# NCERT Solutions Class 8 Maths Chapter 5

## NCERT Solutions for Class 8 Mathematics Chapter 5: Data Handling

Mathematics can be a complex subject, as every concept is different from the others. However, it is an important stage for the students of class 8, as it prepares them for the upcoming exams and entrance tests. To understand the concepts and theorems, students need regular practice and a deeper understanding of the topics.

Quick Links

NCERT Class 8 Mathematics Chapter 5: Data Handling starts with introducing the term “frequency”, which reflects the number of times a given data or item exists. Students will get to learn how to make a frequency distribution table. Further, the chapter introduces histograms, pie charts, and probability with pictorial representation. Finally, students will learn to organize the data systematically to make valuable inferences.

Our Extramarks NCERT Solutions for Class 8 Mathematics Chapter 5 are available for the students. It covers all the chapter-end questions along with their comprehensive answers explained in detail with proper illustrations. . Furthermore, it outlines all the relevant formulas neatly with the help of appropriate examples and charts. In addition, these solutions allow students to familiarise themselves with the data handling questions. Questions include filling in the blanks, matching the pair, and true or false, etc..

Extramarks is a well-known online learning platform for students for the best study materials and NCERT solutions. All the questions are provided by considering the new pattern of CBSE so they can get updated content for their exams.

Students can regularly visit our Extramarks website for exam-related updates. Further, they can also refer to the other solutions of secondary classes, including NCERT solutions class 10, NCERT solutions class 11, and NCERT solutions class 12.

## Key Topics Covered in NCERT Solutions for Class 8 Mathematics Chapter 5:

Our Extramarks NCERT Solutions for Class 8 Mathematics Chapter 5 outline all the relevant formulas with the help of extensive answers to the textual exercise examples, relevant charts, and tables. It helps the students to get a clear picture of how they have been derived. In addition, data handling is an essential chapter introducing various concepts and theories to the students.

Some of the key topics featured in NCERT Solutions for Class 8 Mathematics Chapter 5 are:

• Looking for Information
• Organising Data
• Grouping Data
• Circle Graphs or Pie chart
• Chances and Probability

### Looking for Information

In day-to-day life, we come across information such as runs made by a batsman in the last 10 matches or the number of wickets taken by a bowler. All information that is collected in all such cases is referred to as data. It is usually collected in the context of a situation we want to study. Data can be represented graphically to give a clear idea of what it represents. Different types of graphs elaborated in NCERT Solutions for Class 8 Mathematics Chapter 5 are:

• A pictograph representation of data using symbols.
• A bar graph displays the information using bars of uniform width, their heights proportional to the respective values.
• Double bar graphs show two sets of data simultaneously. It is helpful for the comparison of the data.
• Add information on Pie charts as well.

### Organizing Data

Data in an unorganized form is called raw data. To draw meaningful inferences, we need to organize the data systematically.

Terms related to Data Organising explained in NCERT Solutions for Class 8 Mathematics Chapter 5 are:

1. Frequency: It tells us the number of times a particular quantity repeats itself.
2. Frequency Distribution table: It is represented in tallies and numbers.

### Grouping Data

The data regarding the choice of subjects showed the occurrence of each of the entries several times. The information can be displayed graphically using a pictograph or a bar graph. Sometimes, we have to illustrate answers with a large amount of data.

We need to use a grouped frequency distribution table to represent a more significant number of quantities. Some essential terms related to grouped frequency distribution tables are as follows:

• Class Interval or class: If the observations are classified into several groups according to their size, these are known as class intervals.
• Upper-class limit: The highest number in every class interval is its upper-class limit.
• Lower-class limit: The lowest number in every class interval is known as its lower-class limit.
• Width or size or magnitude of the class interval: The difference between the upper-class limit and the lower-class limit is called the size of the class interval.

To understand the different categories of grouped frequency distribution tables, students can refer to NCERT Solutions for Class 8 Mathematics Chapter 5.

### Circle Graphs or Pie chart

A circle graph illustrates the relationship between the whole and its constituents. The whole circle is broken into segments. The size of every sector is proportional to the activity or data it represents.  If we represent the data in a form of a circle, it is known as a pie chart. We use circle graphs or pie charts when we have information on percentages or fractions.

When drawing a pie chart, we need to calculate the respective angles to show them in the pie chart. For example, as a complete circle is 360 degrees, we need to calculate the fraction of 360 degrees for every sector.

### Chances and Probability

Probability means possibility. It’s a branch of mathematics that studies the case of random events. It can be expressed as a number from 0 to 1. Mathematics introduced probability to predict the likelihood of events occurring. Probability is simply the likelihood that something will happen. This is the fundamental probability theory. It is also used in probability distribution.

Different types of probability explained in NCERT Solutions for Class 8 Mathematics Chapter 5 are:

• Random Experiment: A random experiment where the outcome cannot be predicted with certainty.
• Experiments on outcomes: It is a procedure that can be infinitely repeated and has a well-defined set of possible outcomes known as the sample space.
• Equally likely outcomes are the outcomes with the same chance of occurring.
• Probability of an event: It is an occurrence of an event when the outcome of experiments are equal; the probability of an event is represented by:
• P(E)= No. of outcomes that make an event/total no. of outcomes of the experiments.

## NCERT Solutions for Class 8 Mathematics Chapter 5: Exercise & Solutions

NCERT Solutions for Class 8 Mathematics Chapter 5 are essential for the students of the 8th class to clarify their doubts and perform well in their examinations. The answers in NCERT solutions are explained in detail, which gives students an idea of how to attempt a question in the board exam in the right manner

Class 8 Mathematics chapter 5 introduces essential concepts that assume the uprightness of the research data. Our subject-matter experts design the solutions as per the CBSE syllabus for 2020-21, which is prescribed by the board itself. Students can register on our website to access our study material.

The solution has various questions such as fill in the blanks, match the pair, true or false, and objective questions. Students will also witness descriptive-type questions at the end of each solution.

Click the links below for specific questions and solutions.

Along with this, students can also refer to other solutions for primary and secondary classes:

## NCERT Exemplar for Class 8 Mathematics:

Mathematics is a subject that demands a deeper understanding of formulas and theorems. Students face difficulty in remembering them. Therefore, it is beneficial to solve various questions of different difficulty levels. It helps students remember the formulas and their use at various stages of problems.

Class 8 students must be aware of the NCERT Exemplar for Class 8 Mathematics to prepare for their annual exams. All topics and concepts are explained in easily understandable language. It builds a strong foundation on all the concepts by providing the students with comprehensive answers to the textual questions. Most of the questions asked in the annual examination are from NCERT books. Further, exemplars have various questions, including MCQ, descriptive and objective-type questions.

The NCERT Solutions for Class 8 Mathematics Chapter 5 also includes questions from the NCERT exemplar. Students can get solutions or answers for all the questions present in the exercise of each chapter and miscellaneous questions.

### Key Features of NCERT Solutions for Class 8 Mathematics Chapter 5:

Data is the information collected in a context that has to be studied.. There are different ways to represent data, including pictographs, bar graphs, double bar graphs, and grouping the data using the frequency distribution table.. Our Extramarks NCERT Solutions for Class 8 Mathematics Chapter 5 cover all textual questions with their well-explained answers which let students understand the topics and concepts well.

These are the characteristics we included in our NCERT Solutions for Class 8 Mathematics Chapter 5 solutions:

• You can find a detailed solution to each exercise and every question and test yourself easily from the tip to the toe.
• A team of subject matter experts has prepared the study material while ensuring that it is highly accurate and easy to understand.
• All the study materials available on Extramarks are as per the latest guidelines by CBSE.
• Through regular practice, students will be able to increase their understanding of the chapter and build a strong foundation on the concepts.

Q.1 For which of these would you use a histogram to show the data?

(a) The number of letters for different areas in a postman’s bag.

(b) The height of competitors in an athletics meet.

(c) The number of cassettes produced by 5 companies.

(d) The number of passengers boarding trains from 7:00 a.m. to 7:00 p.m. at a station. Give reasons for each.

Ans-

1. The number of letters for different areas in a postman’s bag: Here, we cannot use the histogram as we do not know about the number of letters of different areas. Therefore, we can’t divide the given data into class intervals.
2. The height of competitors in an athletics meet:

Here, a histogram can be drawn as the given data can be divided into class intervals.

1. The number of cassettes produced by 5 companies: As we are not aware about the number of cassettes produced by the given companies, so we cannot divide the data into class intervals. Therefore, a histogram can’t be drawn.
2. The number of passengers boarding trains from 7:00 a.m. to 7:00 p.m. at a station: The histogram can be drawn here as the given data can be divided into class intervals.

Q.2 The shoppers who come to a departmental store are marked as: man (M), woman(W), boy (B) or girl (G). The following list gives the shoppers who came during the first hour in the morning:

W W W G B W W M G G M M W W W W G B M W B G G M W W M M W W W M W B W G M W W W W G W M M W W M W G W M G W M M B G G W

Make a frequency distribution table using tally marks. Draw a bar graph to illustrate it.

Ans-

The frequency distribution table is as follows:

$\begin{array}{|ccc|}\hline \mathrm{Shopper}& \mathrm{Tally}\text{ }\mathrm{marks}& \mathrm{Number}\\ \mathrm{W}& \overline{)\mathrm{llll}}\text{ }\overline{)\mathrm{llll}}\text{ }\overline{)\mathrm{llll}}\text{ }\overline{)\mathrm{llll}}\text{ }\overline{)\mathrm{llll}}\text{ }\mathrm{lll}& 28\\ \mathrm{M}& \overline{)\mathrm{llll}}\text{ }\overline{)\mathrm{llll}}\text{ }\overline{)\mathrm{llll}}& 15\\ \mathrm{B}& \overline{)\mathrm{llll}}& 5\\ \mathrm{G}& \overline{)\mathrm{llll}}\text{ }\overline{)\mathrm{llll}}\text{ }\mathrm{ll}& 12\\ \hline\end{array}$

The bar graph is as follows:

Q.3 The weekly wages (in Rs) of 30 workers in a factory are:

830, 835, 890, 810, 835, 836, 869, 845, 898, 890, 820, 860, 832, 833, 855, 845,

804, 808, 812, 840, 885, 835, 835, 836, 878, 840, 868, 890, 806, 840

Using tally marks make a frequency table with intervals as 800–810, 810–820 and so on.

Ans-

The frequency distribution table is as follows:

 $\begin{array}{l}\text{ }\mathrm{Interval}\\ 800-810\\ 810-820\\ 820-830\\ 830-840\\ 840-850\\ 850-860\\ 860-870\\ 870-880\\ 880-890\\ 890-900\end{array}$ $\begin{array}{l}\mathrm{Tally}\text{ }\mathrm{marks}\\ \text{ }\mathrm{lll}\\ \text{ }\mathrm{ll}\\ \text{ }\mathrm{l}\\ \text{ }\overline{)\mathrm{llll}}\mathrm{llll}\\ \text{ }\overline{)\mathrm{llll}}\\ \text{ }\mathrm{l}\\ \text{ }\mathrm{lll}\\ \text{ }\mathrm{l}\\ \text{ }\mathrm{l}\\ \text{ }\mathrm{lll}\end{array}$ $\begin{array}{l}\mathrm{Frequency}\\ \text{ }3\\ \text{ }2\\ \text{ }1\\ \text{ }9\\ \text{ }5\\ \text{ }1\\ \text{ }3\\ \text{ }1\\ \text{ }1\\ \text{ }4\end{array}$

Q.4 Draw a histogram for the frequency table made for the data in Question 3, and answer the following questions.

(i) Which group has the maximum number of workers?

(ii) How many workers earn Rs 850 and more?

(iii) How many workers earn less than Rs 850?

Ans-

Histogram is as follows:

(i) The maximum number of workers is in the group 830 − 840.

(ii) From the graph, we can observe that the workers who earn more than Rs 850 falls in the group of 850 − 860 or 860 − 870 or 870 − 880 or 880 − 890.

Hence, the total number of workers earning more than 850 = 1 + 3 + 1 + 1 + 4 = 10

(iii) It is clear from the graph that the workers who earn less than Rs 850 falls in the group of 800 − 810 or 810 − 820 or 820 − 830 or 830 − 840 or 840 − 850.

Hence, the total number of workers earning less than 850 = 3 + 2 + 1 + 9 + 5 = 20

Q.5 The number of hours for which students of a particular class watched television during holidays is shown through the given graph.

Answer the following.

1. For how many hours did the maximum number of students watch TV?
2. How many students watched TV for less than 4 hours?
3. How many students spent more than 5 hours in watching TV?

Ans-

(i)The maximum number of students watched TV for 4 − 5 hours.

(ii) The students who watched TV for less than 4 hours falls in the group of 1 − 2 hours or 2 − 3 hours or 3 − 4 hours.

Hence, total number of students who watched TV for less than 4 hours

= 4 + 8 + 22

= 34

(iii) The students who watched TV for more than 5 hours falls in the group of 5 − 6 hours or 6 − 7 hours.

Hence, total number of students who watched TV for more than 5 hours

= 8 + 6

= 14

Q.6 A survey was made to find the type of music that a certain group of young people liked in a city. Adjoining pie chart shows the findings of this survey.

From this pie chart answer the following:

(i) If 20 people liked classical music, how many young people were surveyed?

(ii) Which type of music is liked by the maximum number of people?

(iii) If a cassette company were to make 1000 CD’s, how many of each type would they make?

Ans-

(i) From the given pie chart the number of people who like classical music =10%

Let the number of young people who were surveyed be ‘x’.

According to the question, we get

10% of x = 20

$\begin{array}{l}\frac{10}{100}×\mathrm{x}=20\\ ⇒\mathrm{x}=\frac{20×100}{10}\\ ⇒\mathrm{x}=200\end{array}$

Therefore, 200 young people were surveyed.

(ii) From the given pie chart, it can be observed that light music is liked by maximum number of people.

(iii)

$\begin{array}{l}\mathrm{Number}\text{}\mathrm{of}\text{}\mathrm{CD}’\mathrm{s}\text{}\mathrm{for}\text{}\mathrm{light}\text{}\mathrm{music}=40\text{}\mathrm{%}\\ =\frac{40}{100}×1000=400\\ \mathrm{Number}\text{}\mathrm{of}\text{}\mathrm{CD}’\mathrm{s}\text{}\mathrm{for}\text{}\mathrm{folk}\text{}\mathrm{music}=30\text{}%\\ =\frac{30}{100}×1000=300\\ \mathrm{Number}\text{}\mathrm{of}\text{}\mathrm{CD}’\mathrm{s}\text{}\mathrm{for}\text{}\mathrm{classical}\text{}\mathrm{music}=10\text{}\mathrm{%}\\ =\frac{10}{100}×1000=100\\ \mathrm{Number}\text{}\mathrm{of}\text{}\mathrm{CD}’\mathrm{s}\text{}\mathrm{for}\text{}\mathrm{semi}-\mathrm{classical}\text{}\mathrm{music}=20\text{}\mathrm{%}\\ =\frac{20}{100}×1000=200\end{array}$

Q.7 A group of 360 people were asked to vote or their favourite season from the three seasons rainy, winter and summer.
(i) Which season got the most votes?
(ii) Find the central angle of each sector.
(iii)Draw a pie chart to show this information.

Ans-

 Season No. of votes Summer 90 Rainy 120 Winter 150

(i) Winter season got the maximum votes.
(ii) Here, total number of votes = 90 + 120 + 150 = 360.
The central angles of each sector are shown in the following table:

$\begin{array}{|cccc|}\hline \mathrm{Season}& \mathrm{No}.\text{ }\mathrm{of}\text{ }\mathrm{votes}& \mathrm{In}\text{ }\mathrm{Fraction}& \mathrm{Central}\text{ }\mathrm{angle}\\ \mathrm{Summer}& 90& \frac{90}{360}& \frac{90}{360}×{360}^{\circ }\text{ }=\text{ }{90}^{\circ }\\ \mathrm{Rainy}& 120& \frac{120}{360}& \frac{120}{360}×{360}^{\circ }\text{ }=\text{ }{120}^{\circ }\\ \mathrm{Winter}& 150& \frac{150}{360}& \frac{150}{360}×{360}^{\circ }\text{ }=\text{ }{150}^{\circ }\\ \hline\end{array}$

(iii) The pie chart is as follows:

Q.8 Draw a pie chart showing the following information. The table shows the colours preferred by a group of people.

 Colours Number of People Blue 18 Green 9 Red 6 Yellow 3 Total 36

Ans-

The central angles for a pie chart are calculated in the following table:

$\begin{array}{|cccc|}\hline \mathrm{Language}& \mathrm{No}.\text{of }\mathrm{students}& \mathrm{In}\text{ }\mathrm{Fraction}& \mathrm{Central}\text{ }\mathrm{angle}\\ \mathrm{Blue}& 18& \frac{18}{36}& \frac{18}{36}×{360}^{\circ }={180}^{\circ }\\ \mathrm{Green}& 9& \frac{9}{36}& \frac{9}{36}×{360}^{\circ }={90}^{\circ }\\ \mathrm{Red}& 6& \frac{6}{36}& \frac{6}{36}×{360}^{\circ }={60}^{\circ }\\ \mathrm{Yellow}& 3& \frac{3}{36}& \frac{3}{36}×{360}^{\circ }={30}^{\circ }\\ \hline\end{array}$

The pie chart is as follows:

Q.9 The adjoining pie chart gives the marks scored in an examination by a student in Hindi, English, Mathematics, Social Science and Science. If the total marks obtained by the students were 540, answer the following questions.

(i) In which subject did the student score 105 marks?(Hint: for 540 marks, the central angle = 360°. So, for 105 marks, what is the central angle?)

(ii) How many more marks were obtained by the student in Mathematics than in Hindi?

(iii) Examine whether the sum of the marks obtained in Social Science and Mathematics is more than that in Science and Hindi.

Ans-

(Hint: Just study the central angles)

Total marks obtained by the student are 540.

Hence, 540 marks represent 360º.

Now, we have to find the central angle for 105 marks.

$\begin{array}{l}=\frac{105}{540}×360°\\ =70°\end{array}$

Therefore, the subject in which the student scored 105 marks is Hindi.

(ii) Marks obtained by student in Mathematics = 90 %

$\begin{array}{l}=\frac{90°}{360°}×540\\ =135\end{array}$

Now, the marks obtained by student in Hindi = 70 %

= 105

Therefore, the difference between the marks of Mathematics and Hindi = 135 – 105 = 30.

Hence, 30 more marks were obtained by the student in Mathematics than in Hindi.

(iii) The sum of central angles of Social Science and Mathematics

= 90° + 65°

= 155°

However, the sum of central angles of Science and Hindi

= 80° + 70°

= 150°

We can observe that, the sum of the central angles for Social Science and Mathematics is more than that of Science and Hindi.

Therefore, the student scored more in Social Science and Mathematics than in Science and Hindi.

List the outcomes you can see in these experiments.

(a) Spinning a wheel

(b) Tossing two coins together

(a)The possible outcomes on spinning a wheel are A, B, C, and D.

(b) The possible outcomes when two coins are tossed together are:

HT, TH, HH, TT where, H represents Head and T represents Tail.

Q. When a die is thrown; list the outcomes of an event of getting
(i) (a) a prime number

(b) not a prime number

(ii) (a) a number greater than 5

(b) a number not greater than 5.

Ans-

When a die is thrown, the possible outcomes are: 1, 2, 3, 4, 5 and 6.

Therefore, the outcomes of an event of getting:

(i) (a) a prime number are: 2, 3 and 5

(b) not a prime number are : 1 , 4 and 6

(ii) (a) a number greater than 5 is : 6

(b) a number not greater than 5 are: 1, 2, 3, 4 and 5

Find the

1. Probability of the pointer stopping on D ?

2. Probability of getting an ace from a well shuffled deck of 52 playing cards?

3. Probability of getting a red apple. (See figure below)

Ans-

(i) The possible outcomes on spinning a wheel are :

A, B, C, and D

Therefore, the pointer can stop at one of the regions.

Total number of regions is 5.

The pointer will stop at region D only once.

Hence, probability that the pointer will stop at region D =

$\frac{1}{5}.$

(ii) Total number of cards in a well shuffled deck = 52

Total number of Ace cards = 4.

Therefore, the probability of getting an ace card =

$\frac{4}{52}\text{ }=\frac{1}{13}.$

(iii)Total number of apples = 7

Total number of red apples = 4

Total number of green apples = 3

Therefore, the probability of getting a red apple =

$\frac{4}{7}$

Q. Numbers 1 to 10 are written on ten separate slips (one number on one slip), kept in a box and mixed well. One slip is chosen from the box without looking into it. What is the probability of

(i) getting a number 6?

(ii) getting a number less than 6?

(iii) getting a number greater than 6?

(iv) getting a 1-digit number?

Ans-

Total number of slips = 10

(i) The probability of getting a number 6 =

$\frac{1}{10}$

(ii) The numbers less than 6 are: 1, 2, 3, 4 and 5

Therefore, the probability of getting a number less than 6 =

$\frac{5}{10}=\frac{1}{2}$

(iii) The numbers greater than 6 are: 7, 8, 9 and 10

Therefore, the probability of getting a number greater than 6 =

$\frac{4}{10}=\frac{2}{5}$

(iv) The one digit numbers are: 1, 2, 3, 4, 5, 6, 7, 8 and 9.

Therefore, the probability of getting a 1-digit number =

$\frac{9}{10}$

Q. If you have a spinning wheel with 3 green sectors, 1 blue sector and 1 red sector, what is the probability of getting a green sector? What is the probability of getting a non blue sector?

Ans-

Total number of green sectors =3

Total number of blue sectors = 1

Total number of red sector = 1

Total number of sectors = 5

Therefore, the probability of getting a green sector =

$\frac{3}{5}$

Now, the non blue sectors are the blue and red sectors. Therefore, the probability of getting a non blue sector =

$\frac{4}{5}$

Find the probabilities of the events given in Question 2.

When a die is thrown, the possible outcomes are: 1, 2, 3, 4, 5 and 6

(i) (a) The prime numbers are: 2, 3 and 5

Hence, the probability of getting a prime number =

$\frac{3}{6}=\frac{1}{2}$

(b)The numbers which are not prime are: 1, 4 and 6

Hence, the probability of not getting a prime number =

$\frac{3}{6}=\frac{1}{2}$

(ii) (a) A number greater than 5 is 6.

Therefore, the probability of getting a number greater than 6 =

$\frac{1}{6}$

(b) A number not greater than 5 are: 1, 2, 3, 4 and 5

Therefore, the probability of not getting a number greater than 5 =

$\frac{5}{6}$

Q. The number of students in a hostel, speaking different languages is given below. Display the data in a pie chart.

 Language Hindi English Marathi Tamil Bengali Total Number of Students 40 12 9 7 4 72

Ans-

The central angle for each subject is as follows:

$\begin{array}{|cccc|}\hline \mathrm{Language}& \mathrm{No}.\text{of }\mathrm{students}& \mathrm{In}\text{ }\mathrm{Fraction}& \mathrm{Central}\text{ }\mathrm{angle}\\ \mathrm{Hindi}& 40& \frac{40}{72}& \frac{40}{72}×{360}^{\circ }={200}^{\circ }\\ \mathrm{English}& 12& \frac{12}{72}& \frac{12}{72}×{360}^{\circ }={60}^{\circ }\\ \mathrm{Marathi}& 9& \frac{9}{72}& \frac{9}{72}×{360}^{\circ }={45}^{\circ }\\ \mathrm{Tamil}& 7& \frac{7}{72}& \frac{7}{72}×{360}^{\circ }={35}^{\circ }\\ \mathrm{Bengali}& 4& \frac{4}{72}& \frac{4}{72}×{360}^{\circ }={20}^{\circ }\\ \hline\end{array}$

The pie chart is as follows:

## FAQs (Frequently Asked Questions)

### 1. How to utilise NCERT solutions effectively?

Students should pay close attention to highlighting sections in the chapter when they discuss essential facts. These sections should be taken down and reviewed. Also, it is important to practise solving exercises and practising solved examples. This will allow students to use the NCERT Solutions Class 8 Mathematics Chapter 5 efficiently.

### 2. How many exercises are there in the solutions?

The NCERT Solutions for Class 8 Mathematics Chapter 5 has three exercises as follows:

• Exercise 5.1 Solutions 5 Questions (5 Short Answer Questions)
• Exercise 5.2 Solutions 5 Questions (3 Long Answer Questions and 2 Short Answer Questions)
• Exercise 5.3 Solutions 6 Questions (2 Long Answer Questions and 4 Short Answer Questions)

### 3. Which essential topics were covered in class 8, chapter 5, Mathematics?

The NCERT Solutions for Class 8 Mathematics Chapter 5 covers all the chapter end questions along with their comprehensive answers explained with proper illustrations.. The data handling is elaborated with the help of relevant frequency distribution tables, bar graphs, double bar graphs, and histograms. In addition, the function of probability has been discussed in the last section of the chapter.