Introduction Data Interpretation forms one of the most critical areas of different general and entrance examinations. It accounts 0-0 questions in the MBA entrance examination, 0-5 questions in Bank examination, 0-5 questions in SSC examination etc., therefore, it s understanding and its inherent concepts will help the students to score maximum in different examination. Data Interpretation Data Interpretation is drawing conclusions and inference from a comprehensive data presented numerically in taular form or pictorial form y means of an illustration, graphs, pie charts etc. Thus the act of organizing and interpreting data to get meaningful information is data interpretation. It is an extension of mathematical skill and accuracy. Soundly knowledge of quantitative techniques is prerequisite for good performance in this section. Since, all such questions may require a fair amount of calculations, one should e ale to multiply and divide quickly using shortcut methods. So, here we are providing a quicker and clear concepts, shortcut methods of these chapters in view of their essence in solving the performs of Data Interpretation. A detailed study on the pattern of questions appearing in this section for various competitive examinations has concluded that Data Interpretation is mainly the game of three chapters of arithmetic namely Percentage, Average and Ratio.. Percentage The term per cent means for every hundred. A fraction whose denominator is 00 is called a percentage and the numerator of the fraction is called the rate per cent. It is denoted y the symol %. x x % x 00 00 Percentage is a very useful tool for comparison in the analysis of data. For example, in their captaincy Sourav Ganguly has won 7 matches out of 05 matches and Rahul Dravid has won 64 matches out of 40 matches. This can however, e etter comprehended in a percentage form, which is for 7 Success rate of Ganguly 00 695 05 %. % 64 Success rate of Dravid 00 457 40 %. % This reveals that as a captain, Ganguly is more successful than Dravid. Percentage Equivalent of Important Fractions 6 6 % 4 9 % 0 0% 3 33 % 3 9 % 9 8 % 7 4 % 7 6 6 % 3 5 0% 3 5 60% 4 5% 3 66 % 3 3 4 75% 5 40% 4 5 80% 9 0 45% 7 8 87 % % 9 0 90% 0 55% 00%
4 The Accredited Guide to Data Interpretation & Data Sufficiency Calculation of Percentage. Find 79% of 49. Sol. 79% 80% % 80% 4/ 5 i.e., 4 / 5 49 4 858. 343. % 4.9 79% of 49 ~ 343. 4.9 ~ 339. Find 36% of 345. Interpretation of Data Involving the Percentage Rule To find y how much per cent x is more or less than y (or over y) or compared to y. Value of x Value ofy Required percentage 00 (when x y) Value ofy Sol. 36% ( 300 0 )% 300% 3 345 7035 % / 345 7. 5 0% / 0 345 34.5 % / 00 345 3. 45 Required 36% 7035 7.5 34.5 3.45 ~ 8465 3. How much per cent of 795645 is 64598? 64598 Sol. Required percentage 00% 3.6% 795645 But you are not required to find the exact value as you have to choose only one option and only approximate value is sufficient to solve your prolem. Therefore, approximate percentage 6 78 00 % ~ 6 0 00% 80 3 % 3.33% Value ofy Value of x 00 (when x y) Value ofy The denominator part contains the value with which the comparison is made. Rule To find the percentage change in any value, say x compared to the other value, say y. Required percentage change (c) x y y Ifc is positive, then there is percentage increase in the value of x overy (percentage growth) and ifc is negative, then there is percentage decrease (percentage decline) or negative growth. Rule 3 Two percentage values can t e compared unless the ase values are known. Rule 4 Two percentage values can e compared in terms of percentage values ut not in asolute values when the ase values are same although ase values are not known. 4. Find in how many countries the production of cars has increased y less than 5% in 008 over last year. Countries Figure (in 000 tonne) 007 008 India 700 760 USA 900 940 China 760 860 Japan 0 580 Italy 60 8 Sol. Increment in production of cars in 008 over 007 760 700 in India 00% 8.57% 700 940 900 in USA 00% 4.44% 900 860 760 in China 00% 5.68% 760 580 0 in Japan 00% 5.33% 0 8 60 in Italy 00% 5% 60 Clearly, only in one country USA production of cars has increased y less than 5% in 008 over last year. Note You can solve this type of questions as given elow.
Introduction 5 5. Find in how many years the production of sugar has decreased y more than 30% over the previous year. Years 004 005 006 007 008 Production in million tonne Sol. Method I 974 76 54 76 84 Decrement in the production of sugar over previous year 974 76 in 005 00% 5.5% 974 76 54 in 006 00% 7.8% 76 54 76 in 007 00% 47.3% 54 76 84 in 006 00% 33.3% 76 Thus, in two years 007 and 008, decreased in the production of sugar is more than 30%. Method II In this case, x 30 070. 00 00 Step. 76 974 54 76 76 54 84 76 Step. 074. 07. 0. 5 066. Therefore, in two years (007 and 008), the production of sugar has decreased y more than 30% over the production in the previous year.. Ratio Ratio is compared y division of the measure of two quantities of the same kind. If a, are two quantities of the same kind (a, 0), then the quotient a (which is clearly a numer without any unit is called the ratio ofa and). It is written asa : (read asa is to). The quantitiesa and are called terms of the ratioa :, a is the first term and is the second term. A ratio can e expressed in several ways, i.e., a : is equal to ma : m, since the quotient does not change when we divide (or multiply) the dividend and the divisile y same non-zero numer, saym. For example, : 3 = 4 : 6 = 0 : 30 = 00 : 300 m : 3m. In the ratio : 3, the two terms and 3 have no common factors other than. The ratio expressed in this form is said to e in the simplest form. Usually, a ratio is expressed in the simplest form. Ratio of Equality, Greater Inequality or Lesser Inequality A ratio is said to e a ratio of equality, greater or lesser inequality according as first term also known as antecedent is equal to or greater than or less than to second term also known as consequent. In other words, the ratio a :, where a is called a ratio of equality. (e.g., :, : etc.) the ratio a :, where a is called a ratio of greater inequality. (e.g., 3 :, 4 : 3 etc.) the ratio a :, where a is called a ratio of lesser inequality. (e.g., 3 : 5, 4 : 7 etc.) Rule A ratio of equality is unaltered, a ratio of greater inequality is diminished and a ratio of lesser inequality is increased, if the same positive quantity is added to oth its terms. Let a/ e the given ratio and x e a positive quantity and x. If a a x a, then x If a a x a, then x If a a x a, then x Rule A ratio of equality is unaltered, a ratio of greater inequality is increased and a ratio of lesser inequality is diminished, if same positive quantity is not greater than the smaller term e sutracted from each of its terms. Let a the given ratio, x e a positive quantity and x. If a a x a, then x If a a x a, then x If a a x a, then x (here, students are advised that they should try assuming certain values and check the results.)
6 The Accredited Guide to Data Interpretation & Data Sufficiency Interpretation of Data Involving the Ratio Rule To evaluate a ratio 7/470 (say), where numerator << denominator, it is always etter to reverse it and divide 470 y 7 (reverse operation) as 470 7 ~ 67, Remainder. So, the given ratio ~ 67 Rule To evaluate a ratio 635. (say), where numerator << denominator and also the numerator is a 384 decimal numer, it is always etter to first approximate it to a closest fraction involving integers only and then apply the reverse operation. 6. 35 6 Therefore, ~, then dividing 384 y 6, we get 4 as the result. 384 384 Therefore, given ratio ~ ~ 4 ~ 4 ~ 4 4 96 95 94 Rule 3 To find the highest and the lowest among the ratios (<) when numerator << denominator. Step. Apply reverse operation, i.e., straight aways divide the denominator of the ratio y the numerator to find how many times the denominator is of the numerator. Step. Maximum numer of times will indicate the lowest ratio and minimum numer of times will correspond to the highest ratio. Rule 4 To find the highest and the lowest among the ratio (<) when numerator < denominator. Step. Approximate the given ratio (if the numer of digits in numerator / denominator is more than ). Step. Multiply the numerator y 0 and get the resultant fraction. Step 3. Find only integer value of the resultant fraction. Step 4. If any of the integer value of the resultant fraction are same, then find the next decimal place and so on. Step 5. Compare the value of the resultant fraction. The maximum ratio will have the maximum value. Rule 5 To find the value which constitutes the maximum part (or portion) or minimum part of the total value. Ifa and are the two values constituting the total value( a ), then a is maximum when a a > and a is minimum when a. a 6. Find the highest and the lowest among the following 4 34 9 4,,,. 340 60 57 74 Sol. Step. Apply reverse operation. 340 60 57 74,,,. 4 34 9 4 Step. Numer of times 4 7 9 8 (take only integer values) Maximum value Lowest ratio Minimum value Highest ratio 7. Find the highest and the lowest among the following 673 56 8 90,,,. 77 63 95 998 Sol. Step. Approximated as 67 5 85 90,,,. 7 6 95 99 Hence, ratio. Step. Multiply y 0 670 50 8 900 ;,,,. 7 6 95 99 Step 3. 9.3 8.5 8.9 9.0 673 56 is the highest ratio and is the lowest 77 63
Introduction 7 3. Average Average is a very simple ut effective way of representing an entire group y a single value. Average of a group is defined as Sum of all items in the group Average Numer of items in the group Sum of all the items in the group means sum of the values of all the items in the group. A atsman s performance can e expressed as the average numer of runs scored per innings rather than giving the scores of individual innings. For example, let us say MS Dhoni scored the following runs in 9 different innings in a test series 45, 66, 34, 39, 0, 97, 08, 55 and 85. Then, his average score per innings in that particular test series 45 66 34 39 0 97 08 55 85 7 9 Similarly, if there are students in a class, instead of talking of the height of each individual student, we can talk of average height of the class. The average height of the class of students is equal to the sum of the heights of all the students in the class divided y the numer of students in the class. Average is also called the mean or mean value of all the values. In other words, if x, x, x 3,..., x n e n numers, then their x x K x n average n 8. Find the average numer of ikes sold over the period 004-08. Year 004 005 006 007 008 Numer of ikes 400 0 760 940 00 Sol. Average numer of ikes sold over the period Total numer of ikes sold over the period Total numer of years 400 0 760 940 00 5 3700 740 5 Interpretation of Data Involving the Average Rule If the value of each item is increased y the same valuek, then the average of the group of items will also increase y k. Rule If the value of each item is decreased y the same value k, then the average of the group of items will also decrease y k. Rule 3 If the value of each item is multiplied y the same valuek, then the average of the group of items will also e multiplied y k. Rule 4 If the value of each item is divided y the same value k (k 0), then the average of the group of items will also e divided y k. Rule 5 The average of a group of items will always lie etween the smallest value and the largest value in the group i.e., the average will e greater than the smallest value and less than the largest value in the group. Rule 6 To find the value of which year (or the entry in a tale) is close to the average value of given period. Step. Find the average value of the given period. Step. Find the difference Any value (or entry) Average value Minimum the difference, closer the value to average. Step 3. If the difference is same for any two different values (or entries), then find the percentage deviation Difference over the average i.e., 00 Average Since, difference is same, so more the average, less the percentage deviation, closer the value to the average.
8 The Accredited Guide to Data Interpretation & Data Sufficiency Weighted Average When two groups of items are comined together, then we can talk of the average of the entire group. However, if we know only the average of the two groups individually, we cannot find out the average of the comined group y items. For example, there are two sections A and B of a class where the average height of section A is cm and that of section B is 60 cm. On the asis of this information, we cannot find the average of the entire class (of the two sections together). As discussed earlier, the average height of the entire class is Sum of the total height of the entire class Total numer of students in the entire class In other words, if x is the average of n numers, x is the average of n numers, x 3 is the average of n 3 numers and so on, then average of all ( n n...) numers nx nx n 3x 3 K n n n K 3 Presentation of Data The raw data collected in any investigation is so voluminous that they are unwieldy and incomprehensile. Having collected and edited the data, the next step is to organize them in a condensed form that will highlight the main characteristics, facilitate comparisons and render them suitale for further processing and interpretation. Top management people rarely find time to go through the entire details of any report, it s daily production or the sales forecast. An effective presentation of data enales them to draw upon the information with the least effort and time. Effective presentation of data is roadly classified into the following categories.. Data Tales Tales are often used in reports, magazines and newspapers to present a set of numerical facts. They enale the reader to make comparisons and to draw quick conclusions. It is one of the easiest and most accurate way of presenting data. One of the main purpose of tales is to make complicated information easier to understand. The advantage of presenting data in a tale is that one can see the information at a glance. While answering questions ased on tales, carefully read the tale title and the column headings. The title of the tale will give you a general idea of the type and often the purpose of the information presented. The column headings tell you the specific kind of information given in that column. Both the tale title and the column headings are usually very straight forward. For Example, The data pertaining to the production of motorikes in India is represented in the following tale. Production of Motorikes in India Years Pulsar CBZ Freedom Total 000-0 5397 949 338577 6968 00-0 6464 3896 334583 6833 00-03 65559 3955 395970 380744 003-04 779 3844 35987 49 004-05 04784 798976 94 34684 The aove tale pertains to the data on the production of motorikes, yearwise from 000-0 to 004-05. Further the tale also divides the production of motorikes y categories viz. Pulsar, CBZ and Freedom. Thus, it is possile to get a picture of the production of different types of motorikes in India over a span of five years. If we want to find the contriution of CBZ in terms of the percentage of the total production in 00-03, then its value Production of CBZ in 00-03 00% Total production of motorikes in India in 00-03 3955 380744 00% ~ 3% If we want to find the growth rate of Freedom motorikes in India from 00-0 to 00-03, then its value Production of Freedom in 00-03 Production of Freedom in 00-0 00% Production of Freedom in 00-0 395970 334583 00% 334583 6387 334583 00% 8 ~ %
Introduction 9 Directions (Q. Nos. 9-3) Study the given tale carefully and answer the questions given elow. Numer of employees working in various departments of ABC Ltd. Years Departments Production Marketing Corporate Accounts Research 999 5 45 75 000 5 40 45 6 70 00 4 65 30 90 73 00 470 73 3 05 70 003 0 80 35 3 74 004 5 75 36 30 75 9. In which year did the total numer of employees reach twice the total numer of employees that the factory had in the year 999? (a) 000 () 00 (c) 00 (d) 003 Sol. () Total numer of employees in the year 999 345, 000 44, 00 708, 00 7, 003 8, 004 8 Clearly, figure of year 00 is closed to the doule of figure of the year 999. 0. In which department did the numer of employees remain the same during the year 999 and 004? (a) Production () Corporate (c) Research (d) None of these Sol. (c) Clearly, numer of employees in research department is the same in the year 999 and 004.. What is the approximate percentage increase in the numer of employees in production department from 999-04? (a) 37% () 5% (c) 0% (d) None of these Sol. (a) Numer of employees in production department in 999 in 004 5 5 Required percentage increase 00% 355 00 37 % ~ %. In which year did each department have a larger numer of employees that it had in the immediately preceding year? (a) 00 () 004 (c) 00 (d) 003 Sol. (d) From the tale, it is clear that in the year 003, each department has a larger numer of employees than it had in the immediately preceding year, i.e., 00. 3. Which department had less than 0% of the total numer of employees through all the years shown in the tale? (a) Marketing () Corporate (c) Accounts (d) None of these Sol. (a) Clearly, marketing department had less than 0% of the total numer of employees through all the years shown in the tale.. Line (Cartesian) Graph The line graph simplifies the data interpretation, as it is a pictorial presentation of data and is therefore very useful for determining trends and rate of change. The slope of the line graph helps in comparing the magnitude of change etween any two consecutive points on the graph. Steeper the slope, greater is the change in magnitude etween the two consecutive points. For Example, The following graph shows the profit percentage of WIPRO in various respective years. Profit per cent 60 40 30 0 0 000 00 00 003 004 Years If we want to find to ratio of the per cent profits of the company in the year 00 to 000, then its value Per cent profit in 00 Per cent profit in 000 5 :
0 The Accredited Guide to Data Interpretation & Data Sufficiency Directions (Q. Nos. 4-5) These questions are ased on the following graph. SL enterprises manufactures sprinklers The company s sales data is given elow. Sales (units) Sales (in ` 000) 400 3 300 00 30 00 300 000 000 00 900 900 8 860 800 7 700 600 0 90 00 400 300 00 00 5 00 00 987 988 989 990 99 99 00 4. The maximum percentage price rise was displayed in the year (a) 987 () 988 (c) 990 (d) 99 Sol. () Percentage price rise 00 in 988 00% % 00 00 90 in 990 00% 5.6% 90 5 00 in 99 00%.5% 00 Thus, percentage rise was maximum in the year 988. 5. Find the percentage increase in sales (units) in 989 as compared to previous year. (a) 4% () 6% (c) 8% (d) None of these Sol. () Sales in 988 8 units Sales in 989 900 units 900 8 Required percentage increase 00% 8 8 00% 3. Bar Graphs Sales (Units) Sales (in ` 000) 5.88% ~ 6% Given quantities can e compared y the height or length of a ar graph. A ar graph can have either vertical or horizontal ars. You can compare different quantities or the same quantity at different times. In ar graphs, the data is discrete. Presentation of data in this form makes evaluation of parameters comparatively very easy. For Example, The following graph shows the total profits of WIPRO (in ` crore) in various years. Profit 90 80 70 60 40 30 0 0 0 6 000 00 00 003 004 Year If we want to find the per cent increase in the profit of WIPRO in the year 004 as compared to previous year, then its value 8 0 00% 0 6 00 30 0 % % Directions (Q. Nos. 6-7) Refer to the following ar chart and answer the questions that follow. 4000 000 0000 8000 6000 4000 000 0 Imports 6. The percentage increase in imports etween 995-96 and 999-000 was (a) 5% () 5% (c) 00% (d) 75% Sol. (c) Import in 995-96 `7000 Crore Import in 999-000 `4000 Crore 4000 7000 Required increase 00% 7000 7000 00 00 7000 % % 36 0 Indian's foreign trade in (in ` crore) 995-96 996-97 997-98 998-99 999-000 Exports 8
Introduction 7. If oil imports constituted 0% of the total imports in 997-98, then what percentage of the trade gap was due to oil? (assuming that no oil is exported) (a) 30% () 40% (c) 85% (d) 5% 0 Sol. () Oil imports in 997-98 00 3000 = ` 600 crore Trade gap in 997 98 3000 60 `60 600 Hence, per cent of trade gap due to oil 60 00% 4. Pie Chart 40% A pie chart is a circular chart divided into sectors either in percentagewise or in degreewise. If distriution is percentagewise, then total value of the chart is taken as 00%. If distriution is degreewise, then total value of the chart is taken as 360. The arc length and the area of each sector is proportional to the quantity it represents. For Example, following pie chart shows the expenditure of a family. Others 7% Groiery 0% Savings 5% Education Clothes 5% 8% Moile Electricity ill 9% 6% Total income `0000 If we want to find the difference etween the expenditures on education and clothes sectors, then it s value 5% of 0000 8% of 0000 7% of 0000 ` 3400 As an another example, the pie chart given elow shows the marks otained y a student in different sujects. Mathematics 90 English SSc 60 54 Hindi Science 80 70 Maximum marks of each suject 0 If we want to find the marks got y the student in Mathematics, 90 then its value 0 360 30 Directions (Q. Nos. 8-9) Study to the following pie chart carefully and answer the following questions. National Budget Expenditure in the year 000 (Percentage Distriution) Veterans 6% 8. If India in the year 000, had a total expenditure of ` 0 illion, approximately how many illions did it spend on interest on det? (a) `.4 illion () ` 8.4 illion (c) ` 9.3 illion (d) ` 0.8 illion Sol. (d) Total expenditure `0 illion 00% Interest on det 9% of 0 ` 0.8 illion 9. If ` 9 illion were spent in the year 000 for veterans, what would have een the total expenditure for that year? (a) ` 00 illion () ` 80 illion (c) ` 0 illion (d) ` illion Sol. (d) ` 9 illion were spent for veterans. This represents 6.0% of the total expenditure for the year 9 000. Hence, total expenditure 00 6 ` illion 5. Case Studies Others Interest on det 9% 7% Military International 9% In this form of data presentation, the data is given in the form of a paragraph. The student is required to understand the data presented in the caselet and convert it into a tale for solving the questions. Directions (Q. Nos. 0-) The following caselet is an example of a caselet ased on reasoning. Five friends Anand, Ashish, Aishwarya, Deepak and Mani pursue the following professions in their careers : Human Resource, Law, Chartered Accountancy, Engineering and Foreign Relationship. They live in Ranchi, Patna, Kolkata, Delhi and Meerut ut not in that order. Mani and Aishwarya do not live in Ranchi or Meerut and neither of them is a lawyer or a chartered accountant. Anand and Ashish are neither an expert in foreign relationship nor an engineer and they do not live in Delhi or Ranchi. Deepak is neither a chartered accountant nor a human resource professionals. 59%
The Accredited Guide to Data Interpretation & Data Sufficiency The person living in Ranchi is neither an expert in foreign relationship nor an engineer. Anand does not live in Kolkata and Ashish is not a chartered accountant. Mani is not an expert in foreign relationship. The expert in foreign relationship does not live in Delhi. 0. Who lives in Ranchi? For Example, production of sugarcane in the world can e presented with the help of ar graph and price of sugarcane can e presented y line graph. Now, data represented y these graphs are related to each other in one or other way, World sugarcane production (in million kg) 60 Sol. We can easily answer the aove question y using the information given in the aove caselet. The following tales will result y using the direct clues etween the person and his place and the person and his profession. The question is answered at this point itself since there is only Deepak who could live in Ranchi.. Who is Chartered Accountant? Sol. From the tale we made, it is clear that Anand is chartered Accountant.. Who lives in Delhi? Sol. From the tale, it is clear that either Deepak or Mani lives in Delhi ut we know that, Deepak lives in Ranchi. Hence, Mani lives in Delhi. HR Law CA Engg. Foreign Relationship Anand X X Ashish X X X Aishwarya X X X Deepak X X X X Mani X X X Ranchi Patna Kolkata Delhi Meerut Anand X X X Ashish X X Aishwarya X X X Deepak Mani X X 6. Mixed Graph Among the graphs we have studied, if we have the comination of two or more graphs, then it is called mixed graph. It happens in cases when desired parameter is a function of two or three variales. In such cases, information is presented more than one type of graphs together. 40 30 0 0 0 95-96 96-97 97-98 98-99 99-000 Price of sugarcane in international market (per kg) 70 65 60 55 45 48 64 5 95-96 96-97 97-98 98-99 99-000 Directions (Q. Nos. 3-5) Refer to the following graphs and answer the questions ased on them. 400 00 000 800 600 400 00 0 604.36 57.83 58.93 54 Votes polled in general elections in 984-85 (in millions) 637.8 66 456.53 54.69 555.4 477.56 North South West East Numer of male voters Numer of female voters
Introduction 3 Percentage of votes polled y different parties Congress (I) 47.8 Independent 7.6 Janata 0.69 3. Which region had the highest male-female ratio of voters in 984-85? (a) North () South (c) West (d) East Sol. (a) Male-female ratio of voters in North 637.8 604.36 =.05 in South 54.69 57.83 =.03 BJP 7.49 Others 0.83 Congress (S) 6.45 in East 477.56 456.53 =.046 Thus, the male-female ratio is highest in North. Note No need to calculate for West as the resultant will e less than. 4. As per diagrams aove in 985, the numer of women in 985 per men were (a) 49 () 5 (c) 47 (d) Can t e determined Sol. (d) The data given in the aove graphs shows the numer of voters. It means numer of men and women who are not eligile for voting are not given. Hence, we cannot solve the aove question. 5. What was the total numer of votes polled y BJP in 984-85 elections? (a) 65.67 million () 38.33 million (c) 07.6 million (d) 47.35 million Sol. () Total votes polled in 984-85 elections 4.64 069.5 38.35 934.09 4383.6 million Votes polled in favour of BJP 4383.6 7.49% 38.33 million Before starting any shortcut or calculation technique, students are told aout the level of Expert s difficulty of questions as Data Interpretation mainly depends on the range of its options i.e., Advice if values given in different options are very close to each other, then it requires more time and more accuracy in solving the prolems. On the other hand, if there is wide gap among the values provided in different options of a prolem, then it is easier to solve such type of the prolem. And in such type of the prolem, we generally use approximation rather to find the exact answer. Again, range of approximation also depends, on the range of options provided in a given prolem. For Example, The data pertaining to the production of motorikes in India is represented in the following tale. Years Pulsar CBZ Yamaha Total 003-04 4098 34795 6495 99468 004-05 69 4874 9605 356 005-06 584605 39654 446576 47435 006-07 70564 57064 54 77760 007-08 843456 65704 56398 03558 Find the contriution of CBZ in terms of percentage of the total production in 004-05 year. Example (a) 7.38 () 7.35 (c) 7.3 (d) 7.33 Example of close options to each other (a) 7. () 7.5 (c) 7.35 (d) 7.8 Example of options having gap (a) 6.84 () 8. (c) 7.35 (d) 9.45 Example of options having wide gap (a).3 () 7.35 (c) 0.4 (d) 6.33 From the aove example, it is clear that for a single prolem we have four levels of difficulty. It depends on the range of the given options. So, students are suggested here efore solving the prolem, take a glance of all options and then decide the level of difficulty, level of accuracy and level of approximation.