Numerical Problems
6
Example 1 Convert the 50 m distance on ground to distance on map using a statement scale, where map scale is 1 cm: 10 m.
Solution Distance on the ground = 50 m Map scale: 1 cm represents 10 m or 1 cm: 10 m
We need to find out x, where x will be used to represent 50 m.
If we have x: 50, then x × 10 m = 1 × 50
So, ___ 10
10 x = 50___
10 Therefore, x = 5 cm
Hence, we use 5 cm on the map to represent 50 m on the ground.
Example 2 Convert the 60 km distance on ground to distance on map using a statement scale where map scale is 3 cm: 30 km.
Solution Distance on the ground = 60 km
Map scale: 3 cm represents 30 km or 3 cm: 30 km.
We need to find out x, where x will be used to represent 60 km.
If we have X: 60, then x × 30 km = 3 × 60 km
So, 1__
3 x = 60___
30 Therefore, x = 6 cm
Hence, we use 6 cm on the map to represent 60 km on the ground.
Do yourself
1. Distance 200 m; scale: 5 cm represents 20 m 2. Distance 210 m; scale: 1 km represents 20 m 3. Distance 150 m; scale: 2 cm represents 25 m Answers: (1) 5 cm (2) 10.5 cm (3) 12 cm
Example 3 Draw a graphical scale to represent 50 m distance on ground on map using map scale 1 cm: 1 km.
Solution Given map scale is 1 cm: 1 km. Hence, 1 cm on map represents 1 km.
A line is drawn and divided into sections of 1 cm. Each 1 cm represents 1 km. The values that each division represents are then added to the scale.
Therefore, the division line is marked 0, 1, 2, 3, 4, and so on, and km is written beside the numbers.
Example 4 Calculate the distance on ground represented by a linear scale in which 1 cm represents 60 m.
Solution Given map scale is 1 cm: 60 m. Hence, 1 cm on map represents 60 m.
A line is drawn and divided into sections of 1 cm. Each 1 cm represents 60 m. The values that each division represents are then added to the scale. Therefore, the division line is marked 60, 120, 180, 240, 360, and so on, and m is written beside the numbers.
Do yourself:
What distance is represented by each line, if it is drawn to scale stated?
1. Line 8.5 cm long, if 5 cm represents 10 km 2. Line 10.4 cm long, if 1 cm represents 2 km 3. Line 8 cm long, if 25 cm represents 5 m 4. Line 6.5 cm long, if 5 cm represents 50 m
Answers: (1) 170 km (2) 20.8 km (3) 160 m (4) 650 m
Example 5 If we have 4 cm representing 1 m, then find the length on the map.
Solution Distance on the ground = 4 cm: 1 m So in the same units,
4 cm____________
1 m × 100 cm = _______ 4 100 cm
But the numerator has to be one. Therefore, the RF scale is expressed as 1/25.
Do yourself:
Find out the representative fraction of the following scales.
1. 2 cm represents 50 m 2. 2 cm represents 1 km 3. 1 cm represents 1 km 4. 2 cm represents 10 km
Answers: (1) 1/25,000,000 (2) 1/50,000 (3) 1/1,000,000 (4) 1/5,000,000
Conversion from Verbal Scale to RF Scale
For converting a verbal scale to RF scale, an important criterion is that the unit of length on both sides of fraction must be the same and the numerator must be always 1.
Example 6 Convert the verbal scale “1 inch: 10 miles” to an RF scale.
Solution 1 inch = 10 miles 1 mile = 633,600 inches
Hence, 1 inch = 10 × 63,360 inches RF scale is 1: 63,360.
Example 7 Convert the verbal scale “2 inches: 3 miles” to an RF scale.
Solution 2 inches = 3 miles 1 inch = 1.5 miles
1 mile = 5280 feet 1 foot = 12 inches
Hence, 1 mile × 5280 feet________
1 mile × 12 inches_________
1 foot = 63,360 inches Therefore, 1 inch = 1.5 miles × 63.360 inches____________
1 mile = 90,040 inches RF scale is 1: 90,040.
Example 8 Convert the verbal scale “5 inches: 4 miles” to an RF scale.
Solution 5 inches = 4 miles 1 inch = 0.8 mile
1 mile = 5280 feet 1 foot = 12 inches
Hence, 1 mile × 5280 feet________
1 mile × 12 inches_________
1 foot = 63,360 inches Therefore, 1 inch = 0.8 mile × 63,360 inches____________
1 mile = 50,688 inches RF scale is 1: 50,688.
Conversion from RF Scale to Verbal Scale
Representative fraction to verbal conversions are much easier than vice versa since either unit of verbal scale might not be same or for scales that require same unit on both sides of the fraction, we have to convert only the denominator of the fraction into larger unit.
Example 9 Convert the RF scale 1: 25,000 to a verbal scale.
Solution 1 cm on map = 25,000 cm on ground We know that
100,000 cm = 1 km Therefore,
25,000 cm = 25,000/100,000 km
= 0.25 km
Hence, 1 cm on map is 0.25 km on ground.
Example 10 Convert the RF scale 1: 30,000 to a verbal scale.
Solution
1 mile ground × 63,360 inches____________
1 mile × 1 map
_____________
30,000 ground = 0.47 mile (approx.) Therefore, 1 inch equals 0.47 miles.
Example 11 On a map with an RF scale of 1: 100,000, you measure a length of 6.3 inches. How many miles of pipeline do this represent?
Solution Representative fraction representation for the given problem is
Inches on the map __________________
Miles on the ground = Inches on the map __________________
Miles on the ground or 1/100,000 = 6.3 inches/x
or x = 630,000 inches or x = 630,000/63,360 miles or x = 9.9 miles (approximately)
Example 12 On a map (Map A) having an RF scale of 1: 70,200, road A measures 4.5 inches. On a second map (Map B) of the same area, road B measures 8.5 inches. This map has a verbal scale of 1 inch: 0.25 miles. Which road shows the shortest route?
Solution For the given problem, we will first calculate for Map A.
Map A
1/70,200 = 4.5/x
x = 315,900 miles/63,360
or x = 4.9 miles (approximately) Map A
For Map B, we will first convert the verbal scale into RF scale. The verbal scale is 1 inch: 0.25 miles. Hence, equivalent RF scale will be
1 inch = 0.25 miles 1 mile = 5280 feet 1 foot = 12 inches
Hence, 1 mile × 5280 feet________
1 mile × 12 inches_________
1 foot = 63,360 inches Therefore, 1 inch = 0.25 mile × 63.360 inches____________
1 mile = 15,840 RF scale is 1: 15,840.
Now,
1/15840 = 8.5/x
or x = 134,640 miles/63,360 or x = 2.125 miles
Hence, road B is shorter.
Example 13 In a map of scale 1: 120,000, a path XY measures 2.7 inches.
On an image without a scale, the distance AB measures 7.5 inches. What is the RF scale for the image?
Solution To solve the above problem, first we will calculate the length of the path XY on ground, which will be same for all data sources. For the map,
Distance on map/distance on ground = Distance on map/distance on ground
or 1/120,000 = 2.7/x or x = 324,000
x is the distance on ground for the XY path, which is same for all data sources.
For the image, we have distance on map and distance on ground (calculated from map scale). We have to find out scale for image. For image, distance on map/distance on ground = distance on map/distance onground
or 1/x = 7.5/324,000 or 7.5x = 324,000 or x = 43,200
Hence, the RF scale will be 1: 43,200.
Example 14 A rectangular object measures 5 cm × 4 cm on a map. The map scale is 1: 48,000. What is the area of the object?
Solution The area of object on the map is 5 cm × 4 cm = 20 cm2 Distance on map/distance onground = Distance on map/distanceon ground
or 1/48,000 = 20 cm2/x
or x = 20 cm2/(1 × 48,000)2 × (1 m/100 cm)2 or x = 20 cm2 × 2,304,000,000 × 1 m2/10,000 or x = 4,608,000 m2
or x = 4,608,000/1,000,000 or x = 4.608 km2
Example 15 For the following figure, find the distance between city X and city Y on a map with scale 1: 60,000.
Solution For the given problem, the total distance between city X and city Y is 3 + 2.5 = 5.5 cm. Map scale is 1: 60,000. Hence, distance on map/distanceonground = distance on map/distance onground
or 1/60,000 = 5.5 cm/x
or x = 60,000 × 5.5 cm/100,000 or x = 33,000/100,000
or x = 0.33 km Do yourself:
1. Convert the verbal scale 4 inches: 6.5 miles to RF scale.
2. Convert the RF scale 1: 2,350,000 to verbal scale.
3. Find the scale factor for Map A with scale 1: 10,000 and Map B with scale 1: 20,000.
Notes:
1. Large scales show a small area with a lot of detail, for example, 1:
50 to 1: 10,000.
2. Medium scales show reasonable detail, for example, 1: 25,000 to 1:
100,000.
3. Small scales show a large area with minimal detail, for example, 1:
250,000 and smaller (1: 250,000 is a smaller scale than 1: 25,000).
4. Scale factor = (RF of map 1)/(RF of map 2)
Scale Miles/inch Line width
on ground* Example
1: 2,000,000 ~32.0 2000¢ USGS nation-wide maps
1: 1,000,000 ~16.0 1000¢ National and state maps
1: 500,000 ~8.0 500¢ State or regional maps
1: 250,000 ~4.0 250¢ US Army map series
1: 100,000 ~1.6 100¢ USGS 30 Quads
1: 62,500 5,208 feet 62.5¢ USGS 15 Quads
Contd...
1: 24,000 2,000 feet 24.0¢ USGS 7 Quads
1: 15,840 1,320 feet 15.85¢ Soil
1: 9,600 800 feet Aerial photos
Source <http://id.water.usgs.gov/reference/map_scales.html>