Map Scale Conversion and Distance Calculation Practice Questions — JAMB UTME

Question 1Question →

A map is drawn to a Representative Fraction (RF) scale of $1 : 200,000$ . What is this scale expressed as a statement scale?

1\text{ cm represents } 2\text{ km}

2\text{ cm represents } 1\text{ km}

1\text{ cm represents } 4\text{ km}

1\text{ cm represents } 20\text{ km}

Show answer & explanation

Answer: $1\text{ cm represents } 2\text{ km}$

Answer

The statement scale is

1\text{ cm represents } 2\text{ km}

.

An RF of

1 : 200,000

means

1\text{ cm}

on the map equals

200,000\text{ cm}

on the ground. Dividing

200,000\text{ cm}

by

100,000

converts the ground distance to

2\text{ km}

, yielding the statement scale of

1\text{ cm represents } 2\text{ km}

.

Step-by-Step Solution

1

Understand the Representative Fraction (RF)

1\text{ cm}

on the map represents

200,000\text{ cm}

on the actual ground.

An RF scale of

1 : 200,000

indicates a 1-to-200,000 ratio in identical measurement units.

2

Convert ground centimeters to meters

\frac{200,000\text{ cm}}{100} = 2,000\text{ m}

There are

100\text{ cm}

in

1\text{ m}

.

3

Convert ground meters to kilometers

\frac{2,000\text{ m}}{1,000} = 2\text{ km}

There are

1,000\text{ m}

in

1\text{ km}

.

Key Concept

Converting Representative Fraction (RF) to Statement Scale

Estimated Time:45s

Question 2Question →

On a topographical map drawn to a Representative Fraction (RF) scale of $1 : 80,000$ , the measured distance along a proposed drainage canal between two agricultural communities is $17.5\text{ cm}$ . What is the actual ground distance of the canal in kilometers?

14.0\text{ km}

140.0\text{ km}

1,400.0\text{ km}

7.0\text{ km}

Show answer & explanation

Answer: $14.0\text{ km}$

Answer

The actual ground distance of the drainage canal is

14.0\text{ km}

.

The option showing

14.0\text{ km}

is correct because multiplying the map distance (

17.5\text{ cm}

) by the RF scale factor (

80,000

) gives a ground measurement of

1,400,000\text{ cm}

. Converting to kilometers by dividing by

100,000\text{ cm/km}

yields exactly

14.0\text{ km}

.

Step-by-Step Solution

1

Identify the given map scale and distance

Representative Fraction scale =

1 : 80,000

; Map distance =

17.5\text{ cm}

.

Establishing known values is essential for scale conversion.

2

Calculate the ground distance in centimeters

Ground distance in cm =

17.5\text{ cm} \times 80,000 = 1,400,000\text{ cm}

.

The RF denominator indicates that

1\text{ cm}

on the map represents

80,000\text{ cm}

on the ground.

3

Convert the ground distance from centimeters to kilometers

Ground distance in km =

\frac{1,400,000\text{ cm}}{100,000\text{ cm/km}} = 14.0\text{ km}

.

Since

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

, dividing by

100,000

converts centimeters to kilometers.

Key Concept

Ground Distance Calculation using Representative Fraction (RF)

Estimated Time:1m 30s

Question 3Question →

On a topographic map drawn to a Representative Fraction (RF) scale of $1 : 60,000$ , a linear segment of a river between two hydrological gauging stations measures $22.5\text{ cm}$ . What is the actual ground distance between the two stations in kilometers?

Show answer & explanation

Answer: 13.5

Answer

The actual ground distance between the two stations is

13.5\text{ km}

.

The ground distance is obtained by scaling up the map distance using the Representative Fraction:

22.5\text{ cm} \times 60,000 = 1,350,000\text{ cm}

. Converting to kilometers (

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

) yields

13.5\text{ km}

.

Step-by-Step Solution

1

Calculate the ground distance in centimeters using the Representative Fraction

Ground distance in cm =

22.5\text{ cm} \times 60,000 = 1,350,000\text{ cm}

The RF scale of

1 : 60,000

indicates that

1\text{ cm}

on the map represents

60,000\text{ cm}

on the ground.

2

Convert the ground distance from centimeters to kilometers

Ground distance in km =

\frac{1,350,000\text{ cm}}{100,000\text{ cm/km}} = 13.5\text{ km}

There are

100,000\text{ cm}

in

1\text{ km}

(

100\text{ cm/m} \times 1,000\text{ m/km}

).

Key Concept

Map Scale Conversion and Ground Distance Calculation

Estimated Time:1m 30s

Question 4Question →

A forest reserve occupies an area of $5\text{ cm}^2$ on a topographical map drawn to a scale of $1 : 100,000$ . If the map is enlarged to a new scale of $1 : 50,000$ , what is the area of the forest reserve on the new map?

10\text{ cm}^2

20\text{ cm}^2

2.5\text{ cm}^2

1.25\text{ cm}^2

Show answer & explanation

Answer: $20\text{ cm}^2$

Answer

The area of the forest reserve on the new enlarged map is

20\text{ cm}^2

.

When a map scale is changed from

1 : 100,000

to

1 : 50,000

, the linear dimensions are enlarged by a factor of

\frac{100,000}{50,000} = 2

. Because area is measured in two dimensions, the area scale factor is the square of the linear scale factor (

2^2 = 4

). Therefore, the new area on the map is

5\text{ cm}^2 \times 4 = 20\text{ cm}^2

.

Step-by-Step Solution

1

Determine the linear scale enlargement factor

Linear Factor =

\frac{\text{Old Scale Denominator}}{\text{New Scale Denominator}} = \frac{100,000}{50,000} = 2

Enlarging from

1 : 100,000

to

1 : 50,000

doubles the linear dimensions on the map.

2

Calculate the area scale factor

Area Factor =

(\text{Linear Factor})^2 = 2^2 = 4

Areal change on a map is proportional to the square of the linear scale change.

3

Compute the new area on the enlarged map

New Area =

\text{Original Area} \times \text{Area Factor} = 5\text{ cm}^2 \times 4 = 20\text{ cm}^2

Multiplying the initial map area by the area scale factor yields the new map area.

Key Concept

Relationship between Linear Scale and Area Scale in Map Enlargement

Estimated Time:1m 30s

Question 5Question →

A topographical map has a statement scale of $1\text{ cm to } 2.5\text{ km}$ . What is the Representative Fraction (RF) of this map?

1 : 250,000

1 : 25,000

1 : 2,500

250,000 : 1

Show answer & explanation

Answer: $1 : 250,000$

Answer

1 : 250,000

To convert a statement scale of

1\text{ cm to } 2.5\text{ km}

to a Representative Fraction (RF), both measurements must be brought to the same unit (centimeters). Since

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

, multiplying

2.5\text{ km}

by

100,000

yields

250,000\text{ cm}

. Therefore,

1\text{ cm}

on the map represents

250,000\text{ cm}

on the ground, giving an RF of

1 : 250,000

.

Step-by-Step Solution

1

Identify the given statement scale

The statement scale is

1\text{ cm to } 2.5\text{ km}

.

Converting to a Representative Fraction requires expressing both map distance and ground distance in identical units.

2

Convert ground distance from kilometers to centimeters

2.5\text{ km} = 2.5 \times 100,000\text{ cm} = 250,000\text{ cm}

.

There are

100,000\text{ cm}

in

1\text{ km}

(

1,000\text{ m} \times 100\text{ cm/m}

).

3

Formulate the Representative Fraction

The ratio is

\frac{1\text{ cm}}{250,000\text{ cm}} = 1 : 250,000

.

Representative Fraction is expressed as a ratio

1 : n

without unit labels.

Key Concept

Statement Scale to Representative Fraction Conversion

Question 6Question →

On a topographical map drawn to a scale of $1 : 50,000$ , the straight-line distance between Town X and Town Y is measured as $8\text{ cm}$ . What is the actual ground distance between the two towns in kilometers?

Show answer & explanation

Answer: 4

Answer

The actual ground distance between Town X and Town Y is

4\text{ km}

.

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

8\text{ cm}

) by the representative scale ratio denominator (

50,000

), giving

400,000\text{ cm}

. Dividing this by

100,000

converts the distance into kilometers, yielding

4\text{ km}

.

Step-by-Step Solution

1

Calculate the ground distance in centimeters using the map scale denominator.

Ground distance in cm =

8\text{ cm} \times 50,000 = 400,000\text{ cm}

.

A Representative Fraction of

1 : 50,000

indicates that

1\text{ unit}

on the map represents

50,000\text{ units}

of the same measurement on the actual ground.

2

Convert centimeters into kilometers.

Ground distance in km =

\frac{400,000}{100,000} = 4\text{ km}

.

Since

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

and

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

,

1\text{ km}

contains

100,000\text{ cm}

.

Key Concept

Ground distance calculation using Representative Fraction (RF) scale

Estimated Time:45s

Question 7Question →

On a topographical map drawn to a scale of $1 : 50,000$ , the straight-line distance between two settlements is measured as $14\text{ cm}$ . What is the actual ground distance between the two settlements in kilometers?

7.0\text{ km}

3.5\text{ km}

70.0\text{ km}

28.0\text{ km}

Show answer & explanation

Answer: $7.0\text{ km}$

Answer

The actual ground distance between the two settlements is

7.0\text{ km}

.

The scale

1 : 50,000

means that

1\text{ cm}

on the map represents

50,000\text{ cm}

(or

0.5\text{ km}

) on the ground. Therefore, a distance of

14\text{ cm}

on the map corresponds to

14 \times 0.5\text{ km} = 7.0\text{ km}

on the ground.

Step-by-Step Solution

1

Identify the scale ratio and map distance

Map scale

= 1 : 50,000

; Map distance

= 14\text{ cm}

.

Establishes the given measurements needed for calculation.

2

Convert the scale denominator from centimeters to kilometers

50,000\text{ cm} = \frac{50,000}{100,000}\text{ km} = 0.5\text{ km}

.

Since

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

,

1\text{ cm}

on the map represents

0.5\text{ km}

on the ground.

3

Calculate the ground distance

Ground distance

= 14\text{ cm} \times 0.5\text{ km/cm} = 7.0\text{ km}

.

Multiplying map distance by the ground distance per centimeter yields total real-world distance.

Key Concept

Map Scale and Distance Calculation

Question 8Question →

A regional planning map with a statement scale of $2\text{ cm to } 5\text{ km}$ is enlarged so that its linear dimensions are increased by a factor of $2.5$ . If the distance between two agricultural centers on the enlarged map measures $16\text{ cm}$ , what is the actual ground distance between them?

6.4\text{ km}

16.0\text{ km}

40.0\text{ km}

100.0\text{ km}

Show answer & explanation

Answer: $16.0\text{ km}$

Answer

The actual ground distance between the two agricultural centers is

16.0\text{ km}

.

The original statement scale of

2\text{ cm to } 5\text{ km}

simplifies to

1\text{ cm to } 2.5\text{ km}

. Enlarging the map's linear dimensions by a factor of

2.5

means that details become

2.5

times larger, so

1\text{ cm}

on the new map represents

\frac{2.5\text{ km}}{2.5} = 1.0\text{ km}

. Therefore, a measured distance of

16\text{ cm}

on the enlarged map corresponds to

16\text{ cm} \times 1.0\text{ km/cm} = 16.0\text{ km}

on the ground.

Step-by-Step Solution

1

Determine the unit scale of the original map.

Since

2\text{ cm}

represents

5\text{ km}

,

1\text{ cm}

represents

\frac{5\text{ km}}{2} = 2.5\text{ km}

.

Establishing the distance represented by

1\text{ cm}

on the original map provides the baseline linear scale.

2

Calculate the new linear scale after map enlargement.

The new scale denominator is

\frac{2.5\text{ km}}{2.5} = 1.0\text{ km}

per centimeter (

1\text{ cm to } 1\text{ km}

).

Enlarging a map by a linear factor of

2.5

means each centimeter on the new map represents a smaller ground distance by a factor of

2.5

.

3

Compute the actual ground distance using the enlarged map measurement.

\text{Ground Distance} = 16\text{ cm} \times 1.0\text{ km/cm} = 16.0\text{ km}.

Multiplying the measured distance on the enlarged map by the new ground equivalence yields the total actual distance.

Key Concept

Map Scale Conversion during Linear Enlargement

Estimated Time:2m 0s

Question 9Question →

On a topographical map with a Representative Fraction (RF) of $1 : 50,000$ , a straight coastline segment between two coastal lighthouses measures $14.4\text{ cm}$ . What is the actual ground distance between the two lighthouses in kilometers?

Show answer & explanation

Answer: 7.2

Answer

The actual ground distance between the two lighthouses is 7.2 km.

To find actual ground distance from a Representative Fraction (

1 : 50,000

), multiply the measured map distance (

14.4\text{ cm}

) by the scale denominator (

50,000

) to get

720,000\text{ cm}

. Converting to kilometers by dividing by

100,000

yields

7.2\text{ km}

.

Step-by-Step Solution

1

Calculate the ground distance in centimeters using the Representative Fraction denominator.

14.4\text{ cm} \times 50,000 = 720,000\text{ cm}

The scale ratio

1 : 50,000

means

1\text{ cm}

on the map represents

50,000\text{ cm}

on the ground.

2

Convert the ground distance from centimeters to meters.

720,000\text{ cm} \div 100 = 7,200\text{ m}

There are

100\text{ cm}

in

1\text{ meter}

.

3

Convert the ground distance from meters to kilometers.

7,200\text{ m} \div 1,000 = 7.2\text{ km}

There are

1,000\text{ meters}

in

1\text{ kilometer}

.

Key Concept

Converting map distance to actual ground distance using Representative Fraction (RF) scale

Estimated Time:1m 0s

Question 10Question →

A surveyor measures a straight stretch of road between two agricultural settlements on a topographical map drawn to a scale of $1 : 75,000$ . If the distance between the two settlements on the map is $16\text{ cm}$ , what is the actual ground distance in kilometers?

Show answer & explanation

Answer: 12

Answer

The actual ground distance between the two settlements is

12\text{ km}

.

A Representative Fraction scale of

1 : 75,000

means

1\text{ cm}

on the map represents

75,000\text{ cm}

on the ground. Converting

75,000\text{ cm}

into kilometres gives

\frac{75,000}{100,000} = 0.75\text{ km}

. Multiplying the measured map length of

16\text{ cm}

by

0.75\text{ km/cm}

yields an actual distance of

12\text{ km}

.

Step-by-Step Solution

1

Convert scale denominator from centimetres to kilometres

1\text{ cm on map} = 75,000\text{ cm on ground} = 0.75\text{ km}

Dividing centimetres by

100,000

(

100\text{ cm/m} \times 1,000\text{ m/km}

) converts centimetres to kilometres.

2

Calculate the actual ground distance

16\text{ cm} \times 0.75\text{ km/cm} = 12\text{ km}

Multiplying the measured distance on the map by the ground distance represented per centimetre gives the total ground distance.

Key Concept

Map Scale Conversion and Ground Distance Calculation

Question 11Question →

A topographical map with a Representative Fraction (RF) of $1 : 100,000$ undergoes map reduction such that its new surface area is $\frac{1}{4}$ of the original map area. If the straight-line distance between two towns measured on the reduced map is $15\text{ cm}$ , what is the actual ground distance between the two towns in kilometers?

7.5\text{ km}

15.0\text{ km}

30.0\text{ km}

60.0\text{ km}

Show answer & explanation

Answer: $30.0\text{ km}$

Answer

The actual ground distance between the two towns is

30.0\text{ km}

.

Reducing a map's area to

\frac{1}{4}

of its original size means its linear dimensions are reduced by a factor of

\sqrt{\frac{1}{4}} = \frac{1}{2}

. The new scale denominator becomes

100,000 \times 2 = 200,000

, resulting in a Representative Fraction of

1 : 200,000

. Multiplying the measured map distance of

15\text{ cm}

by

200,000

gives

3,000,000\text{ cm}

, which equals

30.0\text{ km}

on the ground.

Step-by-Step Solution

1

Determine the linear scale reduction factor from the areal reduction ratio

Linear reduction factor =

\sqrt{\frac{1}{4}} = \frac{1}{2}

Linear scale changes as the square root of surface area changes.

2

Calculate the Representative Fraction (RF) of the reduced map

New scale denominator =

100,000 \times 2 = 200,000

, giving a new scale of

1 : 200,000

Reducing linear dimensions by half doubles the scale denominator of the representative fraction.

3

Calculate the actual ground distance using the new map distance and new scale

Ground distance =

15\text{ cm} \times 200,000 = 3,000,000\text{ cm} = 30.0\text{ km}

Converting map distance in centimeters to ground distance in kilometers by dividing

3,000,000\text{ cm}

by

100,000\text{ cm/km}

.

Key Concept

Linear versus Areal Map Scale Transformation and Ground Distance Calculation

Estimated Time:2m 0s

Question 12Question →

On a topographical map drawn to a scale of $1 : 40,000$ , the measured length of a railway line is $17.5\text{ cm}$ . If the map is enlarged so that a statement scale of $1\text{ cm to } 250\text{ m}$ applies, what is the length of the railway line on the new enlarged map, in centimeters?

Show answer & explanation

Answer: 28

Answer

The length of the railway line on the enlarged map is

28\text{ cm}

.

To find the distance on the enlarged map, first calculate the actual ground distance by multiplying the original map distance (

17.5\text{ cm}

) by the scale denominator (

40,000

), yielding

700,000\text{ cm}

or

7,000\text{ m}

. Next, divide this ground distance by the ground distance represented per centimeter on the new map (

250\text{ m}

), which yields

28\text{ cm}

.

Step-by-Step Solution

1

Calculate the actual ground distance using the original map scale and measured distance.

Ground distance =

17.5\text{ cm} \times 40,000 = 700,000\text{ cm} = 7,000\text{ m}

.

Multiplying the map distance by the scale factor gives the true ground distance.

2

Calculate the corresponding distance on the enlarged map using the new statement scale.

New map distance =

\frac{7,000\text{ m}}{250\text{ m/cm}} = 28\text{ cm}

.

Dividing the total ground distance by the ground distance represented per centimeter on the new map yields the map measurement.

Key Concept

Ground distance calculation and map scale conversion during map enlargement.

Question 13Question →

A proposed power transmission line measures $14.4\text{ cm}$ on Map X, which is drawn to a Representative Fraction (RF) scale of $1 : 250,000$ . The same power line is subsequently transferred to Map Y, where its total path is measured across two contiguous sections: a straight stretch of $8.0\text{ cm}$ and a curved section of $10.0\text{ cm}$ . What is the denominator of the Representative Fraction (RF) scale of Map Y?

Show answer & explanation

Answer: 200000

Answer

The denominator of the Representative Fraction (RF) scale of Map Y is 200,000 (representing a scale of

1 : 200,000

).

Because ground distance remains constant, the actual distance is

14.4\text{ cm} \times 250,000 = 3,600,000\text{ cm}

. On Map Y, the total route length is

8.0\text{ cm} + 10.0\text{ cm} = 18.0\text{ cm}

. Dividing

3,600,000\text{ cm}

by

18.0\text{ cm}

gives a scale denominator of

200,000

, which corresponds to a scale of

1 : 200,000

.

Step-by-Step Solution

1

Calculate the actual ground distance using the length on Map X and its scale

Ground distance =

14.4\text{ cm} \times 250,000 = 3,600,000\text{ cm}

(or

36\text{ km}

)

Map scale defines the relationship between map distance and ground distance:

\text{Ground distance} = \text{Map distance} \times \text{Scale denominator}

.

2

Determine the total measured length of the route on Map Y

Total distance on Map Y =

8.0\text{ cm} + 10.0\text{ cm} = 18.0\text{ cm}

The entire feature consists of two contiguous sections that must be added together.

3

Calculate the RF scale denominator of Map Y

RF denominator =

\frac{3,600,000\text{ cm}}{18.0\text{ cm}} = 200,000

The scale denominator is the ratio of ground distance to map distance when both are expressed in the same unit.

Key Concept

Ground distance conservation across maps of differing scales and multi-segment scale denominator calculation.

Estimated Time:2m 30s

Question 14Question →

A proposed pipeline route between two water reservoirs measures $18\text{ cm}$ on a regional map drawn to a statement scale of $5\text{ cm to } 2\text{ km}$ . If this map is reduced to a Representative Fraction (RF) scale of $1 : 100,000$ , what will be the length of the pipeline route on the reduced map?

7.2\text{ cm}

45.0\text{ cm}

2.88\text{ cm}

0.72\text{ cm}

Show answer & explanation

Answer: $7.2\text{ cm}$

Answer

7.2\text{ cm}

The original statement scale of

5\text{ cm to } 2\text{ km}

means each centimeter on the map corresponds to

0.4\text{ km}

on the ground. An

18\text{ cm}

line therefore represents a total ground distance of

18 \times 0.4\text{ km} = 7.2\text{ km}

. On the new map with an RF scale of

1 : 100,000

,

1\text{ cm}

represents

100,000\text{ cm}

(or

1\text{ km}

). Converting the

7.2\text{ km}

ground distance onto the new map gives

\frac{7.2\text{ km}}{1\text{ km/cm}} = 7.2\text{ cm}

.

Step-by-Step Solution

1

Determine the ground distance represented by

1\text{ cm}

on the original map scale.

Since

5\text{ cm} = 2\text{ km}

,

1\text{ cm} = \frac{2\text{ km}}{5} = 0.4\text{ km}

(or

40,000\text{ cm}

, giving an RF scale of

1 : 40,000

).

Converting statement scale to ground distance per centimeter establishes the actual linear distance on Earth.

2

Calculate the actual ground distance of the pipeline route.

\text{Ground Distance} = 18\text{ cm} \times 0.4\text{ km/cm} = 7.2\text{ km}

(or

720,000\text{ cm}$).

Multiplying the map measurement by the ground equivalence per centimeter yields the true distance.

3

Calculate the new map length on the reduced map of scale

1 : 100,000

.

On an RF scale of

1 : 100,000

,

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

. Therefore, \text{New Map Length} = \frac{7.2\text{ km}}{1\text{ km/cm}} = 7.2\text{ cm}$.

Dividing the true ground distance by the new scale factor gives the updated map distance.

Key Concept

Map scale conversion between statement scale and Representative Fraction (RF), and linear distance scaling.

Question 15Question →

A topographical map displays a statement scale of $2\text{ cm}$ to $1\text{ km}$ . Which of the following represents this scale as a Representative Fraction (R.F.)?

50,000 : 1

1 : 50,000

1 : 2,500,000

1 : 500

Show answer & explanation

Answer: $1 : 50,000$

Answer

The Representative Fraction (R.F.) of the map is

1 : 50,000

.

The correct Representative Fraction is

1 : 50,000

. Converting

1\text{ km}

into centimeters yields

100,000\text{ cm}

. Expressing the scale as a ratio gives

2\text{ cm} : 100,000\text{ cm}

, which simplifies to

1 : 50,000

when both terms are divided by

2

.

Step-by-Step Solution

1

Convert the ground distance from kilometers to centimeters so both sides of the scale ratio share identical units.

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

.

Representative Fraction requires unitless comparison, so both map distance and ground distance must be in the same measurement units.

2

Express the relationship as a ratio of map distance to ground distance.

\text{Scale} = 2\text{ cm} : 100,000\text{ cm}

.

A Representative Fraction is written in the form

1 : n

, where

1

represents map distance and

n

represents ground distance.

3

Divide both terms of the ratio by the map distance value (

2

) to reduce the numerator/antecedent to

1

.

\frac{2}{2} : \frac{100,000}{2} = 1 : 50,000

.

Simplifying the antecedent to unity gives the final standard Representative Fraction format.

Key Concept

Conversion of Statement Scale to Representative Fraction (R.F.)

Estimated Time:45s

Question 16Question →

On a regional map (Map X), a straight stretch of a major highway between two towns measures $15\text{ cm}$ . Map X is drawn to a statement scale of $1\text{ cm to } 2\text{ km}$ . If Map X is reduced to produce Map Y such that Map Y has a Representative Fraction (RF) scale of $1 : 500,000$ , what is the measured length of the same highway on Map Y?

6.0\text{ cm}

37.5\text{ cm}

2.4\text{ cm}

60.0\text{ cm}

Show answer & explanation

Answer: $6.0\text{ cm}$

Answer

The measured length of the highway on Map Y is

6.0\text{ cm}

.

The ground distance between the two towns is

30\text{ km}

(

15\text{ cm} \times 2\text{ km/cm}

). On Map Y, where

1\text{ cm}

represents

500,000\text{ cm}

(

5\text{ km}

), the measured length is

\frac{30\text{ km}}{5\text{ km/cm}} = 6.0\text{ cm}

.

Step-by-Step Solution

1

Calculate the actual ground distance using Map X.

Ground Distance

= 15\text{ cm} \times 2\text{ km/cm} = 30\text{ km}

.

Map X's scale of

1\text{ cm to } 2\text{ km}

means each centimeter on the map represents

2\text{ km}

on the ground.

2

Convert the Representative Fraction (RF) scale of Map Y to a statement scale in kilometers.

Map Y Scale

= 1 : 500,000 \Rightarrow 1\text{ cm to } \frac{500,000}{100,000}\text{ km} = 1\text{ cm to } 5\text{ km}

.

There are

100,000\text{ cm}

in

1\text{ km}

, so dividing

500,000

by

100,000

converts the scale to kilometers.

3

Calculate the distance on Map Y.

Map Y Distance

= \frac{30\text{ km}}{5\text{ km/cm}} = 6.0\text{ cm}

.

Dividing the actual ground distance by the ground equivalent per centimeter of Map Y gives the distance on Map Y.

Key Concept

Map Scale Reduction and Ground Distance Calculation

Estimated Time:1m 30s

Question 17Question →

A straight road connecting a farm settlement to a regional market town measures $8.5\text{ cm}$ on a map with a scale of $1 : 20,000$ . What is the actual ground distance between the two locations in kilometers?

Show answer & explanation

Answer: 1.7

Answer

The actual ground distance between the farm settlement and the market town is

1.7\text{ km}

.

Multiplying

8.5\text{ cm}

by the scale ratio denominator (

20,000

) gives a real-world distance of

170,000\text{ cm}

. Converting to kilometers (

170,000 \div 100,000

) yields

1.7\text{ km}

.

Step-by-Step Solution

1

Calculate the ground distance in centimeters

170,000\text{ cm}

Ground distance equals map distance multiplied by scale denominator (

8.5\text{ cm} \times 20,000

).

2

Convert centimeters into kilometers

1.7\text{ km}

Since

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

, divide

170,000\text{ cm}

by

100,000

.

Key Concept

Map Scale Distance Calculation

Question 18Question →

A rectangular agricultural zone measures $6\text{ cm}$ by $4\text{ cm}$ on a topographical map drawn to a Representative Fraction (R.F.) scale of $1 : 100,000$ . If the map is enlarged so that the area of the agricultural zone on the new map becomes $96\text{ cm}^2$ , what is the Representative Fraction (R.F.) scale of the enlarged map?

1 : 50,000

1 : 25,000

1 : 200,000

1 : 400,000

Show answer & explanation

Answer: $1 : 50,000$

Answer

The Representative Fraction (R.F.) scale of the enlarged map is

1 : 50,000

.

The correct answer of

1 : 50,000

is derived by recognizing that the original map area of

24\text{ cm}^2

increases to

96\text{ cm}^2

, representing a four-fold areal enlargement. Because linear dimensions change as the square root of the areal change, the linear scale factor is

\sqrt{4} = 2

. Enlarging a map by a factor of 2 doubles linear dimensions on paper, making the scale larger by reducing the scale denominator from 100,000 to 50,000.

Step-by-Step Solution

1

Calculate the area of the agricultural zone on the original map.

Original Area

= 6\text{ cm} \times 4\text{ cm} = 24\text{ cm}^2

.

Knowing the original map area allows us to determine the area enlargement ratio.

2

Determine the area scale change ratio.

Area Ratio

= \frac{\text{New Area}}{\text{Original Area}} = \frac{96\text{ cm}^2}{24\text{ cm}^2} = 4

.

The area of the zone has been enlarged 4 times.

3

Calculate the linear scale enlargement factor.

Linear Factor

= \sqrt{\text{Area Ratio}} = \sqrt{4} = 2

.

Linear scale changes as the square root of areal scale changes.

4

Calculate the scale denominator for the enlarged map.

New Scale Denominator

= \frac{\text{Original Scale Denominator}}{\text{Linear Factor}} = \frac{100,000}{2} = 50,000

.

Enlarging a map increases its detail and linear dimensions, which reduces the scale denominator proportionally.

Key Concept

Map Enlargement and Linear vs Areal Scale Relationship

Question 19Question →

On a topographical map drawn to a Representative Fraction (RF) scale of $1 : 100,000$ , the measured distance between a railway station and a local airstrip is $7.5\text{ cm}$ . What is the actual ground distance between the two locations in kilometers?

Show answer & explanation

Answer: 7.5

Answer

The actual ground distance between the railway station and the airstrip is

7.5\text{ km}

.

Given an RF scale of

1 : 100,000

, every

1\text{ cm}

on the map corresponds to

100,000\text{ cm}

on the ground, which equals

1\text{ km}

. Multiplying the measured map length of

7.5\text{ cm}

by

1\text{ km}

per centimeter gives an actual ground distance of

7.5\text{ km}

.

Step-by-Step Solution

1

Interpret the Representative Fraction (RF) scale.

An RF scale of

1 : 100,000

means

1\text{ cm}

on the map represents

100,000\text{ cm}

on the ground.

RF scale expresses map distance to ground distance in identical units.

2

Convert the ground distance scale unit from centimeters to kilometers.

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

.

There are

100,000\text{ cm}

in

1\text{ km}

(

100\text{ cm/m} \times 1,000\text{ m/km}

).

3

Calculate the total actual ground distance.

\text{Ground Distance} = 7.5\text{ cm} \times 1\text{ km/cm} = 7.5\text{ km}$.

Multiplying the map length by the real-world distance represented per unit length determines the true distance.

Key Concept

Map Scale Conversion and Ground Distance Calculation

Question 20Question →

A proposed high-voltage transmission line connecting a hydro-electric station to an urban sub-station measures $14.5\text{ cm}$ on a topographical map drawn to a scale of $1 : 50,000$ . If the map undergoes a photographic reduction such that its total surface area becomes $\frac{1}{4}$ of its original size, what is the actual ground distance, in kilometers, represented by a length of $5.8\text{ cm}$ measured on the newly reduced map?

Show answer & explanation

Answer: 5.8

Answer

The actual ground distance represented by 5.8 cm on the reduced map is 5.8 kilometers.

Reducing the surface area of a map to one-fourth reduces its linear dimensions to one-half, because linear scale factor equals the square root of the area scale factor. With linear dimensions halved, each centimeter on the new map represents twice as much ground distance as before, changing the scale from 1 : 50,000 to 1 : 100,000. At a scale of 1 : 100,000, 1 cm represents 1 km. Therefore, 5.8 cm on the reduced map corresponds to an actual ground distance of 5.8 km.

Step-by-Step Solution

1

Determine the linear scale factor from the area reduction ratio

Linear factor = \sqrt{\frac{1}{4}} = \frac{1}{2}

Map area varies as the square of the linear scale factor. Reducing area to one-fourth reduces linear dimensions by half.

2

Calculate the Representative Fraction (R.F.) of the reduced map

New R.F. = 1 : (50,000 \times 2) = 1 : 100,000

Reducing the physical dimensions of a map by half doubles the denominator of its scale ratio, making it a smaller scale map.

3

Convert the new R.F. scale to kilometers per centimeter

1\text{ cm} = 100,000\text{ cm} = 1.0\text{ km}

Dividing the scale denominator (100,000) by 100,000 converts centimeters to kilometers.

4

Compute the ground distance for a map length of 5.8 cm

5.8\text{ cm} \times 1.0\text{ km/cm} = 5.8\text{ km}

Multiplying map distance by the ground equivalence per centimeter yields actual distance.

Key Concept

Linear vs Area Scale Conversion and Ground Distance Calculation

Map Scale Conversion and Distance Calculation

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept