Unit 6: Data Interpretation and Data Sufficiency - Subjective Questions

PEA308 — Advanced Analytical Skills-Ii • Practice Questions with Detailed Answers

20 questions

A bar graph represents the annual production of two factories, A and B, over five years.

Year	Factory A	Factory B
2020	120	100
2021	150	140
2022	180	160
2023	210	200
2024	240	220

The production figures are in thousand units. Calculate the total production of each factory, the percentage by which Factory A's total exceeds Factory B's total, and the compound annual growth rate of Factory A from 2020 to 2024.

A grouped bar graph shows the revenue of four branches in 2022 and 2023.

Branch	2022 revenue	2023 revenue
North	80	100
South	120	150
East	100	90
West	150	180

Revenue is measured in lakh rupees. Compare the overall revenues of the two years, identify the branches with the largest absolute increase, and determine the percentage change for the East branch.

Explain how a bar graph should be interpreted systematically. Distinguish between a simple bar graph, a grouped bar graph, and a stacked bar graph, and describe two common sources of misleading interpretation.

The following table gives the number of units of four products sold in 2022 and 2023.

Product	2022	2023
P	500	650
Q	400	460
R	600	540
S	300	450

Calculate the total percentage growth in sales, the percentage change for each product, and identify the fastest-growing product.

The marks of four students in three subjects are shown below. Student C's Mathematics marks are missing.

Student	Mathematics	Physics	English
A	78	82	74
B	85	76	80
C	72	$x$	84
D	88	90	86

If Student C's average is 78, find $x$ . Then calculate the class average in Physics and rank the students according to their total marks.

A production table contains the following data for three machines.

Machine	Total output	Operating hours	Defective units
M1	480	8	12
M2	550	10	11
M3	630	9	21

Compare the machines using gross output per hour, defect rate, and good output per hour. Also calculate the overall defect rate.

A pie chart represents a monthly household income of Rs. 60,000 distributed as follows: Housing $30\%$ , Food $20\%$ , Transport $15\%$ , Education $10\%$ , Savings $15\%$ , and Others $10\%$ . Find the amount spent or saved under each category, the central angle for Education, and the ratio of Savings to Others.

Two pie charts show the market shares of five companies in the previous year and the current year. The previous year's total market was 700 thousand units, while the current year's total market is 800 thousand units.

Company	Previous share	Current share
A	$20\%$	$25\%$
B	$25\%$	$20\%$
C	$20\%$	$15\%$
D	$15\%$	$10\%$
E	$20\%$	$30\%$

Calculate the actual sales of every company in both years and identify the company with the highest percentage growth.

A pie chart divides 2,500 employees among five departments. The central angles for Production, Sales, Finance, and Human Resources are respectively $72^\circ$ , $108^\circ$ , $54^\circ$ , and $90^\circ$ . The remaining sector represents Research. Determine the angle, percentage share, and number of employees in Research. Also find the combined number of employees in Production and Sales.

A line graph records the monthly visitors, in thousands, to a website as follows: January 120, February 150, March 135, April 180, May 210, and June 195. Determine the average monthly visitors, the month with the largest absolute increase, the percentage increase during that month, and the overall percentage change from January to June.

A company's monthly profit, in lakh rupees, for six consecutive months is $10, 14, 13, 18, 21,$ and $24$ . Calculate the three-month moving averages and explain what they reveal about the profit trend.

A line graph compares the output of two production lines over five weeks. Line A records $20, 30, 40, 50,$ and $60$ units, while Line B records $50, 45, 40, 35,$ and $30$ units. Compare their trends, identify their point of intersection, and calculate the total and average output of each line.

A mixed graph uses bars for units sold and a line for average selling price. The data are:

Year	Units sold	Price per unit
2020	1,000	Rs. 50
2021	1,200	Rs. 55
2022	1,500	Rs. 52
2023	1,400	Rs. 60

Calculate annual revenue, identify the year with the highest revenue, and determine the percentage increase in revenue from 2020 to 2023. Explain why the year with the highest units sold need not have the highest revenue.

A mixed pie-chart and tabular presentation describes 4,000 online orders. Regional shares are North $35\%$ , South $25\%$ , East $20\%$ , and West $20\%$ . Their respective return rates are $2\%, 3\%, 4\%,$ and $1\%$ . Calculate the number of orders and returned orders for every region, and determine the overall return rate.

A mixed graph shows the output and capacity utilization of three plants. Plant A produces 800 units at $80\%$ utilization, Plant B produces 1,000 units at $100\%$ utilization, and Plant C produces 1,200 units at $75\%$ utilization. Determine the installed capacity and unused capacity of each plant, and calculate the combined utilization rate.

A histogram has the following equal-width class intervals and frequencies:

Class interval	Frequency
$0$ - $10$	5
$10$ - $20$	9
$20$ - $30$	12
$30$ - $40$	8
$40$ - $50$	6

Estimate the mean, median, and mode of the distribution.

A histogram is to be drawn for unequal class intervals with the following frequencies:

Class interval	Frequency
$0$ - $5$	10
$5$ - $15$	30
$15$ - $20$	20
$20$ - $40$	40

Calculate the frequency density of each interval, identify the modal class from the histogram, and find the percentage of observations between 5 and 20.

Distinguish between a bar graph and a histogram. Explain the role of class boundaries, class width, frequency, frequency density, and bar area in constructing and interpreting a histogram.

For the question "What is the value of $x$ ?", examine the following statements and determine whether the data are sufficient.

Statement I: $x^2=49$ .
Statement II: $x>0$ .

State whether Statement I alone, Statement II alone, both statements together, or neither is sufficient. Justify your conclusion.

A positive two-digit number $N$ has tens digit $a$ and units digit $b$ . Determine whether the following statements are sufficient to find $N$ uniquely.

Statement I: The sum of the digits is 11.
Statement II: The number obtained by reversing the digits is 27 greater than the original number.

Test each statement separately and then together.

Question 1

A bar graph represents the annual production of two factories, A and B, over five years.

Year	Factory A	Factory B
2020	120	100
2021	150	140
2022	180	160
2023	210	200
2024	240	220

Tap card to reveal answer

Answer 1

Step 1: Total production

Factory A: $120+150+180+210+240=900$ thousand units.
Factory B: $100+140+160+200+220=820$ thousand units.

Step 2: Percentage excess of A over B

$\frac{900-820}{820}\times 100=9.76\%$

Thus, Factory A's total production exceeds Factory B's total by approximately $9.76\%$ .

Step 3: Compound annual growth rate of Factory A

There are four annual growth intervals from 2020 to 2024.

$\text{CAGR}=\left(\frac{240}{120}\right)^{\frac{1}{4}}-1$

$\text{CAGR}=2^{\frac{1}{4}}-1\approx 0.1892$

Therefore, the CAGR of Factory A is approximately $18.92\%$ per year.

Tap card to flip back

Question 2

A grouped bar graph shows the revenue of four branches in 2022 and 2023.

Branch	2022 revenue	2023 revenue
North	80	100
South	120	150
East	100	90
West	150	180

Revenue is measured in lakh rupees. Compare the overall revenues of the two years, identify the branches with the largest absolute increase, and determine the percentage change for the East branch.

Tap card to reveal answer

Answer 2

Overall revenue in 2022

$80+120+100+150=450\text{ lakh}$

Overall revenue in 2023

$100+150+90+180=520\text{ lakh}$

The absolute overall increase is:

$520-450=70\text{ lakh}$

The percentage increase is:

$\frac{70}{450}\times 100=15.56\%$

Branch-wise absolute changes

North: $100-80=20$ lakh.
South: $150-120=30$ lakh.
East: $90-100=-10$ lakh.
West: $180-150=30$ lakh.

South and West have the largest absolute increase of $30$ lakh each.

For the East branch:

$\frac{90-100}{100}\times 100=-10\%$

Therefore, East experienced a $10\%$ decrease, while total revenue increased by approximately $15.56\%$ .

Tap card to flip back

Question 3

Tap card to reveal answer

Answer 3

Systematic interpretation of a bar graph

Read the title to identify the subject and time period.
Examine the horizontal and vertical axes and their units.
Check the scale and determine whether it begins at zero.
Use the legend to identify different data series.
Compare bar heights and calculate absolute or percentage differences when required.

Types of bar graphs

Simple bar graph: Uses one bar for each category and represents a single data series.
Grouped bar graph: Places two or more bars side by side for each category, allowing direct comparison among multiple series.
Stacked bar graph: Divides each bar into components. The complete bar represents a total, while its segments represent constituent parts.

Misleading features

A truncated vertical axis can exaggerate small differences between values.
Unequal bar widths or inconsistent scales can create a false visual comparison.

A correct interpretation must therefore use the numerical scale rather than relying only on the apparent bar sizes.

Tap card to flip back

Question 4

The following table gives the number of units of four products sold in 2022 and 2023.

Product	2022	2023
P	500	650
Q	400	460
R	600	540
S	300	450

Calculate the total percentage growth in sales, the percentage change for each product, and identify the fastest-growing product.

Tap card to reveal answer

Answer 4

Total sales

In 2022: $500+400+600+300=1800$ units.
In 2023: $650+460+540+450=2100$ units.

The overall percentage growth is:

$\frac{2100-1800}{1800}\times 100=16.67\%$

Product-wise percentage changes

Product P: $\frac{650-500}{500}\times 100=30\%$
Product Q: $\frac{460-400}{400}\times 100=15\%$
Product R: $\frac{540-600}{600}\times 100=-10\%$
Product S: $\frac{450-300}{300}\times 100=50\%$

Product S is the fastest-growing product with a $50\%$ increase. Product R is the only product that recorded a decline.

Tap card to flip back

Question 5

The marks of four students in three subjects are shown below. Student C's Mathematics marks are missing.

Student	Mathematics	Physics	English
A	78	82	74
B	85	76	80
C	72	$x$	84
D	88	90	86

If Student C's average is 78, find $x$ . Then calculate the class average in Physics and rank the students according to their total marks.

Tap card to reveal answer

Answer 5

Student C's total marks must be:

$78\times 3=234$

Therefore:

$72+x+84=234$

$x=234-156=78$

Physics class average

$\frac{82+76+78+90}{4}=\frac{326}{4}=81.5$

Total marks of each student

A: $78+82+74=234$
B: $85+76+80=241$
C: $72+78+84=234$
D: $88+90+86=264$

Ranking

D with $264$ marks.
B with $241$ marks.
A and C jointly with $234$ marks each.

Thus, the missing Physics mark is $78$ , and the Physics class average is $81.5$ .

Tap card to flip back

Question 6

A production table contains the following data for three machines.

Machine	Total output	Operating hours	Defective units
M1	480	8	12
M2	550	10	11
M3	630	9	21

Compare the machines using gross output per hour, defect rate, and good output per hour. Also calculate the overall defect rate.

Tap card to reveal answer

Answer 6

Machine M1

Gross output per hour: $480/8=60$
Defect rate: $(12/480)\times 100=2.5\%$
Good output: $480-12=468$
Good output per hour: $468/8=58.5$

Machine M2

Gross output per hour: $550/10=55$
Defect rate: $(11/550)\times 100=2\%$
Good output: $550-11=539$
Good output per hour: $539/10=53.9$

Machine M3

Gross output per hour: $630/9=70$
Defect rate: $(21/630)\times 100=3.33\%$
Good output: $630-21=609$
Good output per hour: $609/9\approx 67.67$

M3 has the highest gross and good output per hour, while M2 has the lowest defect rate.

Overall defect rate

$\frac{12+11+21}{480+550+630}\times 100$

$=\frac{44}{1660}\times 100\approx 2.65\%$

Therefore, the overall defect rate is approximately $2.65\%$ .

Tap card to flip back

Question 8

Company	Previous share	Current share
A	$20\%$	$25\%$
B	$25\%$	$20\%$
C	$20\%$	$15\%$
D	$15\%$	$10\%$
E	$20\%$	$30\%$

Calculate the actual sales of every company in both years and identify the company with the highest percentage growth.

Tap card to reveal answer

Answer 8

Previous-year sales in thousand units

A: $20\%$ of $700=140$
B: $25\%$ of $700=175$
C: $20\%$ of $700=140$
D: $15\%$ of $700=105$
E: $20\%$ of $700=140$

Current-year sales in thousand units

A: $25\%$ of $800=200$
B: $20\%$ of $800=160$
C: $15\%$ of $800=120$
D: $10\%$ of $800=80$
E: $30\%$ of $800=240$

Percentage growth

A: $((200-140)/140)\times 100=42.86\%$
B: $((160-175)/175)\times 100=-8.57\%$
C: $((120-140)/140)\times 100=-14.29\%$
D: $((80-105)/105)\times 100=-23.81\%$
E: $((240-140)/140)\times 100=71.43\%$

Company E has the highest percentage growth, approximately $71.43\%$ .

Tap card to flip back

Question 9

Tap card to reveal answer

Answer 9

The sum of the four given angles is:

$72^\circ+108^\circ+54^\circ+90^\circ=324^\circ$

Therefore, the angle for Research is:

$360^\circ-324^\circ=36^\circ$

Research percentage

$\frac{36}{360}\times 100=10\%$

Research employees

$10\%\times 2500=250$

The combined angle for Production and Sales is:

$72^\circ+108^\circ=180^\circ$

This is half of the complete circle. Hence, the combined number of employees is:

$\frac{180}{360}\times 2500=1250$

Research has 250 employees, while Production and Sales together have 1,250 employees.

Tap card to flip back

Question 10

Tap card to reveal answer

Answer 10

Average monthly visitors

$\frac{120+150+135+180+210+195}{6}=\frac{990}{6}=165$

The average is 165 thousand visitors.

Month-to-month changes

February: $150-120=30$
March: $135-150=-15$
April: $180-135=45$
May: $210-180=30$
June: $195-210=-15$

The largest absolute increase occurred from March to April and was 45 thousand visitors.

Its percentage value is:

$\frac{45}{135}\times 100=33.33\%$

Overall percentage change from January to June

$\frac{195-120}{120}\times 100=62.5\%$

Thus, visitors increased overall by $62.5\%$ .

Tap card to flip back

Question 12

Tap card to reveal answer

Answer 12

Trend comparison

Line A increases continuously by $10$ units per week.
Line B decreases continuously by $5$ units per week.
The lines move in opposite directions.

Both lines record $40$ units in week 3, so this is their point of intersection.

Line A

$20+30+40+50+60=200$

$\text{Average}=\frac{200}{5}=40$

Line B

$50+45+40+35+30=200$

$\text{Average}=\frac{200}{5}=40$

Although the trends are opposite, both production lines have the same total output of 200 units and the same average output of 40 units per week.

Tap card to flip back

Question 13

A mixed graph uses bars for units sold and a line for average selling price. The data are:

Year	Units sold	Price per unit
2020	1,000	Rs. 50
2021	1,200	Rs. 55
2022	1,500	Rs. 52
2023	1,400	Rs. 60

Tap card to reveal answer

Answer 13

Revenue is calculated using:

$\text{Revenue}=\text{Units sold}\times\text{Price per unit}$

Annual revenues

2020: $1000\times 50=\text{Rs. }50000$
2021: $1200\times 55=\text{Rs. }66000$
2022: $1500\times 52=\text{Rs. }78000$
2023: $1400\times 60=\text{Rs. }84000$

The highest revenue was Rs. 84,000 in 2023.

Percentage increase from 2020 to 2023

$\frac{84000-50000}{50000}\times 100=68\%$

The highest number of units was sold in 2022, but the highest revenue occurred in 2023. This is because the price per unit increased from Rs. 52 to Rs. 60, more than compensating for the reduction of 100 units in sales.

Tap card to flip back

Question 14

Tap card to reveal answer

Answer 14

Regional orders and returns

North orders: $4000\times 35\%=1400$
North returns: $1400\times 2\%=28$
South orders: $4000\times 25\%=1000$
South returns: $1000\times 3\%=30$
East orders: $4000\times 20\%=800$
East returns: $800\times 4\%=32$
West orders: $4000\times 20\%=800$
West returns: $800\times 1\%=8$

Total returned orders

$28+30+32+8=98$

Overall return rate

$\frac{98}{4000}\times 100=2.45\%$

East has the largest number of returns, while West has the lowest return rate. The overall return rate is $2.45\%$ , which is a weighted rate based on regional order volumes.

Tap card to flip back

Question 15

Tap card to reveal answer

Answer 15

Installed capacity is determined from:

$\text{Installed capacity}=\frac{\text{Actual output}}{\text{Utilization rate}}$

Plant A

$\text{Capacity}=\frac{800}{0.80}=1000$

Unused capacity: $1000-800=200$ units.

Plant B

$\text{Capacity}=\frac{1000}{1.00}=1000$

Unused capacity: $1000-1000=0$ units.

Plant C

$\text{Capacity}=\frac{1200}{0.75}=1600$

Unused capacity: $1600-1200=400$ units.

Combined utilization

Total output:

$800+1000+1200=3000$

Total capacity:

$1000+1000+1600=3600$

$\text{Combined utilization}=\frac{3000}{3600}\times 100=83.33\%$

Plant C has the greatest output but also the greatest unused capacity. The combined utilization rate is approximately $83.33\%$ .

Tap card to flip back

Question 16

A histogram has the following equal-width class intervals and frequencies:

Class interval	Frequency
$0$ - $10$	5
$10$ - $20$	9
$20$ - $30$	12
$30$ - $40$	8
$40$ - $50$	6

Estimate the mean, median, and mode of the distribution.

Tap card to reveal answer

Answer 16

Estimated mean

The class midpoints are $5, 15, 25, 35,$ and $45$ .

$\sum fm=(5\times5)+(9\times15)+(12\times25)+(8\times35)+(6\times45)$

$\sum fm=25+135+300+280+270=1010$

The total frequency is $40$ . Therefore:

$\bar{x}=\frac{1010}{40}=25.25$

Estimated median

The cumulative frequencies are $5, 14, 26, 34,$ and $40$ . Since $N/2=20$ , the median class is $20$ - $30$ .

$\text{Median}=20+\left(\frac{20-14}{12}\right)\times10=25$

Estimated mode

The modal class is $20$ - $30$ because it has the highest frequency.

$\text{Mode}=20+\left(\frac{12-9}{2(12)-9-8}\right)\times10$

$\text{Mode}=20+\frac{30}{7}\approx24.29$

Thus, the estimated mean, median, and mode are $25.25, 25,$ and $24.29$ , respectively.

Tap card to flip back

Question 17

A histogram is to be drawn for unequal class intervals with the following frequencies:

Class interval	Frequency
$0$ - $5$	10
$5$ - $15$	30
$15$ - $20$	20
$20$ - $40$	40

Calculate the frequency density of each interval, identify the modal class from the histogram, and find the percentage of observations between 5 and 20.

Tap card to reveal answer

Answer 17

For unequal class widths, histogram height must represent frequency density:

$\text{Frequency density}=\frac{\text{Frequency}}{\text{Class width}}$

Frequency densities

$0$ - $5$ : $10/5=2$
$5$ - $15$ : $30/10=3$
$15$ - $20$ : $20/5=4$
$20$ - $40$ : $40/20=2$

The interval $15$ - $20$ has the greatest frequency density, so it is the modal class in the histogram.

The total frequency is:

$10+30+20+40=100$

The frequency from 5 to 20 is:

$30+20=50$

Therefore, the required percentage is:

$\frac{50}{100}\times 100=50\%$

The area of each histogram bar, rather than its height alone, is proportional to frequency.

Tap card to flip back

Question 18

Distinguish between a bar graph and a histogram. Explain the role of class boundaries, class width, frequency, frequency density, and bar area in constructing and interpreting a histogram.

Tap card to reveal answer

Answer 18

Bar graph and histogram comparison

A bar graph represents discrete or categorical data, whereas a histogram represents a continuous frequency distribution.
Bars in a bar graph normally have gaps; histogram bars touch because the class intervals are continuous.
The order of categories in a bar graph can often be changed, but histogram intervals must remain in numerical order.
Bar width in a simple bar graph has no statistical meaning. Histogram width represents the class interval.

Important histogram concepts

Class boundaries: The exact limits that make adjacent classes continuous, such as converting $10$ - $19$ and $20$ - $29$ into $9.5$ - $19.5$ and $19.5$ - $29.5$ .
Class width: The difference between the upper and lower class boundaries.
Frequency: The number of observations in a class.
Frequency density: Calculated as $\text{frequency}/\text{class width}$ and used as bar height when widths are unequal.
Bar area: Proportional to frequency because $\text{area}=\text{class width}\times\text{frequency density}$ .

Consequently, the tallest histogram bar is not necessarily the class with the greatest frequency when class widths differ.

Tap card to flip back

Question 19

For the question "What is the value of $x$ ?", examine the following statements and determine whether the data are sufficient.

Statement I: $x^2=49$ .
Statement II: $x>0$ .

State whether Statement I alone, Statement II alone, both statements together, or neither is sufficient. Justify your conclusion.

Tap card to reveal answer

Answer 19

Testing Statement I alone

From:

$x^2=49$

we obtain:

$x=7\text{ or }x=-7$

Since two values are possible, Statement I alone is not sufficient.

Testing Statement II alone

The condition $x>0$ allows infinitely many positive values. Therefore, Statement II alone is not sufficient.

Using both statements together

Statement I gives $x=7$ or $x=-7$ , while Statement II requires $x$ to be positive. Hence:

$x=7$

Therefore, both statements together are sufficient, but neither statement alone is sufficient.

Tap card to flip back

Question 20

A positive two-digit number $N$ has tens digit $a$ and units digit $b$ . Determine whether the following statements are sufficient to find $N$ uniquely.

Statement I: The sum of the digits is 11.
Statement II: The number obtained by reversing the digits is 27 greater than the original number.

Test each statement separately and then together.

Tap card to reveal answer

Answer 20

The original number is:

$N=10a+b$

The reversed number is:

$10b+a$

Statement I alone

$a+b=11$

Several digit pairs satisfy this equation, such as $(2,9)$ , $(3,8)$ , $(4,7)$ , and $(5,6)$ . Therefore, Statement I alone is not sufficient.

Statement II alone

$10b+a=10a+b+27$

$9b-9a=27$

$b-a=3$

Several pairs satisfy this relation, so Statement II alone is not sufficient.

Both statements together

We have:

$a+b=11$

$b-a=3$

Adding the equations gives:

$2b=14\Rightarrow b=7$

Then:

$a=11-7=4$

Therefore:

$N=10(4)+7=47$

Thus, both statements together are sufficient, but neither statement alone is sufficient.

Tap card to flip back

1 / 20

Unit5

Unit 6: Data Interpretation and Data Sufficiency - Subjective Questions

Did this save you a night before the exam?