Unit 6: Direction Sense and Data Interpretation - Subjective Questions

Cardinal directions:

• North (N): Usually represented at the top of a map.
• South (S): Opposite to north.
• East (E): To the right when facing north.
• West (W): To the left when facing north.

Intercardinal directions:

• North-East (NE): Between north and east.
• North-West (NW): Between north and west.
• South-East (SE): Between south and east.
• South-West (SW): Between south and west.

The opposite pairs are North-South, East-West, North-East-South-West, and North-West-South-East. A complete turn measures , so opposite directions differ by .

A turn changes a person's facing direction according to its angle.

• A right turn is a clockwise rotation of .
• A left turn is an anticlockwise rotation of .
• A half-turn in either direction is a rotation of .

For example, suppose a person initially faces north:

• After one right turn, the person faces east.
• After another right turn, the person faces south.
• A left turn from south makes the person face east.

Therefore, turns should be tracked in their stated order, starting from the initial direction.

Represent the starting point as .

• Ravi first walks km north.
• A right turn from north points east, so he walks km east.
• A right turn from east points south, so he walks km south.

The northward and southward movements cancel:

Only the eastward displacement remains:

Thus, Ravi is km east of the starting point and is finally facing south.

Meena's movements form a right-angled triangle:

• Horizontal displacement m east
• Vertical displacement m north

Using the Pythagorean theorem:

Since both displacements are positive toward east and north, she lies in the north-east direction from the starting point.

Therefore, Meena is m away in the north-east direction.

The eight principal directions are separated by angles of .

Starting direction: South-West.

A clockwise turn of equals three steps of :

• South-West to West
• West to North-West
• North-West to North

The person now faces north.

An anticlockwise turn of equals two steps of :

• North to North-West
• North-West to West

Therefore, the person finally faces west.

At sunrise, the sun is in the east, so a person's shadow falls toward the west.

Arun's shadow is on his left. Therefore, west must be to Arun's left. This happens when Arun faces north:

• Left of north is west.
• Right of north is east.

Since Bina is standing face to face with Arun, she faces the direction opposite to north, which is south.

Hence:

• Arun faces north.
• Bina faces south.

Track each movement carefully:

1. m south
2. A left turn from south points east, so m east
3. A left turn from east points north, so m north
4. A right turn from north points east, so m east

The vertical movements cancel:

Total eastward movement is:

Therefore:

• Final position: m east of
• Final facing direction: East
• Shortest distance from : m

Data interpretation is the process of examining data presented in tables, charts, or graphs and drawing meaningful conclusions through comparison and calculation.

The main steps are:

1. Read the title to identify the subject and period represented.
2. Check labels and legends to understand categories, colors, or symbols.
3. Identify units, such as rupees, percentages, tonnes, or thousands.
4. Read the question carefully and locate only the required data.
5. Select the correct operation, such as addition, subtraction, average, ratio, or percentage.
6. Perform the calculation using consistent units.
7. Verify the result by estimating its expected size and checking for transcription errors.

Accurate interpretation requires both correct reading of the display and correct mathematical reasoning.

Average represents the central value obtained by distributing a total equally among all observations.

For the values and :

Percentage expresses a quantity as a fraction of .

If a student scores out of :

Thus, an average gives a representative numerical value, whereas a percentage shows the relative size of a part compared with its whole.

First, calculate the total sales:

There are five months, so the average is:

The increase from the first month to the fifth month is:

The percentage increase is calculated using the first month's sales as the base:

Therefore:

• Average monthly sales: units
• Percentage increase:

Calculate the employee total for each department:

• Sales:
• Finance:
• Technology:
• Human Resources:

Total employees:

The highest departmental total is , shared by Sales and Technology.

Total number of women:

Percentage of employees who are women:

Therefore:

• Total employees:
• Highest number of employees: Sales and Technology, with each
• Percentage of women: approximately

Advantages of tabulation:

• It organizes large amounts of data into rows and columns.
• It makes exact values easy to locate.
• It supports comparison across several categories.
• Totals, averages, ratios, and percentages can be calculated directly.
• It reduces lengthy textual descriptions.

Limitations of tabulation:

• Trends may not be as visually clear as they are in graphs.
• Large tables can become difficult to scan.
• Incorrect row or column headings may cause misinterpretation.
• Tables are less effective for showing continuous change over time.
• Readers must often perform calculations before drawing conclusions.

Thus, tables are best for presenting precise values, while graphs are often better for displaying patterns and trends.

Total production is:

Average daily production:

The maximum production is units on Friday.

The increase from Monday to Friday is:

Percentage by which Friday exceeds Monday:

Therefore:

• Average production: units per day
• Maximum production: Friday
• Friday's production exceeds Monday's by:

A simple bar graph represents one data series. Each category has a single bar, and the height or length of the bar indicates its value. It is suitable for comparing one variable across categories, such as the number of books sold in different months.

A multiple bar graph represents two or more related data series. Each category contains a group of adjacent bars, with different patterns or colors identified through a legend. It is useful for comparisons such as the sales of two products across several years.

Key distinction:

• Simple bar graph: one variable or data series
• Multiple bar graph: two or more related variables

Both graphs normally use equal-width bars and a uniform numerical scale.

The amount for each category is found using:

• Food:
• Rent:
• Education:
• Transport:
• Savings:
• Others:

The central angle for Education is:

Therefore, Education accounts for and forms an angle of in the pie chart.

A complete circle represents the total data and measures . If a category has value out of total , then:

Therefore:

Since percentage share is:

The relationship between angle and percentage is:

For :

The category's value is:

Thus, the sector represents of the total and has a value of .

The data show an increasing trend from 2020 to 2022, followed by a decline in 2023 and a strong rise in 2024.

Overall increase from 2020 to 2024:

Overall percentage growth:

Year-to-year changes are:

• 2020 to 2021: increase of lakh
• 2021 to 2022: increase of lakh
• 2022 to 2023: decrease of lakh
• 2023 to 2024: increase of lakh

Therefore:

• Overall growth:
• Decline: From 2022 to 2023
• Amount of decline: lakh

Bar graph:

• Uses rectangular bars of equal width.
• Compares quantities across separate categories.
• Best for comparing sales by product, students by class, or production by factory.

Pie chart:

• Divides a circle into sectors.
• Shows each category as a proportion of a whole.
• Best when all categories form one total and percentage shares are important.

Line graph:

• Plots data points and connects them with line segments.
• Displays movement or trends, usually over time.
• Best for showing changes in temperature, prices, revenue, or population.

In summary, bar graphs emphasize comparison, pie charts emphasize composition, and line graphs emphasize trend.

Total enrolment in 2023:

Total enrolment in 2024:

Increase in total enrolment:

Percentage increase:

Percentage change by course:

• Mathematics:
• Science:
• Commerce:
• Arts:

Therefore:

• Total in 2023:
• Total in 2024:
• Overall increase: approximately
• Greatest percentage increase: Arts, with

The known percentages total:

Therefore, the percentage who walk is:

Numbers of commuters:

• Bus:
• Train:
• Car:
• Bicycle:
• Walking:

Pie-chart angles are calculated by multiplying each percentage by :

• Bus:
• Train:
• Car:
• Bicycle:
• Walking:

Comparison of bus users with walkers:

Thus, the number of bus users is seven times the number of walkers.