Additional Mathematics Project Work 2013 (PAHANG)

SEKOLAH SAINS SULTAN HAJI AHMAD SHAH PEKAN

ADDITIONAL MATHEMATICS PROJECT WORK 2013

Index Number In Our Daily Life

NAME

: MUHAMMAD AMIRUL AFIF BIN SAKRI

CLASS

: 5 JASA

TEACHER

: SIR HAMDAN BIN ALANG

I/C NUMBER : 1

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OBJECTIVE We students taking Additional Mathematics are required to carry out a project work while we are in Form 5. This year the Curriculum Development Division, Ministry of Education has prepared four tasks for us. We are to choose and complete only ONE task based on our area of interest. This project can be done in groups or individually, but each of us are expected to submit an individually written report . Upon completion of the Additional Mathematics Project Work, we are to gain valuable experiences and able to :  apply and adapt a variety of problem-solving strategies to solve routine and non-routine problems  experience classroom environments which are challenging, interesting, meaningful and hence improve our thinking skills  experience classroom environments where knowledge and skills are applied in meaningful ways in solving real-life problems  experience classroom environments where expressing ones mathematical thinking, reasoning and communication are highly encouraged and expected  experience classroom environments that stimulates and enhances effective learning  acquire effective mathematical communication through oral and writing, and to use the language of mathematics to express mathematical ideas correctly and precisely  enhance acquisition of mathematical knowledge and skills through problem-solving in ways that increases interest and confidence  prepare for the demands of our future undertakings and in workplace  realise that mathematics is an important and powerful tool in solving real-life problems and hence develop positive attitude towards mathematics  train ourselves not only to be independent learners but also to collaborate, to cooperate, and to share knowledge in an engaging and healthy environment  use technology especially the ICT appropriately and effectively  train ourselves to appreciate the intrinsic values of mathematics and to become more creative and innovative  realise the importance and the beauty of mathematics

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FOREWARD First of all, I would like to say Alhamdulillah, for giving me the strength and health to do this project work. Not forgotten my parents for providing everything, such as money to buy anything that are related to this project work, computer, internet, books and also their advice. They also supported me and encouraged me to complete this task that I will not procrastinate in doing it. Then I would like to thank my teacher, Sir Hamdan bin Alang for guiding me and my friends throughout this project. We had some difficulties in doing this task, but he taught us patiently until we knew what to do. Last but not least, my friends who were doing this project with me and sharing our ideas. They were helpful that when we combined and discussed together, we had this task done.

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CONTENT CONTENTS

PAGES

OBJECTIVE

2

FOREWARD

3

INTRODUCTION

5

PART 1

6 - 13

PART 2

14 - 17

PART 3

18 - 19

FURTHER EXPLORATION

20 - 27

REFLECTION

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INTRODUCTION HISTORY OF INDEX NUMBER Index numbers are meant to study the change in the effects of such factors which cannot be measured directly. According to Bowley, “Index numbers are used to measure the changes in some quantity which we cannot observe directly”. For example, changes in business activity in a country are not capable of direct measurement but it is possible to study relative changes in business activity by studying the variations in the values of some such factors which affect business activity, and which are capable of direct measurement. Index numbers are commonly used statistical device for measuring the combined fluctuations in a group related variables. If we wish to compare the price level of consumer items today with that prevalent ten years ago, we are not interested in comparing the prices of only one item, but in comparing some sort of average price levels. We may wish to compare the present agricultural production or industrial production with that at the time of independence. Here again, we have to consider all items of production and each item may have undergone a different fractional increase (or even a decrease). How do we obtain a composite measure? This composite measure is provided by index numbers which may be defined as a device for combining the variations that have come in group of related variables over a period of time, with a view to obtain a figure that represents the „net‟ result of the change in the constitute variables. Index numbers may be classified in terms of the variables that they are intended to measure. In business, different groups of variables in the measurement of which index number techniques are commonly used are (i) price, (ii) quantity, (iii) value and (iv) business activity. Thus, we have index of wholesale prices, index of consumer prices, index of industrial output, index of value of exports and index of business activity, etc. Here we shall be mainly interested in index numbers of prices showing changes with respect to time, although methods described can be applied to other cases. In general, the present level of prices is compared with the level of prices in the past. The present period is called the current period and some period in the past is called the base period. Index Numbers : Index numbers are statistical measures designed to show changes in a variable or group of related variables with respect to time, geographic location or other characteristics such as income, profession, etc. A collection of index numbers for different years, locations, etc., is sometimes called an index series. Simple Index Number : A simple index number is a number that measures a relative change in a single variable with respect to a base. Composite Index Number : A composite index number is a number that measures an average relative changes in a group of relative variables with respect to a base.

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PART 1 The prices of goods sold in shops vary from one shop to another. Shoppers tend to buy goods which are reasonably priced, giving value for their money. You are required to carry out a survey on three different items based on the following categories : food , detergent and stationary. The survey should be done in three different shops. (a) Collect pictures, newspapers cuttings or photos on items that you have chosen. Design a collage to illustrate the chosen items.

FOODS :

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DETERGENT :

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STATIONARY :

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(b) Record the items and their prices systematically as in Table 1(a), 1(b) and 1(c). Since items may be differently packed, be sure to use consistent measurements for each item selected so that comparison can be done easily and accurately.

Category Food

Item 1. Sugar (1 kg) 2. Flour (1 kg) 3. Butter (250 g) Total Price

Category

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Tunas Manja 3.20 28.90

Price (RM) Mydin 3.40 29.10

Giant 3.60 29.50

14.70

14.50

15.00

46.80 Table 1(b)

47.00

48.10

Price (RM) Mydin 8.90 15.20 0.50 24.60

Giant 9.40 16.00 0.80 26.20

Item

Stationary 1. Pencil (1 dozen) 2. Coloured pen (1 set) 3. Eraser Total Price

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Giant 2.80 1.70 4.70 9.20

Tunas Manja 2.30 1.30 4.20 7.80 Table 1(a)

Item

Detergent 1. Soap (3 bars) 2. Liquid detergent (5 kg) 3. Powder detergent (2.5 kg) Total Price

Category

Price (RM) Mydin 2.50 1.50 4.40 8.40

Tunas Manja 8.50 14.50 0.40 23.40 Table 1(c)

(c) Create at least two suitable graphical representations (the use of ICT is encouraged) to compare and contrast the prices of each category.

BAR CHART 1) Food 5 4.5 4 3.5 3 Sugar 2.5

Flour

2

Butter

1.5 1 0.5 0 Tunas Manja

Mydin

Giant

2) Detergent 35 30 25 20

Soap Liquid detergent

15

Powder detergent

10 5 0 Tunas Manja

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Mydin

Giant

3) Stationary 18 16 14 12 10

Pencil (1 dozen) Coloured pen

8

Eraser 6 4 2 0 Tunas Manja

Mydin

Giant

PIE CHART

Food

Tunas manja Mydin Giant

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Detergent

Tunas Manja Mydin Giant

Stationary

Tunas Manja Mydin Giant

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d) Based on the graphical representations that you have constructed in Part 1(c), interpret, discuss and draw conclusions. Comments on your findings.

Based on the graphical representations that I have constructed in Part 1(c), it is shown that there are large and small differences among the prices of items in each category between the shops. The bar charts show that products from Tunas Manja are much cheaper than products from Mydin and Giant. From the pie charts, we can conclude that shop which is selling the products with higher price is Giant. But we can say that all the products are expensive in Giant because certain products from Giant are reasonable than other shops. As a conclusion, shop which is selling their products with lower price among the shops is Tunas Manja, followed by Mydin, and shop which is selling their products with higher price among the three shops is Giant.

e) Identify an item that has large price difference among the shops. Suggest and discuss possible reasons for the price difference. Coloured pen (1 set) : Tunas Manja – RM14.50 Mydin – RM15.20 Giant – RM16.00 Mean = 14.50 + 15.20 + 16.00 3 = 15.23 Standard deviation : σ=√

= 0.61283 The large price difference of coloured pen among the shops maybe because of the standard of the shop. A high standard shop or supermarket, the items sold intend to be much more expensive than a regular shop. The price difference of the items may also due to the quality of the item present. A better quality means a higher price.

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PART 2 Every year, Shah Pekan organizes a carnival to raise fund for school improvement projects. This year the school plans to install CCTV in order to enhance its security. Last year, during the carnival, my class made and sold nasi lemak. Due to the popularity of this local food, my class has decided to carry out the same project for this year’s carnival. (a) Complete Table 2 with the prices of the given items for the year 2013 Items (For 10 packets of Nasi Lemak)

I

II

III

14

For the flavoured rice : Rice Santan Onion Ginger Salt For the Sambal : Red onion Garlic Dried chillies Ikan bilis Cooking oil Miscellaneous : Fried groundnuts Fried crispy ikan bilis Cucumber 5 eggs Banana leaves (1 bundle)

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Quantity

Price in the year 2012 (RM)

Price in the year 2013 (RM)

1 kg 150 g 10 g 10 g 2g

2.70 1.40 0.15 0.15 0.05

2.80 1.50 0.15 0.15 0.06

10 g 5g 100 g 200 g 100 g

0.05 0.04 1.00 2.00 0.25

0.05 0.04 1.10 2.10 0.27

200 g 150 g 200 g 300 g 300 g Table 2

0.50 1.50 0.20 1.50 1.00

0.60 1.60 0.20 1.60 1.00

(b) Calculate the price index for each of the items in Table 2 for the year 2013 based on the year 2012.

Price Index, I =

Items (For 10 packets of Nasi Lemak)

I

II

III

15

For the flavoured rice : Rice Santan Onion Ginger Salt For the Sambal : Red onion Garlic Dried chillies Ikan bilis Cooking oil Miscellaneous : Fried groundnuts Fried crispy ikan bilis Cucumber 5 eggs Banana leaves (1 bundle)

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𝑃𝑟𝑖𝑐𝑒 𝑎𝑡 𝑎 𝑠𝑝𝑒𝑐𝑖𝑓𝑖𝑐 𝑡𝑖𝑚𝑒 𝑃𝑟𝑖𝑐𝑒 𝑎𝑡 𝑡ℎ𝑒 𝑏𝑎𝑠𝑒 𝑡𝑖𝑚𝑒

×

Price in the year 2012 (RM)

Price in the year 2013 (RM)

Price index for the year 2013 based on the year 2012 (I)

1 kg 150 g 10 g 10 g 2g

2.70 1.40 0.15 0.15 0.05

2.80 1.50 0.15 0.15 0.06

103.70 107.14 100 100 120

10 g 5g 100 g 200 g 100 g

0.05 0.04 1.00 2.00 0.25

0.05 0.04 1.10 2.10 0.27

100 100 110 105 108

200 g 150 g 200 g 300 g 300 g

0.50 1.50 0.20 1.50 1.00

0.60 1.60 0.20 1.60 1.00

120 106.67 100 106.67 100

Quantity

(c) Calculate the composite index for the cost of a packet of nasi lemak in the year 2013 based on the year 2012. Discuss how you obtained your answer.

Ī=

Items (For 10 packets of Nasi Lemak)

I

II

III

For the flavoured rice : Rice Santan Onion Ginger Salt For the Sambal : Red onion Garlic Dried chillies Ikan bilis Cooking oil Miscellaneous : Fried groundnuts Fried crispy ikan bilis Cucumber 5 eggs Banana leaves (1 bundle)

= = 105.34

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∑𝐼𝑖 𝑊𝑖 ∑𝑊𝑖

Price index for the year 2013 based on the year 2012 (I)

Weightage (W)

IiWi

1 kg 150 g 10 g 10 g 2g

103.70 107.14 100 100 120

1.0 0.15 0.01 0.01 0.002

103.70 16.071 1 1 0.24

10 g 5g 100 g 200 g 100 g

100 100 110 105 108

0.01 0.005 0.1 0.2 0.1

1 0.5 11 21 10.8

200 g 150 g 200 g 300 g 300 g

120 106.67 100 106.67 100

0.2 0.15 0.2 0.3 0.3 2.737

24 16.0005 20 32.001 30 288.3125

Quantity

(d) In the year 2012, the nasi lemak was sold at RM2.00 each. Suggest a suitable selling price for a packet of the nasi lemak in the year 2013. Give reasons for your answer.

On 2012: RM2.00 ×

On 2013:

x = RM2.10 Thus, the suitable selling price for a packet of the nasi lemak in the year 2013 is RM2.10. The increase in price is also suitable because of the rise in the price of the ingredients.

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PART 3 In order to upgrade the security, my school has decided to install closed-circuit TV (CCTV). a) Suggest a suitable number of cameras to be installed and their placement. Give your reasons. Illustrate with a plan.

I suggest 4 cameras to be installed at my school. The cameras will be installed at the staff rooms and the office. This action will be done to avoid any robbery case from happen.

STAFF ROOM

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OFFICE

b) Find out the cost of installation of the CCTV. A set of CCTV with 4 cameras : RM2100

c) If your class intends to sponsor 10% of the cost, how many packets of nasi lemak must your class sell? 10% X RM2100 = RM210 Cost for a packet of nasi lemak = RM13.22 10 = RM1.40 Selling price = RM2.10 Profit = RM2.10 – RM1.40 = RM0.70 Packets of nasi lemak must be sell = RM210 RM0.70 = 300 packets

d) Can your class achieve the 10% target? Do you think your class should carry out a different project instead? Yes because many visitors from Pekan will come and visit our school during the carnival. This phenomenon will increase the selling of our nasi lemak. Last year, we had sold about 400 packets of nasi lemak. So, this year our target is 600 packets of nasi lemak will be sold. From the selling, we can sponsor 10% for the cost of installation of the CCTV and we will get profit of about RM210. I think my class should not carry out different project as we can get a lot of profit from the selling of nasi lemak. We need to focus on selling nasi lemak only so we can achieve our target. Great efforts will led to great outcomes.

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FURTHER EXPLORATION Index numbers are being used in many different daily situations, for example, air pollution index, stock market index, gold index and property index. Obtain information from the internet or other reliable sources on the importance of two different types of index numbers of your choice. Elaborate the use and the importance of these index numbers in daily life situations.

1. AIR QUALITY INDEX An air quality index (AQI) is a number used by government agencies to communicate to the public how polluted the air is currently or how polluted it is forecast to become. As the AQI increases, an increasingly large percentage of the population is likely to experience increasingly severe adverse health effects. Different countries have their own air quality indices which are not all consistent. Different countries also use different names for their indices such as Air Quality Health Index, Air Pollution Index and Pollutant Standards Index. Definition and Usage : Air quality is defined as a measure of the condition of air relative to the requirements of one or more biotic species or to any human need or purpose. To compute the AQI requires an air pollutant concentration from a monitor or model. The function used to convert from air pollutant concentration to AQI varies by pollutant, and is different in different countries. Air quality index values are divided into ranges, and each range is assigned a descriptor and a colour code. Standardized public health advisories are associated with each AQI range. The AQI can go up (meaning worse air quality) due to a lack of dilution of air pollutants. Stagnant air, often caused by an anticyclone, temperature inversion, or low wind speeds lets air pollution remain in a local area, leading to high concentrations of pollutants and hazy conditions. An agency might encourage members of the public to take public transportation or work from home when AQI levels are high. Most air contaminants do not have an associated AQI. Many countries monitor groundlevel ozone, particulates, sulphur dioxide, carbon monoxide and nitrogen dioxide and calculate air quality indices for these pollutants.

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Indices by Location :

Hong Kong The Air Pollution Index (API) levels for Hong Kong are related to the measured concentrations of ambient respirable suspended particulate (RSP), sulphur dioxide (SO2), carbon monoxide (CO), ozone (O3) and nitrogen dioxide (NO2) over a 24-hour period based on the potential health effects of air pollutants. An API level at or below 100 means that the pollutant levels are in the satisfactory range over 24 hour period and pose no acute or immediate health effects. However, air pollution consistently at "High" levels (API of 51 to 100) in a year may mean that the annual Hong Kong "Air Quality Objectives" for protecting long-term health effects could be violated. Therefore, chronic health effects may be observed if one is persistently exposed to an API of 51 to 100 for a long time. "Very High" levels (API in excess of 100) means that levels of one or more pollutant(s) is/are in the unhealthy range. The Hong Kong Environmental Protection Department provides advice to the public regarding precautionary actions to take for such levels.

API

Air Pollution Level

0 – 25

Low

26 – 50

Medium

51 – 100

High

Acute health effects are not expected but chronic effects may be observed if one is persistently exposed to such levels.

Very High

People with existing heart or respiratory illnesses may notice mild aggravation of their health conditions. Generally healthy individuals may also notice some discomfort.

Severe

People with existing heart or respiratory illnesses may experience significant aggravation of their symptoms. There may also be widespread symptoms in the healthy population (e.g. eye irritation, wheezing, coughing, phlegm and sore throats).

101 – 200

201 – 500

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Health Implications

None expected.

None expected for the general population.

Canada Air quality in Canada has been reported for many years with provincial Air Quality Indices (AQIs). Significantly, AQI values reflect air quality management objectives, which are based on the lowest achievable emissions rate, and not exclusively concern for human health. The Air Quality Health Index or (AQHI) is a scale designed to help understand the impact of air quality on health. It is a health protection tool used to make decisions to reduce short-term exposure to air pollution by adjusting activity levels during increased levels of air pollution. The Air Quality Health Index also provides advice on how to improve air quality by proposing behavioural change to reduce the environmental footprint. This index pays particular attention to people who are sensitive to air pollution. It provides them with advice on how to protect their health during air quality levels associated with low, moderate, high and very high health risks. The Air Quality Health Index provides a number from 1 to 10+ to indicate the level of health risk associated with local air quality. On occasion, when the amount of air pollution is abnormally high, the number may exceed 10. The AQHI provides a local air quality current value as well as a local air quality maximums forecast for today, tonight, and tomorrow, and provides associated health advice.

Health Risk

Air Quality Health Index

Health Messages

At Risk population

Low

Moderate

High

*General Population

1-3

Enjoy your usual outdoor activities. Ideal air quality for outdoor activities

4-6

Consider reducing or rescheduling No need to modify your usual outdoor activities strenuous activities outdoors if you unless you experience symptoms such as are experiencing symptoms. coughing and throat irritation.

7-10

Reduce or reschedule strenuous Consider reducing or rescheduling strenuous activities outdoors. Children and the activities outdoors if you experience symptoms elderly should also take it easy. such as coughing and throat irritation.

Avoid strenuous activities outdoors. Reduce or reschedule strenuous activities Above Children and the elderly should also outdoors, especially if you experience symptoms Very high 10 avoid outdoor physical exertion. such as coughing and throat irritation.

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Singapore Singapore uses the Pollutant Standards Index to report on its air quality, with details of the calculation similar but not identical to that used in Malaysia and Hong Kong. The PSI chart below is grouped by index values and descriptors, according to the National Environment Agency.

PSI

Descriptor

0 – 50

General Health Effects

None

51 – 100

Moderate

Few or none for the general population

Unhealthy

Mild aggravation of symptoms among susceptible persons i.e. those with underlying conditions such as chronic heart or lung ailments; transient symptoms of irritation e.g. eye irritation, sneezing or coughing in some of the healthy population.

201 – 300

Very Unhealthy

Moderate aggravation of symptoms and decreased tolerance in persons with heart or lung disease; more widespread symptoms of transient irritation in the healthy population.

301 – 400

Hazardous

Early onset of certain diseases in addition to significant aggravation of symptoms in susceptible persons; and decreased exercise tolerance in healthy persons.

101 – 200

Above 400 Hazardous

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PSI levels above 400 may be life-threatening to ill and elderly persons. Healthy people may experience adverse symptoms that affect normal activity.

2. BODY MASS INDEX The body mass index (BMI), or Quetelet index, is a measure for human body shape based on an individual's weight and height. It was devised between 1830 and 1850 by the Belgian polymath Adolphe Quetelet during the course of developing "social physics". Body mass index is defined as the individual's body mass divided by the square of their height. The formulae universally used in medicine produce a unit of measure of kg/m2. BMI can also be determined using a BMI chart, which displays BMI as a function of weight (horizontal axis) and height (vertical axis) using contour lines for different values of BMI or colours for different BMI categories.

The factor for UK/US units is more precisely 703.06957964, but that level of precision is not meaningful for this calculation. To work from stone and pounds first multiply the stone by 14 then add the pounds to give the whole mass in pounds; to work from feet and inches first multiply the feet by 12 then add the inches to give the whole height in inches.

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Usage : While the formula previously called the Quetelet Index for BMI dates to the 19th century, the new term "body mass index" for the ratio and its popularity date to a paper published in the July edition of 1972 in the Journal of Chronic Diseases by Ancel Keys, which found the BMI to be the best proxy for body fat percentage among ratios of weight and height; the interest in measuring body fat being due to obesity becoming a discernible issue in prosperous Western societies. BMI was explicitly cited by Keys as being appropriate for population studies, and inappropriate for individual diagnosis. Nevertheless, due to its simplicity, it came to be widely used for individual diagnosis, despite its inappropriateness. BMI provided a simple numeric measure of a person's thickness or thinness, allowing health professionals to discuss overweight and underweight problems more objectively with their patients. However, BMI has become controversial because many people, including physicians, have come to rely on its apparent numerical authority for medical diagnosis, but that was never the BMI's purpose; it is meant to be used as a simple means of classifying sedentary (physically inactive) individuals, or rather, populations, with an average body composition. For these individuals, the current value settings are as follows: a BMI of 18.5 to 25 may indicate optimal weight; a BMI lower than 18.5 suggests the person is underweight while a number above 25 may indicate the person is overweight; a person may have a BMI below 18.5 due to disease; a number above 30 suggests the person is obese (over 40, morbidly obese). For a given height, BMI is proportional to mass. However, for a given mass, BMI is inversely proportional to the square of the height. So, if all body dimensions double and mass scales naturally with the cube of the height, then BMI doubles instead of remaining the same. These results in taller people having a reported BMI that is uncharacteristically high compared to their actual body fat levels. In comparison, the Ponderal index is based on this natural scaling of mass with the third power of the height. However, many taller people are not just "scaled up" short people, but tend to have narrower frames in proportion to their height. BMI Prime : BMI Prime, a simple modification of the BMI system, is the ratio of actual BMI to upper limit BMI (currently defined at BMI 25). As defined, BMI Prime is also the ratio of body weight to upper body weight limit, calculated at BMI 25. Since it is the ratio of two separate BMI values, BMI Prime is a dimensionless number, without associated units. Individuals with BMI Prime less than 0.74 are underweight; those between 0.74 and 1.00 have optimal weight; and those at 1.00 or greater are overweight. BMI Prime is useful clinically because individuals can tell, at a glance, by what percentage they deviate from their upper weight limits. For instance, a person with BMI 34 has a BMI Prime of 34/25 = 1.36, and is 36% over his or her upper mass limit. In South East Asian populations (see international variation section below) BMI Prime should be calculated using an upper limit BMI of 23 in the denominator instead of 25. Nonetheless, BMI Prime allows easy comparison between populations whose upper limit BMI values differ.

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Categories : A frequent use of the BMI is to assess how much an individual's body weight departs from what is normal or desirable for a person of his or her height. The weight excess or deficiency may, in part, be accounted for by body fat (adipose tissue) although other factors such as muscularity also affect BMI significantly (see discussion below and overweight). The WHO regards a BMI of less than 18.5 as underweight and may indicate malnutrition, an eating disorder, or other health problems, while a BMI greater than 25 is considered overweight and above 30 is considered obese. These ranges of BMI values are valid only as statistical categories :

Category

BMI range – kg/m2

BMI Prime

Very severely underweight

less than 15

less than 0.60

Severely underweight

from 15.0 to 16.0

from 0.60 to 0.64

Underweight

from 16.0 to 18.5

from 0.64 to 0.74

Normal (healthy weight)

from 18.5 to 25

from 0.74 to 1.0

Overweight

from 25 to 30

from 1.0 to 1.2

Obese Class I (Moderately obese)

from 30 to 35

from 1.2 to 1.4

Obese Class II (Severely obese)

from 35 to 40

from 1.4 to 1.6

Obese Class III (Very severely obese)

over 40

over 1.6

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CONCLUSION : After doing research, answering questions, drawing graphs and some problem solving, I saw that the usage of index number is important in daily life. It is not just widely used in markets but also in interpreting the condition of the surrounding especially in conducting an air-pollution survey. In conclusion, index number is a daily life necessity. Without it, surveys can‟t be conducted, the BMI can‟t be calculate and many more. So, we should be thankful to the people who contribute in the idea of index number.

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REFLECTION While I was conducting the project, I have learnt and practice a lot of moral values. These are we should be patient when doing any work or project. This is to ensure our work is completed by the time. We also should be calm and cool down when any source that we are finding for is missing. This sign that we are going to get a conclusion soon. Furthermore, we should always smile when doing our works so that whatever work we are doing can be done easily. Besides that, we should relax and patient when our friends keep forcing us to finish the project so that they can refer to our project when we had done. This act will make us more motivated to complete the project. Lastly, I would like to say :

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