850 g
b) 700 kg
7.2 Approximation and rounding off
An approximate answer is almost correct, but not exact. Approximation is the process of using rounding to quickly determine a fairly accurate answer to a calculation. Approximation is very useful in situations where you have to do a calculation quickly and without paper, a pen or a calculator.
approximation Approximation is the process of using rounding to quickly determine a fairly accurate answer to a calculation.
Rounding off numbers
When we round off numbers, we make them less exact, but easier to work with. We usually round off a number to a specific multiple of 10.
rounding off Rounding off is the process of making a number less exact but easier to work with, by adjusting it to a specific multiple of 10.
When we round off a number to a specific multiple of 10, we have to identify the digit that represents the multiple, and then determine whether than digit remains the same or increases by one.
Worked example 7.4: Rounding off to the nearest ten
A smartphone is advertised for ₦47,689. Round off this number to the nearest ten.

Step 1: Write down half of the required multiple of 10.

Step 2: Determine which digit represents the required multiple of 10.
You may use a place value table to help you with this:
\begin{array}{cccccccccc} \hline \text{billion} & \text{hm} & \text{tm} & \text{million} &\text{hth} & \text{tth} & \text{thousand} & \text{ h } & \text{ t } & \text{ u } \newline \hline & & & & & 4 & 7 & 6 & 8 & 9 \newline \hline \end{array}The 8 represents ten.

Step 3: Consider the digits to the right of the one you identified in Step 2. If they are less than the value in Step 1, the digit you identified remains the same. If they are equal to or more than the value in Step 1, the digit you identified increases by 1.
The digit to the right of the 8 is 9.
The 8 increases to a 9. 
Step 4: Write down the rounded number with the digit you determined in Step 3, and all the digits to the right become zero.
₦47,690
If we round off 47,689 to the nearest ten, we actually want to know whether this number is closer to 47,680 or 47,690. The number 47,689 is 1 unit away from 47,690 and 9 units away from 47,680. That is why it is rounded off to 47,690.
Worked example 7.5: Rounding off to the nearest hundred
A handbag is advertised for ₦2,950. Round off this number to the nearest hundred.

Step 1: Write down half of the required multiple of 10.

Step 2: Determine which digit represents the required multiple of 10.
You may use a place value table to help you with this:
\begin{array}{cccccccccc} \hline \text{billion} & \text{hm} & \text{tm} & \text{million} &\text{hth} & \text{tth} & \text{thousand} & \text{ h } & \text{ t } & \text{ u } \newline \hline & & & & & & 2 & 9 & 5 & 0 \newline \hline \end{array}The 9 represents the hundreds.

Step 3: Consider the digits to the right of the one you identified in Step 2. If they are less than the value in Step 1, the digit you identified remains the same. If they are equal to or more than the value in Step 1, the digit you identified increases by 1.
The digits to the right of the 9 are 50.
The 9 must increase by 1. 
Step 4: If the digit that must increase is a 9, it changes to a 0. The digit to its left increases by 1.
The 9 must change to a 0.
The 2 to the left of the 9 must change to a 3. 
Step 5: Write down the rounded number with the digits you determined in Step 4, and all the digits to the right become zeroes.
₦3,000
If we round off 2,950 to the nearest hundred, we actually want to know whether this number is closer to 2,900 or 3,000. The number 2,950 is exactly halfway between 2,900 and 3,000. By convention, we round up to the higher number.
Worked example 7.6: Rounding off to the nearest thousand
On a specific day, the population of Nigeria is 201,195,431. Round off this number to the nearest thousand.

Step 1: Write down half of the required multiple of 10.

Step 2: Determine which digit represents the required multiple of 10.
You may use a place value table to help you with this:
\begin{array}{cccccccccc} \hline \text{billion} & \text{hm} & \text{tm} & \text{million} &\text{hth} & \text{tth} & \text{thousand} & \text{ h } & \text{ t } & \text{ u } \newline \hline & 2 & 0 & 1 & 1 & 9 & 5 & 4 & 3 & 1 \newline \hline \end{array}The 5 represents the thousands.

Step 3: Consider the digits to the right of the one you identified in Step 2. If they are less than the value in Step 1, the digit you identified remains the same. If they are equal to or more than the value in Step 1, the digit you identified increases by 1.
The digits to the right of the 5 are 431.
431<500
The 5 remains the same. 
Step 4: Write down the rounded number with the digit you determined in Step 3, and all the digits to the right of it as zeroes.
201,195,000
If we round off 201,195,431 to the nearest thousand, we actually want to know whether this number is closer to 201,195,000 or 201,196,000. The number 201,195,431 is 431 units away from 201,195,000 and 569 units away from 201,196,000. That is why it is rounded off to 201,195,000.
Exercise 7.7: Round off numbers
Round off each of the following numbers to the required multiple of 10.

26,548 to the nearest ten
26,550

48,849 to the nearest hundred
48,800

37,627 to the nearest thousand
38,000

58,503 to the nearest thousand
59,000

29,618 to the nearest thousand
30,000

5,319,432 to the nearest thousand
5,319,000

19,553 to the nearest thousand
20,000

2,905,047 to the nearest hundred
2,905,000

5,095 to the nearest ten
5,100

512,909,050 to the nearest hundred
512,909,100
Using approximation in calculations
The aim of using approximation in calculations is to round off numbers to values that you can work with more easily when you are using mental arithmetic. When we work with approximations, we use the symbol instead of an equals sign. For example, .
Worked example 7.7: Using approximation in calculations
Determine the approximate value of .

Step 1: Round the numbers to multiples of ten that will allow an easy calculation.

Step 2: Do the calculation using mental arithmetic.
\begin{align} 475\times 17 &\approx 500 \times20 \\ &=10,000 \end{align}
Exercise 7.8: Calculate approximate values
Determine the approximate answer to each of the following.

\begin{align} 38+52 &\approx 40 +50 \\ &\approx 90 \end{align}

\begin{align} 543+127 &\approx 540 +130 \\ &\approx 670 \end{align}

\begin{align} 6923 &\approx 70  20 \\ &\approx 50 \end{align}

\begin{align} 285162 &\approx 290  160 \\ &\approx 130 \end{align}

\begin{align} 1,403582 &\approx 1,400  600 \\ &\approx 800 \end{align}

\begin{align} 13 \times 72 &\approx 10 \times 70 \\ &\approx 700 \end{align}

\begin{align} 233 \times 36 &\approx 200 \times 40 \\ &\approx 8,000 \end{align}

\begin{align} 7\frac{1}{5} \times 2\frac{7}{8} &\approx 7 \times 3 \\ &\approx 21 \end{align}

\begin{align} \frac{825}{41} &\approx \frac{800}{40} \\ &\approx 20 \end{align}

\begin{align} \frac{3,612}{123} &\approx \frac{3,600}{120} \\ &\approx 30 \end{align}
7.3 Practical applications
The advantages of being able to estimate and approximate have been mentioned already. This section provides more opportunity for you to practise.
Exercise 7.9: Evaluate test answers
Each of the following is an incorrect answer that was given by a student in a test. Say what mistake the student made and give the correct answer.

The cost of 50 rulers at ₦10 each is ₦5,000.
There is one zero too many. It must be ₦500.

The capacity of a teaspoon is 5 L.
The wrong unit was used. It must be 5 ml.

The width of the classroom is 90 m.
There should not be a zero. It must be 9 m.

The mass of a woman is 27 kg.
The numbers were swapped. It must be 72 kg.

Multiplication instead of division was done. It must be
Keep the following in mind when you use approximation in calculations:
 Round off to the smallest multiple of ten that will allow an easy calculation. For example, if you round off to the nearest ten, your answer will be closer to the actual value than if you round off to the nearest hundred.
 When you compare values, round off the same quantities to the same multiple of ten. For example, if you compare the capacity of containers, round off all the capacity values to the nearest ten.
 When you divide similar quantities, first put them in the same unit. For example, if you divide two masses, first make sure they are both in grams or both in kilograms.
Worked example 7.8: Using approximation in calculations
A shop owner has to choose the supplier from which she will order sugar cubes. One supplier offers 50 450 g packs of sugar cubes at ₦19,599. Another supplier offers 20 450 g packs of sugar cubes at ₦5,999. She wants to do a quick calculation to make a decision. She needs to approximate the price of one pack.

Step 1: Decide what operation must be done: addition, subtraction, multiplication or division.
Since the mass of the packs of sugar cubes are the same at both suppliers, the shop owner must determine the cost per pack.
She must divide the total cost by the number of packs.

Step 2: Round the numbers to multiples of ten that will allow an easy calculation.
Round both costs to the nearest thousand:
₦19,599 ≈ ₦20,000
₦5,999 ≈ ₦6,000The numbers of packs are already given as a multiple of ten for both suppliers.

Step 3: Do the calculation with the approximate values.
₦20,000/50=₦400 for one pack
₦6,000/20=₦300 for one packThe price for one pack is lower for the supplier that offers 20 packs at ₦5,999.
Exercise 7.10: Use estimation and approximation to solve problems

A litre of petrol costs ₦153. What is the approximate cost of filling up a small car with a tank of 52 L?
\begin{align} ₦153\times52 &\approx ₦150\times50 \\ &\approx ₦7,500 \\ \end{align}

A fabric shop has a special offer of ₦7,475 for 9.75 m of Ankara print. Give the approximate cost per metre.
\begin{align} \frac{₦7,475}{9.75} &\approx \frac{₦7,500}{10} \\ &\approx ₦750 \end{align}

A student's steps are 60 cm long. He takes 8,315 steps from his front door to the school gate. Approximately how far does he live from the school?
\begin{align} 60\text{ cm}\times8,315 &\approx 60\text{ cm}\times8,000 \\ &\approx 480,000\text{ cm} \\ &\approx 4,800\text{ m} \\ &\approx 4,8\text{ km} \end{align}

A university student rents a flat at ₦82,250 per month. She works out that her rent for the year is going to be ₦98,700 and lets her mother know. Her mother immediately sees that the amount cannot be correct. Do a rough check to determine what is wrong with the calculation.
\begin{align} ₦82,250\times 12 &\approx ₦82,000 \times 10 \\ &\approx ₦820,000 \\ \end{align}
The student probably left out a zero. The amount should be ₦987,000.

A soft drink bottle has a capacity of 33 cl. At the factory, the bottles are filled from a container with a capacity of 21.3 kl. Approximately how many bottles can be filled from the container?
\begin{align} \frac{21.3\text{ kl}}{33\text{ cl}} &= \frac{2,130,000\text{ cl}}{33\text{ cl}} \\ &\approx \frac{2,100,000}{30} \\ &\approx 70,000\text{ bottles} \end{align}

A 5 L container of vegetable oil is sold at ₦2,899. Approximately how much money do you need for 31 L of vegetable oil?.
\begin{align} \frac{₦2,899}{5}\times 31 &\approx \frac{₦3,000}{5}\times 30 \\ &\approx ₦600\times 30 \\ &\approx ₦18,000 \end{align}

A man gets 17 bowls of rice from a sack of 51.15 kg. What is the approximate mass of rice needed for 5 bowls?
\begin{align} \frac{51.15\text{ kg}}{17}\times 5 &\approx \frac{50\text{ kg}}{20}\times 5 \\ &\approx \frac{5}{2}\text{ kg}\times 5 \\ &\approx \frac{25}{2}\text{ kg} \\ &\approx 12.5\text{ kg} \end{align}

A supermarket has special offers on milk powder. One offer is two 1 kg tins at ₦7,595. Another offer is four 400 g tins at ₦6,399. Do a quick calculation to determine which offer is the cheapest.
Offer 1:
\begin{align} \frac{₦7,595}{2\times1\text{ kg}} &\approx \frac{₦7,600}{2} \\ &\approx ₦3,800\text{/kg} \\ \end{align}Offer 2:
\begin{align} \frac{₦6,399}{4\times400\text{ g}} &\approx \frac{₦6,400}{1.6} \\ &\approx ₦4,000\text{/kg} \\ \end{align}Offer 1 is the cheapest.

An internet provider charges ₦9,309 per month for an 8 Mbps line. At the end of a year, they let their customers know that there will be a price increase of 10.25%. Work out what the approximate new price will be.
\begin{align} \frac{10.25}{100}\times₦9,309+₦9,309 &\approx \frac{10}{100}\times₦9,000+₦9,000 \\ &\approx \frac{₦9,000}{10}+₦9,000 \\ &\approx ₦900+₦9,000 \\ &\approx ₦9,900 \end{align}

A caterer has to give a quote for the food at a wedding. The mother of the bride tells her that they are expecting adult 323 guests. She thinks about three quarters of the families will bring their children along. Most families have four children. The food will cost ₦3,960 per adult guest and about half price for a child. Work out a rough quote.
\begin{align} \text{Number of families}&\approx \frac{323}{2} \\ &\approx \frac{320}{2} \\ &\approx 160\\ \end{align} \begin{align} \text{Number of children}&\approx \frac{3}{4}\times160 \times 4\\ &\approx 3\times160 \\ &\approx 480\\ \end{align} \begin{align} \text{Cost of food}&\approx 320\times ₦4,000+480\times \frac{₦4,000}{2} \\ &\approx 320\times ₦4,000+480\times ₦2,000 \\ &\approx ₦1,280,000+₦960,000 \\ &\approx ₦2,260,000\\ \end{align}
7.4 Summary
 Estimation is the process of making a guess about the size or cost of something, without doing the actual measurement or calculation.
 Distance is the length of the space between two points.
 The dimensions of an object are the measurable lengths that we use to determine the size of the object, such as its length, breadth and height.
 Commonly used units of measurement for distances and dimensions are millimetre, centimetre, metre and kilometre. 1,000 mm = 100 cm = 1 m = 0.001 km.
 The amount of liquid that a container can hold is called the capacity of the container.
 Commonly used units of measurement for capacity are millilitre, centilitre, litre and kilolitre. 1,000 ml = 100 cl = 1 L = 0.001 kl.
 The mass of an object is a measure of how much matter is in that object.
 Commonly used units of measurement for mass are milligram, gram, kilogram and tonne. 1,000,000 mg = 1,000 g = 1 kg = 0.001 t.
 Approximation is the process of using rounding to determine quickly a fairly accurate answer to a calculation.
 Rounding off is the process of making a number less exact but easier to work with, by adjusting it to a specific multiple of 10.
 When we round off a number to a specific multiple of 10, we have to identify the digit that represents the
multiple. If the digits to the right of it give a value less than half of the required multiple of 10, the digit
we identified remains the same. Otherwise, it increases by one. For example:
 56,456 rounded to the nearest thousand is 56,000
 56,856 rounded to the nearest thousand is 57,000
 The aim of using approximation in calculations is to round off numbers to values that are more easily done when using mental arithmetic. When we work with approximations, we use the symbol instead of an equals sign.