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Scale: maps and plans

Ratio, proportion and percent, grade 8 Lesson 6 of 12Learning

Scale: maps and plans

Recap: scale
  • 1 m = 100 cm
  • 1 km = 1000 m = 100,000\displaystyle 100{,}000 cm

A map’s scale is the ratio of a length on the map to the length it stands for in reality, in the same unit.

  • On a map at a scale of 1:50,000\displaystyle 1:50{,}000, every 1 cm on the map stands for 50,000\displaystyle 50{,}000 cm in reality – half a kilometre.
  • The first number is the length on the map, the second the real length.
From the map to reality

On a map at a scale of 1:200,000\displaystyle 1:200{,}000 two cities are 7 cm apart.
What is the real distance between them?

  • Every cm on the map is 200,000\displaystyle 200{,}000 cm in reality, so 7 cm are 7×200,000=1,400,000\displaystyle {7\times200{,}000}={1{,}400{,}000} cm.
  • A kilometre is 100,000\displaystyle 100{,}000 cm: 1,400,000:100,000=14\displaystyle {1{,}400{,}000:100{,}000}={14} km.
From reality to the map

On the same map, how many cm stand for 30 km?

  • 30 km are 30×100,000=3,000,000\displaystyle {30\times100{,}000}={3{,}000{,}000} cm.
  • Every 200,000\displaystyle 200{,}000 cm in reality is 1 cm on the map: 3,000,000:200,000=15\displaystyle {3{,}000{,}000:200{,}000}={15} cm.

To reality you multiply, to the map you divide: the real length is the bigger one.

Exercise 1

On a map at a scale of 1:25,000\displaystyle 1:25{,}000 a trail is 12 cm long.
How long is the trail in reality, in kilometres?

    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

    Exercise 2

    The straight-line distance between Haifa and Eilat is about 350 km.
    How far apart are they on a map at a scale of 1:1,000,000\displaystyle 1:1{,}000{,}000?

      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

      Finding the scale

      On a map, 4 cm stand for 2 km in reality.
      What is the map’s scale?

      • Write both lengths in the same unit: 2 km are 200,000\displaystyle 200{,}000 cm.
      • The ratio: 4:200,000\displaystyle 4:200{,}000.
      • Divide by 4, so that the first number is 1: 1:50,000\displaystyle 1:50{,}000.
      Exercise 3

      Find the scale, so that its first number is 1.

      a
      6 cm on the map stand for 3 km

        / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

        / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

        b
        5 cm on the plan stand for 20 m

          / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

          / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

          Comparing maps

          The same 10 km road, on two maps:

          • On a map at a scale of 1:50,000\displaystyle 1:50{,}000 it is 20 cm.
          • On a map at a scale of 1:200,000\displaystyle 1:200{,}000 it is 5 cm.

          The bigger the second number, the smaller each real kilometre is on the map. A 1:200,000\displaystyle 1:200{,}000 map covers a large area on a small sheet; a 1:50,000\displaystyle 1:50{,}000 map shows smaller details – trails and houses – so hikers use it.

          Area at a scale

          This is the plan of a room at a scale of 1:100\displaystyle 1:100, and a square’s side is 1 cm:

          • On the plan the room is 4 cm by 3 cm, and its area is 12 cm².
          • In reality the room is 4 m by 3 m, and its area is 12 m².
          • A m² is 100×100=10,000\displaystyle {100\times100}={10{,}000} cm², so the real area is 120,000\displaystyle 120{,}000 cm².

          The lengths grew 100 times, but the area grew 100×100=10,000\displaystyle {100\times100}={10{,}000} times: each of the two sides of every square grew 100 times.

          Exercise 4

          A rectangular garden is 5 cm by 3 cm on a plan at a scale of 1:200\displaystyle 1:200.
          What is the garden’s real area, in m²?

            / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

            / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

            Exercise 5

            Two maps of the same island: one at a scale of 1:20,000\displaystyle 1:20{,}000, the other at a scale of 1:80,000\displaystyle 1:80{,}000.
            On which map is the island bigger?
            How many times as long are the lengths on it as on the other map?

              / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

              / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power