All practice questions

Question 1341Question →

A river basin covers a rectangular section measuring $4.5\text{ cm}$ by $8\text{ cm}$ on Map X, which has a representative fraction of $1 : 25,000$ . Map X is enlarged to create Map Y, where the same river basin occupies an area of $144\text{ cm}^2$ . Map Y is then reduced to construct Map Z, such that the denominator of Map Z's representative fraction is $4$ times that of Map Y. What is the area of the river basin on Map Z in $\text{cm}^2$ ?

Show answer & explanation

Answer: 9

Answer

The area of the river basin on Map Z is

9\text{ cm}^2

.

To find the area on Map Z, first calculate the basin's area on Map X:

4.5\text{ cm} \times 8\text{ cm} = 36\text{ cm}^2

. The area ratio from Map X to Map Y is

144 / 36 = 4

, which corresponds to a linear enlargement factor of

\sqrt{4} = 2

. Map Y therefore has a scale of

1 : 12,500

. Multiplying Map Y's RF denominator by

4

gives Map Z a scale of

1 : 50,000

. Comparing Map Y (

1 : 12,500

) to Map Z (

1 : 50,000

) reveals a linear reduction factor of

1/4

. The area reduction factor is

(1/4)^2 = 1/16

. Multiplying Map Y's area by

1/16

yields

144\text{ cm}^2 \times (1/16) = 9\text{ cm}^2

. Alternatively, comparing Map X (

1 : 25,000

) directly to Map Z (

1 : 50,000

) shows a linear factor of

1/2

, giving an area factor of

(1/2)^2 = 1/4

, and

36\text{ cm}^2 \times (1/4) = 9\text{ cm}^2

.

Step-by-Step Solution

1

Calculate the initial map area on Map X.

A_X = 4.5\text{ cm} \times 8\text{ cm} = 36\text{ cm}^2

The product of the rectangular map dimensions gives the original map area.

2

Determine the linear scale factor and representative fraction of Map Y.

Area enlargement factor =

144 / 36 = 4

; Linear enlargement factor =

\sqrt{4} = 2

. Map Y scale =

1 : (25,000 / 2) = 1 : 12,500

.

Area change ratio is the square of the linear scale factor (

n^2

). Enlarging a map decreases its RF denominator by the linear factor.

3

Calculate the representative fraction of Map Z.

Map Z RF denominator =

12,500 \times 4 = 50,000

, so scale of Map Z =

1 : 50,000

.

Reducing a map increases the RF denominator proportionately.

4

Determine the final map area on Map Z.

Linear reduction factor from Y to Z =

12,500 / 50,000 = 1/4

. Area reduction factor =

(1/4)^2 = 1/16

. Final area

A_Z = 144\text{ cm}^2 \times (1/16) = 9\text{ cm}^2

.

Applying the squared linear scale reduction factor to Map Y's area yields Map Z's area.

Key Concept

Map Enlargement, Reduction, and Area-to-Linear Scale Ratio Relationships

Estimated Time:3m 0s

Question 1342Question →

Benue Manufacturing Enterprise transfers finished goods from its factory to its retail outlet at a transfer price that includes a mark-up of $25\%$ on cost. On 1 January 2025, the provision for unrealized profit was $\text{₦}2,000$ . On 31 December 2025, the stock of finished goods valued at transfer price was $\text{₦}25,000$ . What is the amount of closing provision for unrealized profit (in $\text{₦}$ ) to be deducted from finished goods inventory in the Statement of Financial Position?

Show answer & explanation

Answer: 5000

Answer

The closing provision for unrealized profit on finished goods inventory is \text{₦}5,000.

The closing inventory of finished goods is held at transfer price (cost + profit). To find the unrealized profit provision, the

25\%

mark-up on cost is converted to a

20\%

margin on transfer price (

\frac{25}{125} = \frac{1}{5}

). Applying

20\%

to the closing inventory value of

\text{₦}25,000

gives a closing provision of

\text{₦}5,000

.

Step-by-Step Solution

1

Convert mark-up on cost to margin on transfer price

Margin on transfer price = 20%

Since closing inventory is valued at transfer price, the profit element must be calculated using the margin on transfer price, where Margin = Mark-up / (1 + Mark-up) = 0.25 / 1.25 = 0.20.

2

Calculate closing provision for unrealized profit

Closing Provision = ₦5,000

Multiply the transfer price of closing inventory (₦25,000) by the profit margin rate (20%) to find the profit fraction contained in unsold inventory.

Key Concept

Provision for Unrealized Profit on Closing Inventory

Question 1343Question →

A commercial printing firm acquires heavy-duty industrial printing equipment under a hire purchase agreement. The cash price of the machinery is $₦4,500,000$ . The contract mandates an initial down payment of $20\%$ of the cash price, with the remaining balance paid in $18$ equal monthly installments of $₦225,000$ each. What is the total interest charge paid on this transaction in Naira?

Show answer & explanation

Answer: 450000

Answer

The total interest charged on the hire purchase transaction is ��450,000.

The total interest (or carrying charge) in a hire purchase transaction represents the difference between the total hire purchase price (deposit plus all installment payments) and the original cash price. Here, the initial deposit is 20% of ₦4,500,000, which equals ₦900,000. Total installment payments over 18 months amount to 18 × ₦225,000 = ₦4,050,000. Thus, the total hire purchase price is ₦900,000 + ₦4,050,000 = ₦4,950,000. Subtracting the cash price of ₦4,500,000 yields a total interest charge of ₦450,000.

Step-by-Step Solution

1

Calculate the initial deposit amount

₦900,000

The deposit is specified as 20% of the cash price of ₦4,500,000.

2

Calculate total monthly installment payments

₦4,050,000

18 monthly installments of ₦225,000 each equal 18 * 225,000 = ₦4,050,000.

3

Calculate the total hire purchase price

₦4,950,000

Hire Purchase Price equals Initial Deposit + Total Installments (₦900,000 + ₦4,050,000).

4

Deduct cash price from total hire purchase price to determine interest

₦450,000

Interest Charge = Hire Purchase Price - Cash Price (₦4,950,000 - ₦4,500,000).

Key Concept

Hire Purchase Price and Carrying/Interest Charge Calculation

Question 1344Question →

Vanguard Manufacturing Company extracted the following closing inventory balances at the end of its accounting period:

Inventory Category	Amount (₦)
Raw materials	65,000
Work-in-progress	35,000
Finished goods (at transfer value)	100,000

Finished goods are transferred from the factory to the trading account at cost plus a $25\%$ mark-up. What is the total carrying value of inventories (in ₦) to be presented under current assets in the statement of financial position?

Show answer & explanation

Answer: 180000

Answer

The total carrying value of inventories to be presented under current assets in the statement of financial position is ₦180,000.

The total carrying amount of inventories presented under current assets is ₦180,000. This is calculated by eliminating the ₦20,000 unrealized profit provision (25/125 × ₦100,000) from finished goods to reduce it to its cost of ₦80,000, then adding raw materials (₦65,000) and work-in-progress (₦35,000).

Step-by-Step Solution

1

Calculate the provision for unrealized profit included in the closing finished goods inventory.

Provision for unrealized profit = ₦20,000.

Since finished goods are transferred at cost plus a 25% mark-up, the unrealized profit margin contained in the transfer price is 25/125 (or 20%).

2

Deduct the provision for unrealized profit from the finished goods inventory at transfer value.

Net carrying value of finished goods = ₦80,000.

Inventories must be stated at original prime/factory cost on the balance sheet by removing the internal unrealized manufacturing profit.

3

Sum all inventory components (raw materials, work-in-progress, and net finished goods).

Total inventory carrying amount = ₦180,000.

All three inventory elements are aggregated and presented as a single total or sub-itemized under current assets.

Key Concept

Balance sheet presentation of manufacturing inventories requires deducting the provision for unrealized profit from closing finished goods to ensure all inventories are valued at actual cost under current assets.

Question 1345Question →

The Earth completes a full rotation of $360^\circ$ on its axis every 24 hours. How many degrees of longitude does the Earth rotate through in 3 hours?

Show answer & explanation

Answer: 45

Answer

45 degrees

Because the Earth completes one full

360^\circ

rotation in 24 hours, its speed of rotation is

15^\circ

per hour (

360^\circ \div 24

). In a time span of 3 hours, it turns through

15^\circ \times 3 = 45^\circ

of longitude.

Step-by-Step Solution

1

Calculate the rotation rate of the Earth per hour.

15 degrees per hour

The Earth turns 360 degrees in 24 hours, so 360 divided by 24 equals 15 degrees per hour.

2

Calculate the total longitudinal rotation in 3 hours.

45 degrees

Multiplying the rate of 15 degrees per hour by 3 hours gives 45 degrees.

Key Concept

Earth's Rotational Angular Speed and Longitude

Estimated Time:45s

Question 1346Question →

Two weather observation stations, Station A and Station B, record simultaneous solar observations. Station A is located at longitude $12^\circ\text{W}$ where the local solar time is $02:00\text{ PM}$ . At the exact same moment, the local solar time at Station B is $07:00\text{ PM}$ . What is the longitude of Station B in degrees East ( $^\circ\text{E}$ )?

Show answer & explanation

Answer: 63

Answer

The longitude of Station B is

63^\circ\text{E}

.

The local time difference between Station A (

02:00\text{ PM}

) and Station B (

07:00\text{ PM}

) is

5\text{ hours}

. Since

1\text{ hour}

equals

15^\circ

of longitude, a

5\text{-hour}

difference equals

75^\circ

of angular distance (

5 \times 15^\circ

). Station B's local time is ahead of Station A, indicating that Station B is located further East. Starting from

12^\circ\text{W}

and moving

75^\circ

East,

12^\circ

is required to reach

0^\circ

(the Prime Meridian), leaving

63^\circ

into the Eastern Hemisphere. Therefore, Station B is located at

63^\circ\text{E}

.

Step-by-Step Solution

1

Calculate the time difference between Station A and Station B.

Station B (

19:00

) is

5\text{ hours}

ahead of Station A (

14:00

).

Comparing local solar times gives the time separation between the two longitudes.

2

Convert the time difference into angular distance in degrees.

5\text{ hours} \times 15^\circ/\text{hour} = 75^\circ

angular distance.

The Earth rotates

360^\circ

in 24 hours, which corresponds to

15^\circ

of longitude per hour.

3

Determine the longitude of Station B from Station A across the Prime Meridian.

75^\circ\text{ East} - 12^\circ\text{W} = 63^\circ\text{E}

.

Station B is later in solar time, so it lies East of Station A. Moving East from

12^\circ\text{W}

uses

12^\circ

to reach

0^\circ

(Prime Meridian) and the remaining

63^\circ

enters the Eastern Hemisphere.

Key Concept

Calculating longitude from local time difference across meridians

Estimated Time:1m 30s

Question 1347Question →

Two meteorological stations are situated along different meridians. Station Alpha is positioned at longitude $38^\circ\text{W}$ , while Station Beta is positioned at longitude $82^\circ\text{E}$ . What is the difference in local solar time between these two stations, in hours?

Show answer & explanation

Answer: 8

Answer

The difference in local solar time between the two stations is 8 hours.

Station Alpha is located in the Western Hemisphere (

38^\circ\text{W}

) and Station Beta is located in the Eastern Hemisphere (

82^\circ\text{E}

). The total angular distance separating them across the Greenwich Meridian is

38^\circ + 82^\circ = 120^\circ

. Dividing

120^\circ

by the Earth's rotation speed of

15^\circ

per hour gives a time difference of 8 hours.

Step-by-Step Solution

1

Calculate the total longitudinal angular distance between

38^\circ\text{W}

and

82^\circ\text{E}

.

Angular distance =

38^\circ + 82^\circ = 120^\circ

.

Because the two locations lie in opposite hemispheres relative to the Prime Meridian (

0^\circ

), their longitudinal degree values must be added together.

2

Convert the total angular distance to time difference in hours.

Time difference =

120^\circ \div 15^\circ\text{/hour} = 8\text{ hours}

.

The Earth completes a

360^\circ

rotation in 24 hours, which corresponds to

15^\circ

per hour.

Key Concept

Calculation of longitudinal difference across the Prime Meridian and conversion to local solar time difference.

Question 1348Question →

On a regional conservation map drawn to a Representative Fraction (R.F.) scale of $1 : 25,000$ , the distance between two forest ranger posts along a straight trail measures $16.4\text{ cm}$ . What is the actual ground distance between the two ranger posts in kilometers?

Show answer & explanation

Answer: 4.1

Answer

The actual ground distance between the two ranger posts is 4.1 kilometers.

The scale ratio 1 : 25,000 indicates that 1 centimeter on the map corresponds to 25,000 centimeters (or 0.25 kilometers) on the ground. Multiplying the measured map length of 16.4 centimeters by 0.25 kilometers per centimeter gives an actual ground distance of 4.1 kilometers.

Step-by-Step Solution

1

Determine ground distance in centimeters

16.4\text{ cm} \times 25,000 = 410,000\text{ cm}

Each unit on the map represents 25,000 identical units on the ground.

2

Convert centimeters to meters

410,000\text{ cm} \div 100 = 4,100\text{ m}

There are 100 centimeters in 1 meter.

3

Convert meters to kilometers

4,100\text{ m} \div 1,000 = 4.1\text{ km}

There are 1,000 meters in 1 kilometer.

Key Concept

Ground distance calculation using Representative Fraction (R.F.) map scale

Question 1349Question →

In a semi-urban local government area in Kano State, Nigeria, the total age dependency ratio is recorded as $80\%$ . If the population of children (aged $0-14$ years) is $90{}000$ and the population of elderly persons (aged $65$ years and above) is $18{}000$ , what is the total number of economically active individuals (aged $15-64$ years) in this local government area?

Show answer & explanation

Answer: 135000

Answer

The total number of economically active individuals (aged 15–64 years) is

135{}000

.

The total age dependency ratio is defined as the ratio of dependents (people aged

0-14

and

65+

) to the working-age population (aged

15-64

), multiplied by

100

. Summing the youth (

90{}000

) and elderly (

18{}000

) yields

108{}000

total dependents. Dividing

108{}000

by the dependency ratio (

0.80

) gives exactly

135{}000

economically active individuals.

Step-by-Step Solution

1

Calculate the total dependent population

108{}000

dependents

The dependent population consists of youth under 15 years (

90{}000

) and elderly individuals aged 65 years and over (

18{}000

).

2

Set up the age dependency ratio equation

80 = \frac{108{}000}{\text{Pop}_{15-64}} \times 100

The dependency ratio measures the relationship between the dependent cohort and the working-age cohort (

15-64

years).

3

Rearrange the equation to solve for the working-age population

\text{Pop}_{15-64} = \frac{108{}000 \times 100}{80} = 135{}000

Dividing the total dependent population by the dependency ratio expressed as a decimal (

0.80

) yields the total economically active population.

Key Concept

Age Dependency Ratio and Cohort Analysis

Estimated Time:2m 0s

Question 1350Question →

Chioma, a sole trader, started the financial year with an opening capital of $₦850,000$ . During the year ended 31 December 2025, she earned a net profit of $₦320,000$ and introduced additional capital of $₦100,000$ . She also withdrew $₦60,000$ in cash and goods costing $₦40,000$ (with a selling price of $₦55,000$ ) for personal use. What is Chioma's closing capital at the end of the financial year in Naira ( $₦$ )?

Show answer & explanation

Answer: 1170000

Answer

The closing capital at the end of the financial year is

₦1,170,000

.

The closing capital is calculated as

₦850,000 \text{ (opening capital)} + ₦100,000 \text{ (additional capital)} + ₦320,000 \text{ (net profit)} - ₦100,000 \text{ (total drawings at cost price)} = ₦1,170,000

. Goods withdrawn for personal use must strictly be recorded at their cost price (

₦40,000

), giving total drawings of

₦60,000 + ₦40,000 = ₦100,000

.

Step-by-Step Solution

1

Determine the correct valuation of drawings for goods withdrawn and calculate total drawings.

Total drawings =

₦60,000 \text{ (cash)} + ₦40,000 \text{ (cost price of goods)} = ₦100,000

.

Goods taken for personal use must always be valued at cost price, not selling price, to avoid incorrectly recognizing unearned profit.

2

Substitute all values into the owner's equity statement formula.

\text{Closing Capital} = ₦850,000 + ₦100,000 + ₦320,000 - ₦100,000 = ₦1,170,000$.

Opening capital, additional capital introduced, and net profit increase equity, whereas total drawings reduce equity.

Key Concept

Treatment of Owner's Capital, Cash Drawings, and Goods Withdrawn at Cost Price

Question 1351Question →

A cargo vessel positioned at longitude $45^\circ\text{W}$ reports a local solar time of $10:30\text{ AM}$ . At that precise moment, a coastal harbor records a local solar time of $03:30\text{ PM}$ . What is the longitude of the coastal harbor in degrees East ( $^\circ\text{E}$ )?

Show answer & explanation

Answer: 30

Answer

The longitude of the coastal harbor is 30 degrees East (30°E).

Because the harbor is experiencing a later local solar time (03:30 PM) than the vessel (10:30 AM), it must lie to the East of the vessel. The 5-hour time difference equates to an angular distance of 75°. Crossing from 45°W past the Greenwich Meridian (0°) leaves 30° in the Eastern Hemisphere, placing the harbor at 30°E.

Step-by-Step Solution

1

Calculate time difference

5 hours

The difference between 10:30 AM and 03:30 PM (15:30) is 5 hours.

2

Convert time difference to longitudinal angular distance

75 degrees

Earth rotates 15 degrees per hour, so 5 hours corresponds to 5 × 15° = 75°.

3

Calculate position relative to Prime Meridian

30 degrees East

Moving East from 45°W by 75° requires traversing 45° to reach 0° (Greenwich Meridian), and the remaining 30° extends into the Eastern Hemisphere.

Key Concept

Longitude and Time Calculations across Hemispheres

Question 1352Question →

A Head Office in Lagos supplies goods to its Port Harcourt branch at an invoice price that includes a mark-up of $33\frac{1}{3}\%$ on cost. At the end of the accounting period, the physical inventory count at the branch shows closing stock valued at $\text{₦}24,000$ at invoice price. What is the value of the stock reserve in Naira required to eliminate the unrealized profit from the closing stock?

Show answer & explanation

Answer: 6000

Answer

The stock reserve required to reduce the closing inventory to cost price is ₦6,000.

To determine the stock reserve on closing stock priced at invoice price, convert the mark-up on cost (33 1/3% or 1/3) to profit margin on selling price, which is 1/4 (25%). Multiplying 25% by the closing inventory value of ₦24,000 gives ₦6,000.

Step-by-Step Solution

1

Convert mark-up on cost to profit margin on selling/invoice price

Profit margin on invoice price = 25% (or 1/4)

Margin = Mark-up / (100% + Mark-up) = (1/3) / (4/3) = 1/4

2

Calculate unrealized profit (stock reserve) on closing inventory

Stock reserve = ₦6,000

Stock reserve = 1/4 of ₦24,000 = ₦6,000

Key Concept

Calculation of Branch Stock Reserve on Closing Inventory at Invoice Price

Question 1353Question →

On 31st December 2025, the Kano Branch Current Account in the Head Office ledger of Zenith Commercial Ventures showed a debit balance of $\text{₦}250,000$ . On the same date, the Head Office Current Account in the Kano Branch ledger showed a credit balance of $\text{₦}195,000$ .

Upon reconciliation, the following timing differences were discovered:
1. Goods worth $\text{₦}35,000$ dispatched by the Head Office to Kano Branch on 28th December 2025 were not received by the branch until 4th January 2026.
2. Cash remittance of $\text{₦}20,000$ sent by Kano Branch to the Head Office on 30th December 2025 was received by the Head Office on 5th January 2026.

What is the true reconciled balance of the inter-entity current account as at 31st December 2025?

Show answer & explanation

Answer: 230000

Answer

The true reconciled inter-entity current account balance as at 31st December 2025 is

\text{₦}230,000

.

To calculate the true reconciled balance, adjust both books for unrecorded timing differences. The Head Office ledger shows a debit balance of

\text{₦}250,000

; crediting this account for cash in transit of

\text{₦}20,000

yields

\text{₦}230,000

. Simultaneously, the Branch ledger shows a credit balance of

\text{₦}195,000

; crediting this account for goods in transit of

\text{₦}35,000

yields

\text{₦}230,000

. Both sets of books reconcile to

\text{₦}230,000

.

Step-by-Step Solution

1

Identify the unadjusted ledger balances in both books

Head Office ledger (Branch Current Account): Debit balance of

\text{₦}250,000

. Branch ledger (Head Office Current Account): Credit balance of

\text{₦}195,000

.

Before reconciliation, timing differences cause discrepancies between inter-entity balances.

2

Adjust the Head Office books for Cash in Transit

Deduct Cash in Transit (

\text{₦}20,000

) from the Branch Current Account debit balance:

\text{₦}250,000 - \text{₦}20,000 = \text{₦}230,000

debit.

The Head Office has not yet credited the Branch Current Account for cash remitted by the branch.

3

Adjust the Branch books for Goods in Transit

Add Goods in Transit (

\text{₦}35,000

) to the Head Office Current Account credit balance:

\text{₦}195,000 + \text{₦}35,000 = \text{₦}230,000

credit.

The branch has not yet credited the Head Office Current Account for goods dispatched by the head office.

Key Concept

Head Office and Independent Branch Inter-Entity Current Account Reconciliation

Estimated Time:1m 30s

Question 1354Question →

In a community with a total population of $50,000$ , a local demographic survey recorded $1,250$ live births in one year. What is the Crude Birth Rate (CBR) per $1,000$ population for this community?

Show answer & explanation

Answer: 25

Answer

The Crude Birth Rate is 25 per 1,000 population.

The Crude Birth Rate is calculated by dividing total live births (

1,250

) by total population (

50,000

) and multiplying by

1,000

, resulting in

25

per

1,000

population.

Step-by-Step Solution

1

Identify the formula for calculating Crude Birth Rate (CBR).

\text{CBR} = \frac{\text{Total Live Births}}{\text{Total Population}} \times 1,000

.

Crude Birth Rate is defined as the number of live births per 1,000 people in a given population in a year.

2

Substitute the given values into the equation and solve.

\text{CBR} = \frac{1,250}{50,000} \times 1,000 = 25

.

Dividing

1,250

by

50,000

gives

0.025

, and multiplying by

1,000

yields

25

.

Key Concept

Crude Birth Rate Calculation

Question 1355Question →

A survey map of a proposed cocoa plantation in southwestern Nigeria is drawn to a scale of $1 : 50,000$ . If the main access road across the plantation measures $12\text{ cm}$ on the map, what is the actual ground distance of the road in kilometers?

Show answer & explanation

Answer: 6

Answer

The actual ground distance of the road is

6\text{ km}

.

To determine the actual ground distance, multiply the measured map distance (

12\text{ cm}

) by the scale denominator (

50,000

) to get

600,000\text{ cm}

. Converting this value into kilometers by dividing by

100,000

gives an actual ground distance of

6\text{ km}

.

Step-by-Step Solution

1

Calculate the ground distance in centimeters

12\text{ cm} \times 50,000 = 600,000\text{ cm}

A Representative Fraction of

1 : 50,000

means

1\text{ cm}

on the map corresponds to

50,000\text{ cm}

on the ground.

2

Convert centimeters to kilometers

600,000\text{ cm} \div 100,000 = 6\text{ km}

Since

1\text{ m} = 100\text{ cm}

and

1\text{ km} = 1,000\text{ m}

,

1\text{ km} = 100,000\text{ cm}

.

Key Concept

Distance calculation using Representative Fraction (R.F.) map scale

Question 1356Question →

An astronomical observatory located at longitude $64^\circ\text{E}$ records a cosmic ray pulse at $04:12\text{ PM}$ local solar time. Simultaneously, a ground monitoring station records the exact same event at $11:48\text{ AM}$ local solar time. What is the longitude of the ground monitoring station in degrees West ( $^\circ\text{W}$ )?

Show answer & explanation

Answer: 2

Answer

The longitude of the ground monitoring station is

2^\circ\text{W}

.

The time difference between

04:12\text{ PM}

(

16:12

) and

11:48\text{ AM}

(

11:48

) is

4\text{ hours and } 24\text{ minutes}

, which corresponds to an angular separation of

66^\circ

(

4 \times 15^\circ + 24/4^\circ

). Because the time at the ground station is earlier, it lies to the west of

64^\circ\text{E}

. Subtracting

66^\circ

from

64^\circ\text{E}

crosses the

0^\circ

Prime Meridian to reach

2^\circ\text{W}

.

Step-by-Step Solution

1

Calculate the difference in local solar time between the two stations.

Time difference =

16:12 - 11:48 = 4\text{ hours and } 24\text{ minutes}

.

Determining the time interval is the first step to finding angular separation.

2

Convert the total time difference into degrees of longitude.

(4 \times 15^\circ) + (24 \div 4^\circ) = 60^\circ + 6^\circ = 66^\circ

.

The Earth rotates

15^\circ

per hour (

1^\circ

every 4 minutes).

3

Determine the relative directional position of the ground station.

The ground station is

66^\circ

west of the observatory.

Earlier local solar time indicates a position further to the west.

4

Calculate the absolute longitude across the Prime Meridian.

66^\circ - 64^\circ\text{E} = 2^\circ\text{W}

.

Moving west from

64^\circ\text{E}

uses

64^\circ

to reach

0^\circ

, leaving

2^\circ

in the Western Hemisphere.

Key Concept

Calculating longitude position across hemispheres using local solar time differences.

Question 1357Question →

On a topographical map with a scale of $1 : 50,000$ , Point X lies on a hilltop at an elevation of $420\text{ m}$ , while Point Y sits near a stream bed at an elevation of $170\text{ m}$ . If the measured distance between Point X and Point Y on the map is $12.5\text{ cm}$ , calculate the slope gradient between the two points. Express your answer as the value of $N$ in the standard gradient ratio $1 : N$ (or $1 \text{ in } N$ ). What is the value of $N$ ?

Show answer & explanation

Answer: 25

Answer

The denominator N in the gradient ratio 1 : N is 25.

The vertical interval (V.I.) between the hilltop at

420\text{ m}

and the stream at

170\text{ m}

is

250\text{ m}

. The horizontal equivalent (H.E.) on the ground is calculated by multiplying the map distance of

12.5\text{ cm}

by the map scale factor of

50,000

, which gives

625,000\text{ cm}

(

6,250\text{ m}

). Dividing V.I. by H.E. yields

250 / 6,250 = 1 / 25

. Thus, the gradient ratio is

1 : 25

, making

N = 25

.

Step-by-Step Solution

1

Calculate Vertical Interval (V.I.)

V.I. = 420 m - 170 m = 250 m

Vertical interval is the difference in height between the highest and lowest points.

2

Convert map distance to ground distance to determine Horizontal Equivalent (H.E.) in meters

H.E. = 12.5 cm × 50,000 = 625,000 cm = 6,250 m

Horizontal equivalent must be in the same units as the vertical interval before computing ratio.

3

Divide V.I. by H.E. to express the gradient as a ratio 1 : N

Gradient = 250 / 6,250 = 1 / 25

Simplifying the fraction V.I. / H.E. yields 1 / 25, giving N = 25.

Key Concept

Slope and Gradient Calculation

Question 1358Question →

A forest reserve covers an area of $18\text{ cm}^2$ on a topographical map drawn to a scale of $1 : 150,000$ . If the map is enlarged so that a statement scale of $1\text{ cm}$ to $0.5\text{ km}$ is used for the new map, what is the area of the forest reserve on the enlarged map in $\text{cm}^2$ ?

Show answer & explanation

Answer: 162

Answer

The area of the forest reserve on the enlarged map is

162\text{ cm}^2

.

The original scale

1 : 150,000

is enlarged to

1 : 50,000

(

1\text{ cm}

to

0.5\text{ km}

), giving a linear enlargement factor of

n = 150,000 / 50,000 = 3

. Since area changes by the square of the linear scale multiplier (

n^2 = 3^2 = 9

), the new area on the map is

18\text{ cm}^2 \times 9 = 162\text{ cm}^2

.

Step-by-Step Solution

1

Convert the new statement scale to a Representative Fraction (R.F.)

New scale R.F. is

1 : 50,000

Both scales must be in the same format to compare denominators directly (

0.5\text{ km} = 50,000\text{ cm}

).

2

Calculate the linear enlargement factor (

n

)

n = \frac{150,000}{50,000} = 3

The linear enlargement factor is found by dividing the original scale denominator by the new scale denominator.

3

Calculate the area enlargement factor (

n^2

)

Area enlargement factor

= 3^2 = 9

Area changes by the square of the linear scale factor.

4

Calculate the area on the enlarged map

18\text{ cm}^2 \times 9 = 162\text{ cm}^2

Multiplying the original area on the map by the area enlargement factor yields the new map area.

Key Concept

Relationship between linear scale factor and area scale factor in map enlargement

Question 1359Question →

A meteorological vessel measures its local solar time to be $3:20\text{ p.m.}$ when Greenwich Mean Time ( $0^\circ$ ) is $11:00\text{ a.m.}$ on the same day. What is the longitude of the vessel in degrees East?

Show answer & explanation

Answer: 65

Answer

The longitude of the vessel is

65^\circ\text{E}

.

The time difference between

3:20\text{ p.m.}

and

11:00\text{ a.m.}

is 4 hours and 20 minutes (

260\text{ minutes}

). Since the Earth rotates

1^\circ

every 4 minutes,

260\text{ minutes} \div 4 = 65^\circ

. Because the vessel's local solar time is ahead of Greenwich Mean Time, the vessel is located east of the Prime Meridian, yielding

65^\circ\text{E}

.

Step-by-Step Solution

1

Calculate time difference

4 hours 20 minutes (260 minutes)

Subtract GMT (11:00 a.m.) from local solar time (3:20 p.m.).

2

Convert time difference to longitude degrees

65 degrees

Divide total time difference in minutes (260) by 4 minutes per degree of Earth's rotation.

3

Determine longitudinal hemisphere

East

Local time is ahead of GMT, which indicates a position to the east of the Prime Meridian.

Key Concept

Calculation of longitude from local time and Greenwich Mean Time (GMT) using Earth's rotation rate

Question 1360Question →

On a topographical map drawn to a scale of $1 : 25,000$ , Point P is situated at a trigonometric station with an elevation of $520\text{ m}$ , while Point Q lies at a stream confluence at an elevation of $370\text{ m}$ . The measured straight-line distance between Point P and Point Q on the map is $15\text{ cm}$ . What is the slope gradient between Point P and Point Q, expressed as the denominator $N$ in the ratio $1 \text{ in } N$ ?

Show answer & explanation

Answer: 25

Answer

The denominator

N

of the slope gradient ratio (

1 \text{ in } N

) is 25.

The Vertical Interval (VI) between Point P and Point Q is

520\text{ m} - 370\text{ m} = 150\text{ m}

. Using the map scale of

1 : 25,000

, the Horizontal Equivalent (HE) is

15\text{ cm} \times 25,000 = 375,000\text{ cm} = 3,750\text{ m}

. Dividing the VI by the HE gives

\frac{150}{3,750} = \frac{1}{25}

. Therefore, the gradient expressed as a ratio is

1 \text{ in } 25

, and the denominator

N

is 25.

Step-by-Step Solution

1

Determine the Vertical Interval (VI)

\text{VI} = 520\text{ m} - 370\text{ m} = 150\text{ m}

Vertical Interval is the difference in elevation between the higher and lower points.

2

Calculate the Horizontal Equivalent (HE) in ground units (meters)

\text{HE} = 15\text{ cm} \times 25,000 = 375,000\text{ cm} = 3,750\text{ m}

Horizontal Equivalent is obtained by converting the measured map distance to actual ground distance using the representative fraction scale.

3

Compute the Gradient Ratio

\text{Gradient} = \frac{\text{VI}}{\text{HE}} = \frac{150\text{ m}}{3,750\text{ m}} = \frac{1}{25}

Gradient is the ratio of Vertical Interval to Horizontal Equivalent, both expressed in identical units.

Key Concept

Slope and Gradient Calculation using Representative Fraction Map Scale

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