Chapter 7 of 8

All 8 chapters
  1. 1Ratio: comparing parts, readunderstood
  2. 2Ratios and fractions, readunderstood
  3. 3Sharing in a ratio, readunderstood
  4. 4Finding one first, readunderstood
  5. 5Direct proportion, readunderstood
  6. 6Rates and speed, readunderstood
  7. 7Maps and scale drawings, readunderstood
  8. 8Inverse proportion, readunderstood

M3.4.7 Chapter 7

Maps and scale drawings

On a map a road is 4 cm long. The scale is 1 : 50 000. How long is the real road?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Use a scale of 1 : 50 000 to find a real distance
  • Draw a plan of a room with 1 cm for each 1 m

A map shows a big place on a small page. The scale tells you how much smaller the map is. A scale of 1 : 50 000 means this. 1 cm on the map is 50 000 cm on the ground.

50 000 cm is hard to picture. Change it to metres. 100 cm make 1 m, so 50 000 ÷ 100 = 500 m. So on this map 1 cm stands for 500 m. A scale bar shows this on the map itself.

Draw it first

Draw a scale bar for 1 : 50 000

  1. Step 1: a bar with one segment, labelled on the map (cm) above and on the ground below. 0 cm matches 0, and 1 cm matches 500 m. Step 2: the bar grows to 4 equal segments. 2 cm match 1 km, 3 cm match 1.5 km and 4 cm match 2 km. Step 3: the label says 1 cm stands for 500 m.001500 mon the map (cm)on the ground

    1 Draw 1 cm on the map. On the ground it is 50 000 cm, which is 500 m.

  2. Step 1: a bar with one segment, labelled on the map (cm) above and on the ground below. 0 cm matches 0, and 1 cm matches 500 m. Step 2: the bar grows to 4 equal segments. 2 cm match 1 km, 3 cm match 1.5 km and 4 cm match 2 km. Step 3: the label says 1 cm stands for 500 m.001500 mon the map (cm)on the ground21 km31.5 km42 km

    2 Add more centimetres. Each one stands for another 500 m, so 2 cm stand for 1 km.

  3. Step 1: a bar with one segment, labelled on the map (cm) above and on the ground below. 0 cm matches 0, and 1 cm matches 500 m. Step 2: the bar grows to 4 equal segments. 2 cm match 1 km, 3 cm match 1.5 km and 4 cm match 2 km. Step 3: the label says 1 cm stands for 500 m.001500 mon the map (cm)on the ground21 km31.5 km42 km1 cm stands for 500 m

    3 Read the bar: 1 cm on the map stands for 500 m on the ground.

Worked example

A map has a scale of 1 : 50 000. A path is 6 cm long on the map. How far is it on the ground, in km?

  1. 1 cm is 50 000 cm, so 6 cm is 6 × 50 000 = 300 000 cm.
  2. Change to metres: 300 000 ÷ 100 = 3 000 m.
  3. Change to kilometres: 3 000 ÷ 1 000 = 3 km.

Answer: 3 km

You can also go the other way. Change the real distance to centimetres. Then divide by the scale to find the length on the map.

Worked example

Two villages are 4.5 km apart. The map has a scale of 1 : 50 000. How far apart are they on the map?

  1. Change to centimetres: 4.5 km = 4 500 m = 450 000 cm.
  2. Divide by the scale: 450 000 ÷ 50 000 = 9.
  3. On the map they are 9 cm apart.

Answer: 9 cm

Two number lines, on the map (cm) above and on the ground below. 0 cm match 0, 2 cm match 1 km, and 9 cm match 4.5 km.on the map (cm)on the ground0021 km94.5 km
Every 2 cm on the map is 1 km on the ground. The villages are 4.5 km apart, so they are 9 cm apart.

A plan is a scale drawing of a room. Choose a scale that fits your page. With 1 cm for each 1 m, a wall 6 m long is drawn 6 cm long. This scale is 1 : 100, because 1 m is 100 cm.

Worked example

A room is 6 m long and 3 m wide. A bed is 2 m long. Draw a plan with 1 cm for each 1 m.

  1. The scale is 1 cm for each 1 m.
  2. The room is 6 m long. 6 ÷ 1 = 6 cm.
  3. The room is 3 m wide. 3 ÷ 1 = 3 cm.
  4. The bed is 2 m long. 2 ÷ 1 = 2 cm.
  5. Draw a rectangle 6 cm by 3 cm. Draw the bed 2 cm long inside it.

Answer: 6 cm by 3 cm

A plan on a grid of 1 cm squares, 6 squares across and 3 down. The width is labelled 6 cm for 6 m and the height is labelled 3 cm for 3 m. The bed is 2 squares by 1 square in the top left corner. The desk is 2 squares by 1 square in the bottom right corner.beddesk6 cm for 6 m3 cm for 3 m
Each square is 1 cm on the plan. It stands for 1 m in the room.

See it three ways

With things

Draw a line 1 cm long and say it is 1 m. A line 5 cm long is then 5 m.

As a picture

Two number lines, on the plan (cm) above and in the room (m) below. 0 cm match 0 m, 1 cm matches 1 m, and 5 cm match 5 m.on the plan (cm)in the room (m)001155

In symbols

1 cm on the plan is 1 m in the room, so 5 cm is 5 m. The scale is 1 : 100.

Try it

Draw a plan of a table

You need: squared paper, a ruler, a table or a desk, and a tape measure or a piece of string.

  1. Measure the length and the width of the table top in centimetres.
  2. Choose a scale. Use 1 cm on the paper for each 10 cm on the table.
  3. Divide each measurement by 10 to get the lengths to draw.
  4. Draw the rectangle on squared paper. Draw a box on it, to the same scale.

What you will see: For example, a table 60 cm by 40 cm is drawn 6 cm by 4 cm. A box 20 cm by 10 cm is drawn 2 cm by 1 cm.

If you cannot try it, this is what happens.
A plan on a grid of 1 cm squares, 6 squares across and 4 down. The width is labelled 6 cm for 60 cm and the height is labelled 4 cm for 40 cm. A box is drawn 2 squares by 1 square.box6 cm for 60 cm4 cm for 40 cm

A wrong idea many people have

On a map with the scale 1 : 50 000, 1 cm stands for 50 000 m.

Why it feels right: The number 50 000 is big, and it is quick to forget which unit it is in.

The test that breaks it: Both numbers in 1 : 50 000 must be in the same unit. If 1 cm stood for 50 000 m, that is 5 000 000 cm. The scale would then be 1 : 5 000 000, which is not the scale we have.

What is true: In the scale 1 : 50 000 both numbers use the same unit. So 1 cm on the map is 50 000 cm on the ground. That is 500 m.

Key words

scale
how much smaller a map or plan is than the real place, such as 1 : 50 000.
scale bar
a line on a map that shows what a length on the map stands for.
plan
a scale drawing of a room or a building, seen from above.

Back to your guess

The answer is B: The real road is 2 km long.

1 cm on the map is 50 000 cm on the ground. 4 cm is 4 × 50 000 = 200 000 cm. That is 2 000 m, which is 2 km. If you read 200 000 as metres you get 200 km, but the scale is in centimetres.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. A map has the scale 1 : 50 000. A road is 7 cm long on the map. How long is the real road?

    A bar cut into 7 equal parts. Each part is 1 cm. The caption says road: 7 cm on the map.road: 7 cm on the map
    Tap your answer.

    1. Not this one. You read 350 000 as metres. The scale is in centimetres. Change 350 000 cm to metres, then to kilometres. Go to M3.4.7. Try another.

    2. Yes, that is right. 7 × 50 000 = 350 000 cm. 350 000 ÷ 100 = 3 500 m. 3 500 ÷ 1 000 = 3.5 km.

    3. Not this one. You divided by 1 000 only. Divide by 100 first to get 3 500 m. Then divide by 1 000. Go to M3.4.7. Try another.

    See the answer

    3.5 km

    7 × 50 000 = 350 000 cm. 350 000 ÷ 100 = 3 500 m. 3 500 ÷ 1 000 = 3.5 km.

    • If you chose A: You read 350 000 as metres. The scale is in centimetres. Change 350 000 cm to metres, then to kilometres. Go to M3.4.7.
    • If you chose C: You divided by 1 000 only. Divide by 100 first to get 3 500 m. Then divide by 1 000. Go to M3.4.7.

  2. A map has the scale 1 : 25 000. A path is 6 cm long on the map. How long is the real path?

    A scale bar with 3 segments. The top row, on the map (cm), reads 0, 1, 2, 3. The bottom row, on the ground, reads 0, 250 m, 500 m, 750 m.001250 m2500 m3750 mon the map (cm)on the ground
    Tap your answer.

    1. Not this one. You read 150 000 as metres. The scale is in centimetres. 150 000 cm is 1 500 m. Go to M3.4.7. Try another.

    2. Not this one. You divided by 1 000 only. First divide by 100 to change centimetres to metres, then by 1 000 for kilometres. Go to M3.4.7. Try another.

    3. Yes, that is right. 1 cm is 25 000 cm, which is 250 m. 6 cm is 6 × 250 = 1 500 m. That is 1.5 km.

    See the answer

    1.5 km

    1 cm is 25 000 cm, which is 250 m. 6 cm is 6 × 250 = 1 500 m. That is 1.5 km.

    • If you chose A: You read 150 000 as metres. The scale is in centimetres. 150 000 cm is 1 500 m. Go to M3.4.7.
    • If you chose B: You divided by 1 000 only. First divide by 100 to change centimetres to metres, then by 1 000 for kilometres. Go to M3.4.7.

  3. A field is 800 m long. The scale of the map is 1 : 50 000. How long is the field on the map?

    Two number lines, on the map (cm) above and on the ground (m) below. 0 cm match 0 m, 1 cm matches 500 m, and a question mark of centimetres matches 800 m.on the map (cm)on the ground (m)001500?800
    Tap your answer.

    1. Yes, that is right. 800 m = 80 000 cm. Divide by the scale: 80 000 ÷ 50 000 = 1.6 cm.

    2. Not this one. That is ten times too long. 800 m is 80 000 cm, and 80 000 ÷ 50 000 = 1.6. Check your zeros. Go to M3.4.7. Try another.

    3. Not this one. You multiplied. Going from the ground to the map, divide by the scale. Go to M3.4.7. Try another.

    See the answer

    1.6 cm

    800 m = 80 000 cm. Divide by the scale: 80 000 ÷ 50 000 = 1.6 cm.

    • If you chose B: That is ten times too long. 800 m is 80 000 cm, and 80 000 ÷ 50 000 = 1.6. Check your zeros. Go to M3.4.7.
    • If you chose C: You multiplied. Going from the ground to the map, divide by the scale. Go to M3.4.7.

  4. Draw a plan of a hall 12 m long and 8 m wide. Use 1 cm for each 2 m. What size is the plan?

    A scale bar with 3 segments. The top row, on the plan (cm), reads 0, 1, 2, 3. The bottom row, in real life, reads 0, 2 m, 4 m, 6 m.0012 m24 m36 mon the plan (cm)in real life
    Tap your answer.

    1. Not this one. That uses 1 cm for each 1 m. Here 1 cm stands for 2 m, so divide each length by 2. Go to M3.4.7. Try another.

    2. Yes, that is right. 1 cm stands for 2 m. 12 ÷ 2 = 6 cm and 8 ÷ 2 = 4 cm.

    3. Not this one. You multiplied by 2. A plan is smaller than the room, so divide by 2. Go to M3.4.7. Try another.

    See the answer

    6 cm by 4 cm

    1 cm stands for 2 m. 12 ÷ 2 = 6 cm and 8 ÷ 2 = 4 cm.

    • If you chose A: That uses 1 cm for each 1 m. Here 1 cm stands for 2 m, so divide each length by 2. Go to M3.4.7.
    • If you chose C: You multiplied by 2. A plan is smaller than the room, so divide by 2. Go to M3.4.7.

  5. Which scale means that 1 cm on the map stands for 1 km on the ground?

    A scale bar with 2 segments. The top row, on the map (cm), reads 0, 1, 2. The bottom row, on the ground, reads 0, 1 km, 2 km.0011 km22 kmon the map (cm)on the ground
    Tap your answer.

    1. Not this one. 1 000 cm is only 10 m. 1 km is 1 000 m, which is much more in centimetres. Go to M3.4.7. Try another.

    2. Yes, that is right. 1 km = 1 000 m = 100 000 cm. So 1 cm on the map stands for 100 000 cm on the ground.

    3. Not this one. 1 000 000 cm is 10 km, ten times too much. 1 km is 100 000 cm. Go to M3.4.7. Try another.

    See the answer

    1 : 100 000

    1 km = 1 000 m = 100 000 cm. So 1 cm on the map stands for 100 000 cm on the ground.

    • If you chose A: 1 000 cm is only 10 m. 1 km is 1 000 m, which is much more in centimetres. Go to M3.4.7.
    • If you chose C: 1 000 000 cm is 10 km, ten times too much. 1 km is 100 000 cm. Go to M3.4.7.

  6. A map has the scale 1 : 50 000. Two towns are 9 cm apart on the map. How many km apart are they?

    See the answer

    4.5 km

    9 × 50 000 = 450 000 cm. That is 4 500 m, which is 4.5 km.

  7. A room is 7 m long and 5 m wide. How long and wide is its plan, if 1 cm stands for 1 m?

    See the answer

    7 cm by 5 cm

    1 cm stands for 1 m. So 7 m is 7 cm and 5 m is 5 cm.

  8. From an earlier chapter

    A bike goes 36 km in 3 hours. What is its speed in km per hour?

    See the answer

    12 km per hour

    36 ÷ 3 = 12. The bike goes 12 km in each hour.

Keep it

Copy this into your exercise book.

On a 1 : 50 000 map, 1 cm is 50 000 cm on the ground. That is 500 m. Multiply to go from map to ground. Divide to go back.

Draw: Draw a scale bar of 4 cm. Mark 0, 500 m, 1 km, 1.5 km and 2 km under the ticks.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

No internet tomorrow? Save the whole book as one file (710 KB). It opens on any phone.