Solving Scale Problems

1
Solving Scale Problems

• The solution will gradually build for each
  problem

2
Some unit conversions

• 1 m 100 cm 1000 mm
• 1 km 1000 m
• 1 foot 12 inches (1 12)
• 1 mile 63,360 inches
• 1 hectare (ha) 10,000m2

3
The real world distance between 2 features is
1600m and the map distance is 8 cm.What is the
map scale ?

• Scale 1 / xxxxx map dist / real world
  distance
• If we know 3 of the figures, then we solve for
  the unknown by cross-multiply then divide
• What are we solving for?
• Map scale is unknown
• 1 / xxx map dist. / real world (RW) dist
• What do we know?
• Map dist 8 cm
• RW distance is 1,600 m 160,000 cm
• 1 / xxx 8 cm / 160,000 cm
• xxx (1160,000 cm)/8 cm
• xxx 20,000
• Scale is 1 20,000

1 Map dist.
xxxx Real world dist.
1 Map dist.
xxxx Real world dist.
1 8cm
xxxx 160,000cm
4
The distance between 2 features on a 110,000 map
is 5 cm. What is the real world distance?

• Scale 1 / xxxx map dist / real world distance
• If we know 3 of the figures, then we solve for
  the unknown by cross-multiply then divide
• What are we solving for?
• Real world (RW) distance
• 1 /xxxx map dist / RW dist
• What do we know?
• Scale of map (1/xxxx) is 1 / 10,000
• Map dist 5 cm
• 1 / 10,000 5 cm / RW dist. cm
• RW dist (cm) (510,000 cm)/1
• RW dist cm 50,000 cm
• In metres, RW dist 500 m

1 Map dist.
xxxx Real world dist.
1 Map dist.
xxxx Real world dist.
1 5 cm
10,000 Real world dist. cm
5
What is the length in cm of one km on a 125,000
map?

• Scale 1 / xxxxx map dist / real world
  distance
• If we know 3 of the figures, then we solve for
  the unknown by cross-multiply then divide
• What are we solving for?
• Length on map (in cm)
• 1 / x map dist. / real world (RW) dist
• What do we know?
• Scale of map (1/x) is 1 / 25,000
• 1 km RW dist (1 km 1,000 m 100,000 cm)
• 1 / 25,000 map dist cm / 100,000 cm
• Map dist (cm) (1100,000 cm)/25,000
• Map dist 4 cm

1 Map dist.
xxxx Real world dist.
1 Map dist.
xxxx Real world dist.
1 Map dist. cm
25,000 100,000cm
6
You have a photo with a scale expressed as 1
¼ mile. What is the scale as a representative
fraction?

• This is really a unit conversion question
• Scale 1 / xxx map / RW
• Scale of photo is 1 ¼ mile
• If you convert top and bottom values to the same
  units, then you have a representative fraction
  (since the units will cancel out)
• So the question really is how many inches in ¼
  mile?
• 1 mile 63,360 inches
• ¼ mile 63,360 inches / 4 15,840 inches
• Scale is 1 15,840 (NO need to show units
  anymore)

1 Map dist.
xxxx Real world dist.
1 1 inch
xxxx ¼ mile
1 1 in
xxxx 15,840 in
1 1
xxxx 15,840
7
A map scale is 1 cm 750 m and a photo covers a
portion of the map area.  The distance between
two road intersections measures 9.8 cm on the
photo and 5.0 cm on the map.  Determine the scale
of the photo.

• Photo scale 1/xxx photo/RW
• We know
• photo measure of 9.8 cm
• If we had the RW distance we could determine the
  photo scale.
• Can we find RW distance?
• Yes, as we have map scale and a map measure
• So first we find RW distance from map, then use
  it to solve for photo scale
• Map scale is
• 1 cm 750 m, or
• 1 cm 75,000 cm, or
• 1 75,000
• RW distance is
• RW 5.0 cm 75,000
• RW 375,000 cm

1 Photo
xxxx RW
1 9.8 cm
xxxx RW
1 5.0 cm
75,000 RW
8
A map scale is 1 cm 750 m and a photo covers a
portion of the map area.  The distance between
two road intersections measures 9.8 cm on the
photo and 5.0 cm on the map.  Determine the scale
of the photo.

• From last slide
• Photo scale is
• RW distance (from map measures) is
• RW 5.0 cm 75,000
• RW 375,000 cm
• Realize that RW distance is the same no matter if
  we calculated from a map or a photo the RW
  distance is the same
• Now we can solve for photo scale (xxxx)
• xxxx (1 375,000cm) / 9.8 cm
• xxxx 38,265
• Scale is 1 38,265

1 9.8 cm
xxxx RW cm
1 9.8 cm
xxxx 375000 cm
9
Solve for Area

• The unit we will use for an area measure is the
  hectare (ha)
• 1 ha 10,000 m2
• You can visualize this as a large square that
  measures 100m on each side.

10
How many hectares are covered by a standard air
photo at a scale of 120,000

• First remember that an air photo measures 9 by
  9
• We want to know what ground area is covered in
  the photo.
• First, since we will be solving for a metric area
  measure (ha), it is best to first convert 9 to
  cm
• Photo edge 9 in
• 9 in 2.54 cm/in
• 22.86 cm
• Second, we find the RW distance in metres
• RW (20,00022.86cm) / 1
• RW 457200 cm 4572 m
• Third, we find area (L W)
• Area 4572m 4572m
• Area 20903184m2
• Area 2090 ha

1 22.86 cm
20000 RW cm
11
How many hectares are covered by a standard air
photo at a scale of 1 ¼ mile

• The photo scale 1 ¼ mi. needs to be converted
  to a representative fraction
• Solve by converting ¼ mi to inches
• 63,360 in/mi / 4 15,840
• Next, convert the photo edge (9) to metric (cm)
• Since we will be solving for a metric measure
  (ha), it is best to first convert 9 to cm
• Photo edge 9 in
• 9 in 2.54 cm/in
• 22.86 cm
• Now we find the RW distance of the photo edge
• RW cm (22.86 cm15840)/1
• RW 362,102 cm 3621 m
• Last we calculate area, in m2, then convert to ha
• Area 3621 m 3621 m 13,111,641 m2
• Area 13,111,641 m2 / 10,000 1,311 ha

1 1
xxxx ¼ mi
1 1
xxxx 15,840
1 22.86 cm
15840 RW cm