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Logical Reasoning · Data Interpretation and Data Sufficiency

Data Interpretation - Formulas, Key Points and Examples

Explanation

🔵
DATA INTERPRETATION\text{DATA INTERPRETATION}
🟢 1. BASIC CONCEPT Data Interpretation means analyzing given data and using it to calculate required values. Data may be presented in the form of:
Tables \text{Tables}
Bar Graphs \text{Bar Graphs}
Line Graphs \text{Line Graphs}
Pie Charts \text{Pie Charts}
Mixed Graphs \text{Mixed Graphs}
🟡 KEY POINT
Read the data carefully before performing any calculation. \text{Read the data carefully before performing any calculation.}
🟢 2. DATA TABLE A table presents information in rows and columns. For example:
Total=Sum of all relevant values \text{Total} = \text{Sum of all relevant values}
If values are:
a, b, c, d a,\ b,\ c,\ d
then:
Total=a+b+c+d \text{Total}=a+b+c+d
🟢 3. TOTAL VALUE To find the total of different categories:
Total=Value1+Value2+Value3+⋯ \text{Total} = \text{Value}_1+\text{Value}_2+\text{Value}_3+\cdots
🟡 KEY POINT
Always include all categories specified in the question. \text{Always include all categories specified in the question.}
🟢 4. DIFFERENCE The difference between two values is:
Difference=Larger Value−Smaller Value \text{Difference} = \text{Larger Value}-\text{Smaller Value}
For two quantities A and B:
Difference=∣A−B∣ \text{Difference}=|A-B|
🟢 5. RATIO To find the ratio of A to B:
Ratio=A:B \text{Ratio} = A:B
or:
Ratio=AB \text{Ratio} = \frac{A}{B}
The ratio should be simplified whenever possible. 🟢 6. PERCENTAGE To find what percentage A is of B:
Percentage=AB×100 \text{Percentage} = \frac{A}{B}\times100
🟡 KEY POINT
The denominator must be the reference value mentioned in the question. \text{The denominator must be the reference value mentioned in the question.}
🟢 7. PERCENTAGE INCREASE If a value changes from an original value to a new value:
Increase=New Value−Original Value \text{Increase} = \text{New Value}-\text{Original Value}
Percentage Increase=IncreaseOriginal Value×100 \text{Percentage Increase} = \frac{\text{Increase}}{\text{Original Value}}\times100
🟢 8. PERCENTAGE DECREASE
Decrease=Original Value−New Value \text{Decrease} = \text{Original Value}-\text{New Value}
Percentage Decrease=DecreaseOriginal Value×100 \text{Percentage Decrease} = \frac{\text{Decrease}}{\text{Original Value}}\times100
🟢 9. AVERAGE The average of a set of values is:
Average=Sum of ValuesNumber of Values \text{Average} = \frac{\text{Sum of Values}}{\text{Number of Values}}
For n values:
Average=x1+x2+x3+⋯+xnn \text{Average} = \frac{x_1+x_2+x_3+\cdots+x_n}{n}
🟢 10. WEIGHTED AVERAGE When different values have different frequencies or weights:
Weighted Average=∑wx∑w \text{Weighted Average} = \frac{\sum wx}{\sum w}
where:
w=Weight w=\text{Weight}
and:
x=Value x=\text{Value}
🟢 11. BAR GRAPH A bar graph represents data using rectangular bars. The length or height of each bar represents the corresponding value.
Bar Length∝Data Value \text{Bar Length}\propto\text{Data Value}
🟡 KEY POINT
Check the scale before reading the value from a bar graph. \text{Check the scale before reading the value from a bar graph.}
🟢 12. BAR GRAPH SCALE If one division represents k units:
Value=Number of Divisions×k \text{Value} = \text{Number of Divisions}\times k
For example, if one division represents 20:
5 divisions=5×20 5\text{ divisions}=5\times20
=100 =100
🟢 13. DOUBLE BAR GRAPH A double bar graph compares two sets of data. For categories A, B and C:
Difference=First Data Set−Second Data Set \text{Difference} = \text{First Data Set}-\text{Second Data Set}
The values should be compared category by category. 🟢 14. LINE GRAPH A line graph shows changes in data over time or across different categories. The change between two consecutive values is:
Change=New Value−Previous Value \text{Change} = \text{New Value}-\text{Previous Value}
🟡 KEY POINT
Look at the direction and scale of the line carefully. \text{Look at the direction and scale of the line carefully.}
🟢 15. TOTAL FROM A LINE GRAPH If the values shown are:
x1,x2,x3,…,xn x_1,x_2,x_3,\ldots,x_n
then:
Total=x1+x2+x3+⋯+xn \text{Total} = x_1+x_2+x_3+\cdots+x_n
🟢 16. PIE CHART A pie chart represents a whole as a circle of:
360∘ 360^\circ
The complete data represents:
100% 100\%
Therefore:
360∘=100% 360^\circ=100\%
🟢 17. PIE CHART ANGLE TO VALUE If the total quantity is T and the sector angle is θ:
Value=θ360∘×T \text{Value} = \frac{\theta}{360^\circ}\times T
🟢 18. PIE CHART VALUE TO ANGLE If a category has value V and the total is T:
Angle=VT×360∘ \text{Angle} = \frac{V}{T}\times360^\circ
🟢 19. PIE CHART ANGLE TO PERCENTAGE
Percentage=θ360∘×100 \text{Percentage} = \frac{\theta}{360^\circ}\times100
🟢 20. PIE CHART PERCENTAGE TO ANGLE
Angle=Percentage100×360∘ \text{Angle} = \frac{\text{Percentage}}{100}\times360^\circ
Therefore:
Angle=Percentage×3.6∘ \text{Angle} = \text{Percentage}\times3.6^\circ
🟢 21. IMPORTANT PIE CHART VALUES For 50%:
50%→180∘ 50\%\rightarrow180^\circ
For 25%:
25%→90∘ 25\%\rightarrow90^\circ
For 20%:
20%→72∘ 20\%\rightarrow72^\circ
For 10%:
10%→36∘ 10\%\rightarrow36^\circ
For 5%:
5%→18∘ 5\%\rightarrow18^\circ
🟢 22. FINDING TOTAL FROM A PIE CHART If a sector represents V and has angle θ:
V=θ360∘×T V = \frac{\theta}{360^\circ}\times T
Therefore:
T=V×360∘θ T = \frac{V\times360^\circ}{\theta}
🟢 23. COMPARISON OF TWO VALUES To find how many times A is B:
Times=AB \text{Times} = \frac{A}{B}
To find how much greater A is than B:
Difference=A−B \text{Difference} = A-B
To find the percentage by which A is greater than B:
Percentage=A−BB×100 \text{Percentage} = \frac{A-B}{B}\times100
🟢 24. RATIO OF TWO CATEGORIES If two categories have values A and B:
Ratio=A:B \text{Ratio}=A:B
If the ratio must be simplified, divide both terms by their HCF.\text{If the ratio must be simplified, divide both terms by their HCF.}
🟢 25. FINDING UNKNOWN VALUE If the total and known values are given:
Unknown=Total−Sum of Known Values \text{Unknown} = \text{Total}-\text{Sum of Known Values}
🟢 26. TOTAL FROM PERCENTAGES If a total T is divided into percentages:
p1%,p2%,p3%,… p_1\%,p_2\%,p_3\%,\ldots
then the corresponding values are:
Value=p100×T \text{Value} = \frac{p}{100}\times T
🟢 27. FINDING PERCENTAGE FROM A TABLE If a category has value A and the total is T:
Percentage=AT×100 \text{Percentage} = \frac{A}{T}\times100
🟢 28. FINDING AVERAGE FROM A TABLE If the values are:
x1,x2,x3,…,xn x_1,x_2,x_3,\ldots,x_n
then:
Average=x1+x2+x3+⋯+xnn \text{Average} = \frac{x_1+x_2+x_3+\cdots+x_n}{n}
🟢 29. AVERAGE WHEN TOTAL IS GIVEN If the total of n observations is T:
Average=Tn \text{Average} = \frac{T}{n}
Therefore:
T=Average×n T=\text{Average}\times n
🟢 30. CHANGE IN AVERAGE If the total changes by ΔT while the number of observations remains n:
Change in Average=ΔTn \text{Change in Average} = \frac{\Delta T}{n}
🟢 31. MISSING VALUE USING AVERAGE If n values have an average A:
Total=nA \text{Total} = nA
If the sum of known values is S:
Missing Value=nA−S \text{Missing Value} = nA-S
🟢 32. COMBINED DATA If two groups have totals T₁ and T₂ and numbers of observations n₁ and n₂:
Combined Average=T1+T2n1+n2 \text{Combined Average} = \frac{T_1+T_2}{n_1+n_2}
Using individual averages A₁ and A₂:
Combined Average=n1A1+n2A2n1+n2 \text{Combined Average} = \frac{n_1A_1+n_2A_2}{n_1+n_2}
🟢 33. PERCENTAGE OF TOTAL If a category has value V and total value is T:
Percentage Share=VT×100 \text{Percentage Share} = \frac{V}{T}\times100
🟢 34. TOTAL PRODUCTION OR SALES If production or sales are given for different years:
Total=Year 1+Year 2+Year 3+⋯ \text{Total} = \text{Year 1}+\text{Year 2}+\text{Year 3}+\cdots
🟢 35. YEAR-TO-YEAR CHANGE For two consecutive years:
Change=Current Year Value−Previous Year Value \text{Change} = \text{Current Year Value}-\text{Previous Year Value}
Percentage change:
Percentage Change=ChangePrevious Year Value×100 \text{Percentage Change} = \frac{\text{Change}}{\text{Previous Year Value}}\times100
🟢 36. DATA SUFFICIENCY IN DI Some questions provide multiple statements or pieces of data. The objective is to determine whether the given information is sufficient to answer the question. 🟡 KEY POINT
Do not calculate unnecessary values. \text{Do not calculate unnecessary values.}
Use only the information required to answer the question. 🟢 37. APPROXIMATION Approximation can be used when the question asks for an approximate value. For example:
49.8≈50 49.8\approx50
99.7≈100 99.7\approx100
🟡 KEY POINT
Use approximation only when the question permits an approximate answer. \text{Use approximation only when the question permits an approximate answer.}
🟢 38. UNIT CONVERSION Always make sure that quantities use the same units before comparing or calculating. For example:
1 km=1000 m 1\text{ km}=1000\text{ m}
1 hour=60 minutes 1\text{ hour}=60\text{ minutes}
1 kg=1000 g 1\text{ kg}=1000\text{ g}
🟢 39. IMPORTANT DI SHORTCUT If a value is increased by x%:
New Value=Old Value(1+x100) \text{New Value} = \text{Old Value}\left(1+\frac{x}{100}\right)
If a value is decreased by x%:
New Value=Old Value(1−x100) \text{New Value} = \text{Old Value}\left(1-\frac{x}{100}\right)
🟢 40. SUCCESSIVE CHANGES If a value changes successively by a% and b%:
Net Change=(a+b+ab100)% \text{Net Change} = \left(a+b+\frac{ab}{100}\right)\%
when both changes are increases. For two decreases:
Net Decrease=(a+b−ab100)% \text{Net Decrease} = \left(a+b-\frac{ab}{100}\right)\%
🔴 IMPORTANT KEY POINTS
Read the title and headings before solving. \text{Read the title and headings before solving.}
Check the units carefully. \text{Check the units carefully.}
Check the scale of graphs. \text{Check the scale of graphs.}
Identify the total before calculating percentages. \text{Identify the total before calculating percentages.}
For percentage, use the correct reference value as the denominator. \text{For percentage, use the correct reference value as the denominator.}
For pie charts, remember 360∘=100%. \text{For pie charts, remember }360^\circ=100\%.
For averages, Average=TotalNumber of Observations. \text{For averages, Average}=\frac{\text{Total}}{\text{Number of Observations}}.
For ratios, simplify the final ratio whenever possible. \text{For ratios, simplify the final ratio whenever possible.}
For differences, subtract the smaller value from the larger value. \text{For differences, subtract the smaller value from the larger value.}
For percentage increase or decrease, compare with the original value. \text{For percentage increase or decrease, compare with the original value.}
Do not confuse total value with average value. \text{Do not confuse total value with average value.}
Use only the data required by the question. \text{Use only the data required by the question.}
Check the final answer against the given data. \text{Check the final answer against the given data.}

Example

🔵
DATA INTERPRETATION — SOLVED EXAMPLES\text{DATA INTERPRETATION — SOLVED EXAMPLES}
🟢 EXAMPLE 1: TABLE — FINDING TOTAL QUESTION: The number of students in four classes is 40, 35, 45 and 50. Find the total number of students. SOLUTION:
Total=40+35+45+50 \text{Total} = 40+35+45+50
=170 =170
🔵 ANSWER:
170 students 170\text{ students}
🟢 EXAMPLE 2: TABLE — FINDING DIFFERENCE QUESTION: A shop sold 450 pens in January and 600 pens in February. Find the difference in sales. SOLUTION:
Difference=600−450 \text{Difference} = 600-450
=150 =150
🔵 ANSWER:
150 pens 150\text{ pens}
🟢 EXAMPLE 3: RATIO QUESTION: A school has 240 boys and 160 girls. Find the ratio of boys to girls. SOLUTION:
Ratio=240:160 \text{Ratio} = 240:160
Dividing by 80:
=24080:16080 =\frac{240}{80}:\frac{160}{80}
=3:2 =3:2
🔵 ANSWER:
3:2 3:2
🟢 EXAMPLE 4: PERCENTAGE OF TOTAL QUESTION: A company sold 800 units, out of which 200 were electronic items. What percentage of the total sales were electronic items? SOLUTION:
Percentage=200800×100 \text{Percentage} = \frac{200}{800}\times100
=25% =25\%
🔵 ANSWER:
25% 25\%
🟢 EXAMPLE 5: AVERAGE QUESTION: The marks obtained by five students are 60, 70, 80, 90 and 100. Find the average marks. SOLUTION:
Average=60+70+80+90+1005 \text{Average} = \frac{60+70+80+90+100}{5}
=4005 = \frac{400}{5}
=80 =80
🔵 ANSWER:
80 marks 80\text{ marks}
🟢 EXAMPLE 6: FINDING MISSING VALUE QUESTION: The average of five numbers is 24. Four of the numbers are 18, 20, 25 and 27. Find the fifth number. SOLUTION:
Total=24×5 \text{Total} = 24\times5
=120 =120
Sum of the known values:
18+20+25+27 18+20+25+27
=90 =90
Therefore:
Missing Value=120−90 \text{Missing Value} = 120-90
=30 =30
🔵 ANSWER:
30 30
🟢 EXAMPLE 7: BAR GRAPH — TOTAL QUESTION: A bar graph shows that a shop sold 120, 150, 180 and 200 units in four months. Find the total number of units sold. SOLUTION:
Total=120+150+180+200 \text{Total} = 120+150+180+200
=650 =650
🔵 ANSWER:
650 units 650\text{ units}
🟢 EXAMPLE 8: BAR GRAPH — MAXIMUM VALUE QUESTION: The sales of five months are 300, 450, 375, 500 and 425 units. Which month has the highest sales? SOLUTION:
300, 450, 375, 500, 425 300,\ 450,\ 375,\ 500,\ 425
The highest value is:
500 500
🔵 ANSWER:
500 units 500\text{ units}
🟢 EXAMPLE 9: BAR GRAPH — DIFFERENCE QUESTION: A company produced 750 units in March and 950 units in April. Find the percentage increase in production. SOLUTION:
Increase=950−750 \text{Increase} = 950-750
=200 =200
Percentage Increase=200750×100 \text{Percentage Increase} = \frac{200}{750}\times100
=2623% =26\frac{2}{3}\%
🔵 ANSWER:
2623% 26\frac{2}{3}\%
🟢 EXAMPLE 10: LINE GRAPH — CHANGE QUESTION: The sales of a company were 500 units in January and 650 units in February. Find the increase in sales. SOLUTION:
Increase=650−500 \text{Increase} = 650-500
=150 =150
🔵 ANSWER:
150 units 150\text{ units}
🟢 EXAMPLE 11: LINE GRAPH — PERCENTAGE INCREASE QUESTION: The production of a factory increased from 2,000 units to 2,500 units. Find the percentage increase. SOLUTION:
Increase=2500−2000 \text{Increase} = 2500-2000
=500 =500
Percentage Increase=5002000×100 \text{Percentage Increase} = \frac{500}{2000}\times100
=25% =25\%
🔵 ANSWER:
25% 25\%
🟢 EXAMPLE 12: LINE GRAPH — PERCENTAGE DECREASE QUESTION: The number of visitors to a website decreased from 5,000 to 4,000. Find the percentage decrease. SOLUTION:
Decrease=5000−4000 \text{Decrease} = 5000-4000
=1000 =1000
Percentage Decrease=10005000×100 \text{Percentage Decrease} = \frac{1000}{5000}\times100
=20% =20\%
🔵 ANSWER:
20% 20\%
🟢 EXAMPLE 13: PIE CHART — FINDING VALUE QUESTION: A pie chart represents a total expenditure of Rs. 72,000. Food represents a sector of 90°. Find the expenditure on food. SOLUTION:
Food Expenditure=90∘360∘×72000 \text{Food Expenditure} = \frac{90^\circ}{360^\circ}\times72000
=14×72000 = \frac{1}{4}\times72000
=18000 =18000
🔵 ANSWER:
Rs. 18,000 Rs.\ 18,000
🟢 EXAMPLE 14: PIE CHART — FINDING PERCENTAGE QUESTION: A category occupies 72° in a pie chart. Find the percentage represented by the category. SOLUTION:
Percentage=72∘360∘×100 \text{Percentage} = \frac{72^\circ}{360^\circ}\times100
=20% =20\%
🔵 ANSWER:
20% 20\%
🟢 EXAMPLE 15: PIE CHART — FINDING ANGLE QUESTION: A category represents 25% of the total data. Find its angle in a pie chart. SOLUTION:
Angle=25100×360∘ \text{Angle} = \frac{25}{100}\times360^\circ
=90∘ =90^\circ
🔵 ANSWER:
90∘ 90^\circ
🟢 EXAMPLE 16: PIE CHART — FINDING TOTAL QUESTION: In a pie chart, 20% of the total students are represented by 80 students. Find the total number of students. SOLUTION:
Total=80×10020 \text{Total} = \frac{80\times100}{20}
=400 =400
🔵 ANSWER:
400 students 400\text{ students}
🟢 EXAMPLE 17: PIE CHART — FINDING UNKNOWN VALUE QUESTION: A pie chart represents 1,200 employees. The sector for sales employees is 120°. Find the number of sales employees. SOLUTION:
Sales Employees=120∘360∘×1200 \text{Sales Employees} = \frac{120^\circ}{360^\circ}\times1200
=13×1200 = \frac{1}{3}\times1200
=400 =400
🔵 ANSWER:
400 employees 400\text{ employees}
🟢 EXAMPLE 18: COMPARISON QUESTION: A company sold 800 products in one year and 1,000 products in the next year. By how many products did the sales increase? SOLUTION:
Increase=1000−800 \text{Increase} = 1000-800
=200 =200
🔵 ANSWER:
200 products 200\text{ products}
🟢 EXAMPLE 19: HOW MANY TIMES QUESTION: A factory produced 1,200 units in January and 300 units in February. How many times the January production is the February production? SOLUTION:
Times=1200300 \text{Times} = \frac{1200}{300}
=4 =4
🔵 ANSWER:
4 times 4\text{ times}
🟢 EXAMPLE 20: PERCENTAGE COMPARISON QUESTION: A shop sold 900 items in June and 1,200 items in July. By what percentage was July's sales greater than June's sales? SOLUTION:
Increase=1200−900 \text{Increase} = 1200-900
=300 =300
Percentage Increase=300900×100 \text{Percentage Increase} = \frac{300}{900}\times100
=3313% =33\frac{1}{3}\%
🔵 ANSWER:
3313% 33\frac{1}{3}\%
🟢 EXAMPLE 21: UNKNOWN VALUE FROM TOTAL QUESTION: The total number of students in four classes is 500. Three classes have 120, 150 and 110 students. Find the number of students in the fourth class. SOLUTION:
Known Total=120+150+110 \text{Known Total} = 120+150+110
=380 =380
Fourth Class=500−380 \text{Fourth Class} = 500-380
=120 =120
🔵 ANSWER:
120 students 120\text{ students}
🟢 EXAMPLE 22: AVERAGE FROM TOTAL QUESTION: A company produced a total of 3,600 units in 6 months. Find the average monthly production. SOLUTION:
Average=Total ProductionNumber of Months \text{Average} = \frac{\text{Total Production}}{\text{Number of Months}}
=36006 = \frac{3600}{6}
=600 =600
🔵 ANSWER:
600 units 600\text{ units}
🟢 EXAMPLE 23: COMBINED AVERAGE QUESTION: The average marks of 20 students is 60 and the average marks of 30 students is 70. Find the combined average. SOLUTION: Total marks of first group:
20×60=1200 20\times60 = 1200
Total marks of second group:
30×70=2100 30\times70 = 2100
Combined total:
1200+2100=3300 1200+2100 = 3300
Total students:
20+30=50 20+30 = 50
Therefore:
Combined Average=330050 \text{Combined Average} = \frac{3300}{50}
=66 =66
🔵 ANSWER:
66 marks 66\text{ marks}
🟢 EXAMPLE 24: DATA TABLE — PERCENTAGE QUESTION: A company's total sales are 2,500 units. Product A accounts for 625 units. Find the percentage of sales represented by Product A. SOLUTION:
Percentage=6252500×100 \text{Percentage} = \frac{625}{2500}\times100
=25% =25\%
🔵 ANSWER:
25% 25\%
🟢 EXAMPLE 25: DATA TABLE — RATIO QUESTION: A school has 360 boys and 240 girls. Find the ratio of boys to girls. SOLUTION:
Ratio=360:240 \text{Ratio} = 360:240
Dividing by 120:
=3:2 =3:2
🔵 ANSWER:
3:2 3:2
🟢 EXAMPLE 26: DATA TABLE — AVERAGE QUESTION: The daily sales of a shop for five days are 200, 250, 300, 350 and 400 units. Find the average daily sales. SOLUTION:
Average=200+250+300+350+4005 \text{Average} = \frac{200+250+300+350+400}{5}
=15005 = \frac{1500}{5}
=300 =300
🔵 ANSWER:
300 units 300\text{ units}
🟢 EXAMPLE 27: PERCENTAGE DECREASE QUESTION: The number of employees in a company decreased from 800 to 680. Find the percentage decrease. SOLUTION:
Decrease=800−680 \text{Decrease} = 800-680
=120 =120
Percentage Decrease=120800×100 \text{Percentage Decrease} = \frac{120}{800}\times100
=15% =15\%
🔵 ANSWER:
15% 15\%
🟢 EXAMPLE 28: SUCCESSIVE DATA CHANGE QUESTION: The sales of a company increased by 10% in the first year and by 20% in the second year. Find the overall percentage increase. SOLUTION:
Net Increase=(10+20+10×20100)% \text{Net Increase} = \left(10+20+\frac{10\times20}{100}\right)\%
=(30+2)% = (30+2)\%
=32% =32\%
🔵 ANSWER:
32% 32\%
🟢 EXAMPLE 29: MIXED DATA INTERPRETATION QUESTION: A company sold 400, 500, 600 and 700 units in four consecutive months. Find the average monthly sales and the percentage of total sales contributed by the third month. SOLUTION: Total sales:
400+500+600+700 400+500+600+700
=2200 =2200
Average monthly sales:
Average=22004 \text{Average} = \frac{2200}{4}
=550 =550
Percentage contributed by the third month:
Percentage=6002200×100 \text{Percentage} = \frac{600}{2200}\times100
=27311% =27\frac{3}{11}\%
🔵 ANSWER:
Average=550 units \text{Average}=550\text{ units}
Third Month=27311% \text{Third Month}=27\frac{3}{11}\%
🟢 EXAMPLE 30: MIXED PIE CHART PROBLEM QUESTION: A company's total annual expenditure is Rs. 4,00,000. If 15% is spent on transportation, find the amount spent on transportation. SOLUTION:
Transportation Expenditure=15100×400000 \text{Transportation Expenditure} = \frac{15}{100}\times400000
=60000 =60000
🔵 ANSWER:
Rs. 60,000 Rs.\ 60,000