GCSE Maths: Scale drawings and maps (Foundation)
Part of the GCSE Maths loci, bearings and vectors chapter (Foundation). Chapter overview · Higher version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Scale drawings and maps
A map or a scale drawing is a smaller picture of something real. Every length in the picture is the real length divided by the same number. That number is the scale. Architects draw houses on plans, model-makers build toy cars, and walkers follow maps. All of them use scales.
The most common way to write a scale is as a ratio, for example 1 : 50 000. Both numbers are in the same units, so it means: 1 cm on the map is 50 000 cm in real life (or 1 mm is 50 000 mm, or 1 inch is 50 000 inches). You can read any ratio scale this way. A scale of 1 : 100 means that 1 cm on the drawing is 100 cm = 1 m in real life.
Sometimes a scale is written in words, such as “1 cm represents 2 km”. To turn this into a ratio, change both lengths to the same unit (centimetres) and write the map length first: 1 cm : 2 km = 1 cm : 200 000 cm = 1 : 200 000.
Changing units. Lengths on a map are measured in centimetres, but real distances are given in metres or kilometres. Remember: 100 cm = 1 m, 1000 m = 1 km, so 100 000 cm = 1 km. To go from centimetres to metres divide by 100. To go from centimetres to kilometres divide by 100 000.
The two rules. To go from the map to real life, multiply by the scale number (real life is bigger). To go from real life to the map, divide by the scale number (the map is smaller). Do the calculation in centimetres, then change the answer into a sensible unit. A bigger second number in the scale means that the map shows more land in less space: 1 : 50 000 shows a larger area than 1 : 25 000, so it has less detail.
Key ideas
- A scale 1 : n means 1 cm on the drawing is n cm in real life.
- Map to real life: multiply by n. Real life to map: divide by n.
- 100 cm = 1 m and 100 000 cm = 1 km.
- To write “1 cm to 2 km” as a ratio, change both to cm: 1 : 200 000.
- Always give the unit in your answer.
Key words. A scale compares a length on a map or drawing with the real length. A scale of 1 : 50 000 means that 1 cm on the map is 50 000 cm in real life. A ratio compares two amounts in the same units.
Method. Multiply the length on the map by the scale number to get centimetres, then change to sensible units.
- Say what the scale means. A scale of 1 : 50 000 means that every 1 cm on the map stands for 50 000 cm in real life.1 cm on the map = 50 000 cm real
- Multiply the map length by 50 000. The real distance is 50 000 times as long as the length on the map.4.5 × 50 000 = 225 000 cm
- Change the centimetres into km. There are 100 cm in 1 m and 1000 m in 1 km, so there are 100 000 cm in 1 km. Divide by 100 000.225 000 cm ÷ 100 000 = 2.25 km
Check. Go backwards: 2.25 km = 225 000 cm and 225 000 ÷ 50 000 = 4.5 cm, the length we started with. ✓
Watch out. Multiply by the scale number to go from the map to real life. Divide to go the other way. Change the units at the end.
Key words. A scale compares a length on a map or drawing with the real length. A scale of 1 : 50 000 means that 1 cm on the map is 50 000 cm in real life. A ratio compares two amounts in the same units.
Method. Change the real length to centimetres, then divide by the scale number to get the length on the map.
- Change the real length into centimetres. 1 km = 1000 m = 100 000 cm, so multiply by 100 000.3.5 km × 100 000 = 350 000 cm
- Say what the scale means. A scale of 1 : 25 000 means that every 25 000 cm in real life is only 1 cm on the map.25 000 cm real = 1 cm on the map
- Divide by 25 000. The map length is 25 000 times smaller than the real length.350 000 ÷ 25 000 = 14 cm
Check. Go forwards: 14 × 25 000 = 350 000 cm, which is the real length. ✓
Watch out. Going from real life to the map, you divide by the scale number (the map is smaller).
Key words. A scale compares a length on a map or drawing with the real length. A scale of 1 : 50 000 means that 1 cm on the map is 50 000 cm in real life. A ratio compares two amounts in the same units.
Method. Write the map length and the real length in the same unit (cm), make a ratio, and divide so the first number is 1.
- Change the real length into cm. Both parts of a ratio must be in the same unit. 1 km = 1000 m = 100 000 cm, so multiply by 100 000.3 km × 100 000 = 300 000 cm
- Write the ratio map : real. The scale compares the map length with the real length, in that order.6 : 300 000
- Divide both sides by 6 so the first number is 1. A scale is written as 1 : n. Dividing both parts by the same number keeps the ratio the same.6 ÷ 6 = 1 and 300 000 ÷ 6 = 50 000
Scale = 1 : 50 000
Check. 1 cm on the map stands for 50 000 cm = 500 m in real life. So 6 cm on the map should be 3 km. ✓
Watch out. Change to the same units first. A ratio like 2 : 4 km is wrong because the units do not match.
Key words. A scale compares a length on a map or drawing with the real length. A scale of 1 : 50 000 means that 1 cm on the map is 50 000 cm in real life. A ratio compares two amounts in the same units.
Method. Change the real length to centimetres, then divide by the scale number to get the length on the plan.
- Change the real length into centimetres. 1 m = 100 cm, so multiply by 100.4.20000000000000017763568394002504646778106689453125 m × 100 = 420.000000000000017763568394002504646778106689453125 cm
- Say what the scale means. A scale of 1 : 50 means that every 50 cm in real life is only 1 cm on the plan.50 cm real = 1 cm on the plan
- Divide by 50. The plan length is 50 times smaller than the real length.420.000000000000017763568394002504646778106689453125 ÷ 50 = 8.4000000000000003552713678800500929355621337890625 cm
Check. Go forwards: 8.4000000000000003552713678800500929355621337890625 × 50 = 420.000000000000017763568394002504646778106689453125 cm, which is the real length. ✓
Watch out. Going from real life to the map, you divide by the scale number (the map is smaller).
Watch out
Do not mix up the two directions. If your answer for a real distance is smaller than the length on the map, you divided instead of multiplying. Also watch the units: 4.5 cm on a 1 : 50 000 map is 225 000 cm, which is 2.25 km, not 225 km.Exercise 22A Scale and ratio
Non-calculatorKey words. A scale compares a length on a map or drawing with the real length. A scale of 1 : 50 000 means that 1 cm on the map is 50 000 cm in real life. A ratio compares two amounts in the same units.
Method. A scale 1 : n means 1 cm on the drawing is n cm in real life.
- Read the scale. The second number tells you the real length for every 1 cm on the drawing.1 : 10 means 1 cm : 10 cm
- Write the answer. No unit change is needed.1 cm on the drawing = 10 cm in real life
Check. 10 ÷ 1 = 10, so the real length is 10 times the drawing length. ✓
Key words. A scale compares a length on a map or drawing with the real length. A scale of 1 : 50 000 means that 1 cm on the map is 50 000 cm in real life. A ratio compares two amounts in the same units.
Method. 1 : 100 means 1 cm on the drawing is 100 cm in real life. Then change 100 cm to metres.
- Read the scale. The second number is the real length in cm.1 cm on the drawing = 100 cm in real life
- Change to metres. 100 cm = 1 m, so divide by 100.100 cm ÷ 100 = 1 m
Check. So a 5 cm line on this drawing is 5 m in real life. ✓
Key words. A scale compares a length on a map or drawing with the real length. A scale of 1 : 50 000 means that 1 cm on the map is 50 000 cm in real life. A ratio compares two amounts in the same units.
Method. 1 : 1000 means 1 cm on the map is 1000 cm in real life. Change 1000 cm to metres.
- Read the scale. The second number is the real length in cm.1 cm on the map = 1000 cm in real life
- Change to metres. 100 cm = 1 m, so divide by 100.1000 cm ÷ 100 = 10 m
Check. 10 m = 1000 cm. ✓
Key words. A scale compares a length on a map or drawing with the real length. A scale of 1 : 50 000 means that 1 cm on the map is 50 000 cm in real life. A ratio compares two amounts in the same units.
Method. 1 : 100 000 means 1 cm on the map is 100 000 cm in real life. Change 100 000 cm to kilometres.
- Read the scale. The second number is the real length in cm.1 cm on the map = 100 000 cm in real life
- Change to kilometres. 100 cm = 1 m and 1000 m = 1 km, so divide by 100 000.100 000 cm ÷ 100 000 = 1 km
Check. So 3 cm on this map is 3 km. ✓
Key words. A scale compares a length on a map or drawing with the real length. A scale of 1 : 50 000 means that 1 cm on the map is 50 000 cm in real life. A ratio compares two amounts in the same units.
Method. Write the map length and the real length in the same unit (cm), make a ratio, and divide so the first number is 1.
- Change the real length into cm. Both parts of a ratio must be in the same unit. 1 m = 100 cm, so multiply by 100.5 m × 100 = 500 cm
- Write the ratio map : real. The scale compares the map length with the real length, in that order.1 : 500
- The first number is already 1. So the scale is complete.Scale = 1 : 500
Check. 1 cm on the map stands for 500 cm = 5 m in real life. So 1 cm on the map should be 5 m. ✓
Watch out. Change to the same units first. A ratio like 2 : 4 km is wrong because the units do not match.
Key words. A scale compares a length on a map or drawing with the real length. A scale of 1 : 50 000 means that 1 cm on the map is 50 000 cm in real life. A ratio compares two amounts in the same units.
Method. Write the map length and the real length in the same unit (cm), make a ratio, and divide so the first number is 1.
- Change the real length into cm. Both parts of a ratio must be in the same unit. 1 m = 100 cm, so multiply by 100.20 m × 100 = 2000 cm
- Write the ratio map : real. The scale compares the map length with the real length, in that order.1 : 2000
- The first number is already 1. So the scale is complete.Scale = 1 : 2000
Check. 1 cm on the map stands for 2000 cm = 20 m in real life. So 1 cm on the map should be 20 m. ✓
Watch out. Change to the same units first. A ratio like 2 : 4 km is wrong because the units do not match.
Key words. A scale compares a length on a map or drawing with the real length. A scale of 1 : 50 000 means that 1 cm on the map is 50 000 cm in real life. A ratio compares two amounts in the same units.
Method. Write the map length and the real length in the same unit (cm), make a ratio, and divide so the first number is 1.
- Change the real length into cm. Both parts of a ratio must be in the same unit. 1 m = 100 cm, so multiply by 100.500 m × 100 = 50 000 cm
- Write the ratio map : real. The scale compares the map length with the real length, in that order.1 : 50 000
- The first number is already 1. So the scale is complete.Scale = 1 : 50 000
Check. 1 cm on the map stands for 50 000 cm = 500 m in real life. So 1 cm on the map should be 500 m. ✓
Watch out. Change to the same units first. A ratio like 2 : 4 km is wrong because the units do not match.
Key words. A scale compares a length on a map or drawing with the real length. A scale of 1 : 50 000 means that 1 cm on the map is 50 000 cm in real life. A ratio compares two amounts in the same units.
Method. Write the map length and the real length in the same unit (cm), make a ratio, and divide so the first number is 1.
- Change the real length into cm. Both parts of a ratio must be in the same unit. 1 km = 1000 m = 100 000 cm, so multiply by 100 000.2 km × 100 000 = 200 000 cm
- Write the ratio map : real. The scale compares the map length with the real length, in that order.1 : 200 000
- The first number is already 1. So the scale is complete.Scale = 1 : 200 000
Check. 1 cm on the map stands for 200 000 cm = 2 km in real life. So 1 cm on the map should be 2 km. ✓
Watch out. Change to the same units first. A ratio like 2 : 4 km is wrong because the units do not match.
Key words. A scale compares a length on a map or drawing with the real length. A scale of 1 : 50 000 means that 1 cm on the map is 50 000 cm in real life. A ratio compares two amounts in the same units.
Method. Multiply the length on the map by the scale number to get centimetres, then change to sensible units.
- Say what the scale means. A scale of 1 : 1000 means that every 1 cm on the map stands for 1000 cm in real life.1 cm on the map = 1000 cm real
- Multiply the map length by 1000. The real distance is 1000 times as long as the length on the map.3 × 1000 = 3000 cm
- Change the centimetres into m. There are 100 cm in 1 m, so divide by 100.3000 cm ÷ 100 = 30 m
Check. Go backwards: 30 m = 3000 cm and 3000 ÷ 1000 = 3 cm, the length we started with. ✓
Watch out. Multiply by the scale number to go from the map to real life. Divide to go the other way. Change the units at the end.
Key words. A scale compares a length on a map or drawing with the real length. A scale of 1 : 50 000 means that 1 cm on the map is 50 000 cm in real life. A ratio compares two amounts in the same units.
Method. Multiply the length on the map by the scale number to get centimetres, then change to sensible units.
- Say what the scale means. A scale of 1 : 5000 means that every 1 cm on the map stands for 5000 cm in real life.1 cm on the map = 5000 cm real
- Multiply the map length by 5000. The real distance is 5000 times as long as the length on the map.8 × 5000 = 40 000 cm
- Change the centimetres into m. There are 100 cm in 1 m, so divide by 100.40 000 cm ÷ 100 = 400 m
Check. Go backwards: 400 m = 40 000 cm and 40 000 ÷ 5000 = 8 cm, the length we started with. ✓
Watch out. Multiply by the scale number to go from the map to real life. Divide to go the other way. Change the units at the end.
Key words. A scale compares a length on a map or drawing with the real length. A scale of 1 : 50 000 means that 1 cm on the map is 50 000 cm in real life. A ratio compares two amounts in the same units.
Method. Multiply the length on the map by the scale number to get centimetres, then change to sensible units.
- Say what the scale means. A scale of 1 : 25 000 means that every 1 cm on the map stands for 25 000 cm in real life.1 cm on the map = 25 000 cm real
- Multiply the map length by 25 000. The real distance is 25 000 times as long as the length on the map.6 × 25 000 = 150 000 cm
- Change the centimetres into km. There are 100 cm in 1 m and 1000 m in 1 km, so there are 100 000 cm in 1 km. Divide by 100 000.150 000 cm ÷ 100 000 = 1.5 km
Check. Go backwards: 1.5 km = 150 000 cm and 150 000 ÷ 25 000 = 6 cm, the length we started with. ✓
Watch out. Multiply by the scale number to go from the map to real life. Divide to go the other way. Change the units at the end.
Key words. A scale compares a length on a map or drawing with the real length. A scale of 1 : 50 000 means that 1 cm on the map is 50 000 cm in real life. A ratio compares two amounts in the same units.
Method. Multiply the length on the map by the scale number to get centimetres, then change to sensible units.
- Say what the scale means. A scale of 1 : 200 000 means that every 1 cm on the map stands for 200 000 cm in real life.1 cm on the map = 200 000 cm real
- Multiply the map length by 200 000. The real distance is 200 000 times as long as the length on the map.2.5 × 200 000 = 500 000 cm
- Change the centimetres into km. There are 100 cm in 1 m and 1000 m in 1 km, so there are 100 000 cm in 1 km. Divide by 100 000.500 000 cm ÷ 100 000 = 5 km
Check. Go backwards: 5 km = 500 000 cm and 500 000 ÷ 200 000 = 2.5 cm, the length we started with. ✓
Watch out. Multiply by the scale number to go from the map to real life. Divide to go the other way. Change the units at the end.
Key words. A scale compares a length on a map or drawing with the real length. A scale of 1 : 50 000 means that 1 cm on the map is 50 000 cm in real life. A ratio compares two amounts in the same units.
Method. Multiply the length on the map by the scale number to get centimetres, then change to sensible units.
- Say what the scale means. A scale of 1 : 50 000 means that every 1 cm on the map stands for 50 000 cm in real life.1 cm on the map = 50 000 cm real
- Multiply the map length by 50 000. The real distance is 50 000 times as long as the length on the map.12 × 50 000 = 600 000 cm
- Change the centimetres into km. There are 100 cm in 1 m and 1000 m in 1 km, so there are 100 000 cm in 1 km. Divide by 100 000.600 000 cm ÷ 100 000 = 6 km
Check. Go backwards: 6 km = 600 000 cm and 600 000 ÷ 50 000 = 12 cm, the length we started with. ✓
Watch out. Multiply by the scale number to go from the map to real life. Divide to go the other way. Change the units at the end.
Key words. A scale compares a length on a map or drawing with the real length. A scale of 1 : 50 000 means that 1 cm on the map is 50 000 cm in real life. A ratio compares two amounts in the same units.
Method. Multiply the length on the drawing by the scale number to get centimetres, then change to sensible units.
- Say what the scale means. A scale of 1 : 200 means that every 1 cm on the drawing stands for 200 cm in real life.1 cm on the drawing = 200 cm real
- Multiply the drawing length by 200. The real distance is 200 times as long as the length on the drawing.9 × 200 = 1800 cm
- Change the centimetres into m. There are 100 cm in 1 m, so divide by 100.1800 cm ÷ 100 = 18 m
Check. Go backwards: 18 m = 1800 cm and 1800 ÷ 200 = 9 cm, the length we started with. ✓
Watch out. Multiply by the scale number to go from the map to real life. Divide to go the other way. Change the units at the end.
Key words. A scale compares a length on a map or drawing with the real length. A scale of 1 : 50 000 means that 1 cm on the map is 50 000 cm in real life. A ratio compares two amounts in the same units.
Method. Multiply the length on the map by the scale number to get centimetres, then change to sensible units.
- Say what the scale means. A scale of 1 : 50 000 means that every 1 cm on the map stands for 50 000 cm in real life.1 cm on the map = 50 000 cm real
- Multiply the map length by 50 000. The real distance is 50 000 times as long as the length on the map.7.20000000000000017763568394002504646778106689453125 × 50 000 = 360000.0000000000088817841970012523233890533447265625 cm
- Change the centimetres into km. There are 100 cm in 1 m and 1000 m in 1 km, so there are 100 000 cm in 1 km. Divide by 100 000.360000.0000000000088817841970012523233890533447265625 cm ÷ 100 000 = 3.600000000000000088817841970012523233890533447265625 km
Check. Go backwards: 3.600000000000000088817841970012523233890533447265625 km = 360000.0000000000088817841970012523233890533447265625 cm and 360000.0000000000088817841970012523233890533447265625 ÷ 50 000 = 7.20000000000000017763568394002504646778106689453125 cm, the length we started with. ✓
Watch out. Multiply by the scale number to go from the map to real life. Divide to go the other way. Change the units at the end.
Key words. A scale compares a length on a map or drawing with the real length. A scale of 1 : 50 000 means that 1 cm on the map is 50 000 cm in real life. A ratio compares two amounts in the same units.
Method. Change the real length to centimetres, then divide by the scale number to get the length on the map.
- Change the real length into centimetres. 1 m = 100 cm, so multiply by 100.50 m × 100 = 5000 cm
- Say what the scale means. A scale of 1 : 1000 means that every 1000 cm in real life is only 1 cm on the map.1000 cm real = 1 cm on the map
- Divide by 1000. The map length is 1000 times smaller than the real length.5000 ÷ 1000 = 5 cm
Check. Go forwards: 5 × 1000 = 5000 cm, which is the real length. ✓
Watch out. Going from real life to the map, you divide by the scale number (the map is smaller).
Key words. A scale compares a length on a map or drawing with the real length. A scale of 1 : 50 000 means that 1 cm on the map is 50 000 cm in real life. A ratio compares two amounts in the same units.
Method. Change the real length to centimetres, then divide by the scale number to get the length on the map.
- Change the real length into centimetres. 1 km = 1000 m = 100 000 cm, so multiply by 100 000.2 km × 100 000 = 200 000 cm
- Say what the scale means. A scale of 1 : 50 000 means that every 50 000 cm in real life is only 1 cm on the map.50 000 cm real = 1 cm on the map
- Divide by 50 000. The map length is 50 000 times smaller than the real length.200 000 ÷ 50 000 = 4 cm
Check. Go forwards: 4 × 50 000 = 200 000 cm, which is the real length. ✓
Watch out. Going from real life to the map, you divide by the scale number (the map is smaller).
Key words. A scale compares a length on a map or drawing with the real length. A scale of 1 : 50 000 means that 1 cm on the map is 50 000 cm in real life. A ratio compares two amounts in the same units.
Method. Change the real length to centimetres, then divide by the scale number to get the length on the map.
- Change the real length into centimetres. 1 km = 1000 m = 100 000 cm, so multiply by 100 000.15 km × 100 000 = 1 500 000 cm
- Say what the scale means. A scale of 1 : 250 000 means that every 250 000 cm in real life is only 1 cm on the map.250 000 cm real = 1 cm on the map
- Divide by 250 000. The map length is 250 000 times smaller than the real length.1 500 000 ÷ 250 000 = 6 cm
Check. Go forwards: 6 × 250 000 = 1 500 000 cm, which is the real length. ✓
Watch out. Going from real life to the map, you divide by the scale number (the map is smaller).
Key words. A scale compares a length on a map or drawing with the real length. A scale of 1 : 50 000 means that 1 cm on the map is 50 000 cm in real life. A ratio compares two amounts in the same units.
Method. Change the real length to centimetres, then divide by the scale number to get the length on the map.
- Change the real length into centimetres. 1 m = 100 cm, so multiply by 100.360 m × 100 = 36 000 cm
- Say what the scale means. A scale of 1 : 2000 means that every 2000 cm in real life is only 1 cm on the map.2000 cm real = 1 cm on the map
- Divide by 2000. The map length is 2000 times smaller than the real length.36 000 ÷ 2000 = 18 cm
Check. Go forwards: 18 × 2000 = 36 000 cm, which is the real length. ✓
Watch out. Going from real life to the map, you divide by the scale number (the map is smaller).
Key words. A scale compares a length on a map or drawing with the real length. A scale of 1 : 50 000 means that 1 cm on the map is 50 000 cm in real life. A ratio compares two amounts in the same units.
Method. Change the real length to centimetres, then divide by the scale number to get the length on the map.
- Change the real length into centimetres. 1 km = 1000 m = 100 000 cm, so multiply by 100 000.4.5 km × 100 000 = 450 000 cm
- Say what the scale means. A scale of 1 : 25 000 means that every 25 000 cm in real life is only 1 cm on the map.25 000 cm real = 1 cm on the map
- Divide by 25 000. The map length is 25 000 times smaller than the real length.450 000 ÷ 25 000 = 18 cm
Check. Go forwards: 18 × 25 000 = 450 000 cm, which is the real length. ✓
Watch out. Going from real life to the map, you divide by the scale number (the map is smaller).
Key words. A scale compares a length on a map or drawing with the real length. A scale of 1 : 50 000 means that 1 cm on the map is 50 000 cm in real life. A ratio compares two amounts in the same units.
Method. Change the real length to centimetres, then divide by the scale number to get the length on the drawing.
- Change the real length into centimetres. 1 m = 100 cm, so multiply by 100.1.8000000000000000444089209850062616169452667236328125 m × 100 = 180.00000000000000444089209850062616169452667236328125 cm
- Say what the scale means. A scale of 1 : 20 means that every 20 cm in real life is only 1 cm on the drawing.20 cm real = 1 cm on the drawing
- Divide by 20. The drawing length is 20 times smaller than the real length.180.00000000000000444089209850062616169452667236328125 ÷ 20 = 9.0000000000000002220446049250313080847263336181640625 cm
Check. Go forwards: 9.0000000000000002220446049250313080847263336181640625 × 20 = 180.00000000000000444089209850062616169452667236328125 cm, which is the real length. ✓
Watch out. Going from real life to the map, you divide by the scale number (the map is smaller).
Key words. A scale compares a length on a map or drawing with the real length. A scale of 1 : 50 000 means that 1 cm on the map is 50 000 cm in real life. A ratio compares two amounts in the same units.
Method. Write the map length and the real length in the same unit (cm), make a ratio, and divide so the first number is 1.
- Change the real length into cm. Both parts of a ratio must be in the same unit. 1 km = 1000 m = 100 000 cm, so multiply by 100 000.2 km × 100 000 = 200 000 cm
- Write the ratio map : real. The scale compares the map length with the real length, in that order.4 : 200 000
- Divide both sides by 4 so the first number is 1. A scale is written as 1 : n. Dividing both parts by the same number keeps the ratio the same.4 ÷ 4 = 1 and 200 000 ÷ 4 = 50 000
Scale = 1 : 50 000
Check. 1 cm on the map stands for 50 000 cm = 500 m in real life. So 4 cm on the map should be 2 km. ✓
Watch out. Change to the same units first. A ratio like 2 : 4 km is wrong because the units do not match.
Key words. A scale compares a length on a map or drawing with the real length. A scale of 1 : 50 000 means that 1 cm on the map is 50 000 cm in real life. A ratio compares two amounts in the same units.
Method. Write the map length and the real length in the same unit (cm), make a ratio, and divide so the first number is 1.
- Change the real length into cm. Both parts of a ratio must be in the same unit. 1 m = 100 cm, so multiply by 100.250 m × 100 = 25 000 cm
- Write the ratio map : real. The scale compares the map length with the real length, in that order.5 : 25 000
- Divide both sides by 5 so the first number is 1. A scale is written as 1 : n. Dividing both parts by the same number keeps the ratio the same.5 ÷ 5 = 1 and 25 000 ÷ 5 = 5000
Scale = 1 : 5000
Check. 1 cm on the map stands for 5000 cm = 50 m in real life. So 5 cm on the map should be 250 m. ✓
Watch out. Change to the same units first. A ratio like 2 : 4 km is wrong because the units do not match.
Key words. A scale compares a length on a map or drawing with the real length. A scale of 1 : 50 000 means that 1 cm on the map is 50 000 cm in real life. A ratio compares two amounts in the same units.
Method. Write the map length and the real length in the same unit (cm), make a ratio, and divide so the first number is 1.
- Change the real length into cm. Both parts of a ratio must be in the same unit. 1 km = 1000 m = 100 000 cm, so multiply by 100 000.8 km × 100 000 = 800 000 cm
- Write the ratio map : real. The scale compares the map length with the real length, in that order.2 : 800 000
- Divide both sides by 2 so the first number is 1. A scale is written as 1 : n. Dividing both parts by the same number keeps the ratio the same.2 ÷ 2 = 1 and 800 000 ÷ 2 = 400 000
Scale = 1 : 400 000
Check. 1 cm on the map stands for 400 000 cm = 4 km in real life. So 2 cm on the map should be 8 km. ✓
Watch out. Change to the same units first. A ratio like 2 : 4 km is wrong because the units do not match.
Key words. A scale compares a length on a map or drawing with the real length. A scale of 1 : 50 000 means that 1 cm on the map is 50 000 cm in real life. A ratio compares two amounts in the same units.
Method. Write the map length and the real length in the same unit (cm), make a ratio, and divide so the first number is 1.
- Change the real length into cm. Both parts of a ratio must be in the same unit. 1 m = 100 cm, so multiply by 100.6 m × 100 = 600 cm
- Write the ratio map : real. The scale compares the map length with the real length, in that order.3 : 600
- Divide both sides by 3 so the first number is 1. A scale is written as 1 : n. Dividing both parts by the same number keeps the ratio the same.3 ÷ 3 = 1 and 600 ÷ 3 = 200
Scale = 1 : 200
Check. 1 cm on the map stands for 200 cm = 2 m in real life. So 3 cm on the map should be 6 m. ✓
Watch out. Change to the same units first. A ratio like 2 : 4 km is wrong because the units do not match.
Key words. A scale compares a length on a map or drawing with the real length. A scale of 1 : 50 000 means that 1 cm on the map is 50 000 cm in real life. A ratio compares two amounts in the same units.
Method. Multiply the length on the model by the scale number to get centimetres, then change to sensible units.
- Say what the scale means. A scale of 1 : 24 means that every 1 cm on the model stands for 24 cm in real life.1 cm on the model = 24 cm real
- Multiply the model length by 24. The real car length is 24 times as long as the length on the model.17.5 × 24 = 420 cm
- Change the centimetres into m. There are 100 cm in 1 m, so divide by 100.420 cm ÷ 100 = 4.2 m
Check. Go backwards: 4.2 m = 420 cm and 420 ÷ 24 = 17.5 cm, the length we started with. ✓
Watch out. Multiply by the scale number to go from the map to real life. Divide to go the other way. Change the units at the end.
Key words. A scale compares a length on a map or drawing with the real length. A scale of 1 : 50 000 means that 1 cm on the map is 50 000 cm in real life. A ratio compares two amounts in the same units.
Method. Change the real length to centimetres, then divide by the scale number to get the length on the plan.
- Change the real length into centimetres. 1 m = 100 cm, so multiply by 100.5.5999999999999996447286321199499070644378662109375 m × 100 = 559.99999999999996447286321199499070644378662109375 cm
- Say what the scale means. A scale of 1 : 50 means that every 50 cm in real life is only 1 cm on the plan.50 cm real = 1 cm on the plan
- Divide by 50. The plan length is 50 times smaller than the real length.559.99999999999996447286321199499070644378662109375 ÷ 50 = 11.199999999999999289457264239899814128875732421875 cm
Check. Go forwards: 11.199999999999999289457264239899814128875732421875 × 50 = 559.99999999999996447286321199499070644378662109375 cm, which is the real length. ✓
Watch out. Going from real life to the map, you divide by the scale number (the map is smaller).
Key words. A scale compares a length on a map or drawing with the real length. A scale of 1 : 50 000 means that 1 cm on the map is 50 000 cm in real life. A ratio compares two amounts in the same units.
Method. Write the map length and the real length in the same unit (cm), make a ratio, and divide so the first number is 1.
- Change the real length into cm. Both parts of a ratio must be in the same unit. 1 m = 100 cm, so multiply by 100.250 m × 100 = 25 000 cm
- Write the ratio map : real. The scale compares the map length with the real length, in that order.1 : 25 000
- The first number is already 1. So the scale is complete.Scale = 1 : 25 000
Check. 1 cm on the map stands for 25 000 cm = 250 m in real life. So 1 cm on the map should be 250 m. ✓
Watch out. Change to the same units first. A ratio like 2 : 4 km is wrong because the units do not match.
Key words. A scale compares a length on a map or drawing with the real length. A scale of 1 : 50 000 means that 1 cm on the map is 50 000 cm in real life. A ratio compares two amounts in the same units.
Method. Multiply the length on the map by the scale number to get centimetres, then change to sensible units.
- Say what the scale means. A scale of 1 : 25 000 means that every 1 cm on the map stands for 25 000 cm in real life.1 cm on the map = 25 000 cm real
- Multiply the map length by 25 000. The real distance is 25 000 times as long as the length on the map.7.4000000000000003552713678800500929355621337890625 × 25 000 = 185000.0000000000088817841970012523233890533447265625 cm
- Change the centimetres into km. There are 100 cm in 1 m and 1000 m in 1 km, so there are 100 000 cm in 1 km. Divide by 100 000.185000.0000000000088817841970012523233890533447265625 cm ÷ 100 000 = 1.850000000000000088817841970012523233890533447265625 km
Check. Go backwards: 1.850000000000000088817841970012523233890533447265625 km = 185000.0000000000088817841970012523233890533447265625 cm and 185000.0000000000088817841970012523233890533447265625 ÷ 25 000 = 7.4000000000000003552713678800500929355621337890625 cm, the length we started with. ✓
Watch out. Multiply by the scale number to go from the map to real life. Divide to go the other way. Change the units at the end.
Key words. A scale compares a length on a map or drawing with the real length. A scale of 1 : 50 000 means that 1 cm on the map is 50 000 cm in real life. A ratio compares two amounts in the same units.
Method. A ratio scale uses the same unit on both sides. 1 : 2000 means 1 cm : 2000 cm. Change 2000 cm to metres.
- Read the scale correctly. Both numbers in the ratio are in the same unit, so the real length is 2000 cm, not 2000 m.1 cm on the map = 2000 cm in real life
- Change 2000 cm to metres. 100 cm = 1 m, so divide by 100.2000 ÷ 100 = 20 m
- Write the conclusion. Sam used the wrong unit.Sam is wrong. 1 cm on the map is 20 m in real life.
Check. 1 cm = 20 m means 1 cm : 2000 cm, since 20 m = 2000 cm. ✓
Watch out. Never read a ratio scale as 1 cm to n m. It is 1 cm to n cm.
Key words. A scale compares a length on a map or drawing with the real length. A scale of 1 : 50 000 means that 1 cm on the map is 50 000 cm in real life. A ratio compares two amounts in the same units.
Method. Work out what 1 cm stands for on each map. The map where 1 cm stands for the smaller distance is more zoomed in, so it shows more detail.
- Map A. 1 : 25 000 means 1 cm is 25 000 cm. Change to metres by dividing by 100.25 000 ÷ 100 = 250 m
- Map B. 1 : 50 000 means 1 cm is 50 000 cm.50 000 ÷ 100 = 500 m
- Compare. On map A each centimetre covers less real ground, so objects are drawn larger.250 m < 500 m, so Map A shows more detail.
Check. A road 500 m long is 2 cm long on map A but only 1 cm on map B. It is drawn bigger on map A. ✓
Watch out. A smaller second number in the scale means a larger-scale map with more detail.
Key words. A scale compares a length on a map or drawing with the real length. A scale of 1 : 50 000 means that 1 cm on the map is 50 000 cm in real life. A ratio compares two amounts in the same units.
Method. Find the real length of the path first, then use time = distance ÷ speed.
- Find the real length in centimetres. Multiply the map length by the scale number.8.4 × 25 000 = 210 000 cm
- Change to kilometres. 100 000 cm = 1 km, so divide by 100 000.210 000 ÷ 100 000 = 2.1 km
- Find the time in hours. Time = distance ÷ speed.2.1 ÷ 3 = 0.7 hours
- Change to minutes. There are 60 minutes in 1 hour.0.7 × 60 = 42 minutes
Check. At 3 km/h she walks 1 km in 20 minutes, so 2.1 km takes 2.1 × 20 = 42 minutes. ✓
Watch out. Do not forget to change the speed units and the distance units to match (km and km/h).
Key words. A scale compares a length on a map or drawing with the real length. A scale of 1 : 50 000 means that 1 cm on the map is 50 000 cm in real life. A ratio compares two amounts in the same units.
Method. Multiply by the scale number, then change centimetres to kilometres.
- Multiply by the scale number. Map to real life: multiply.2 × 50 000 = 100 000 cm
- Change to kilometres. 100 000 cm = 1000 m = 1 km.100 000 cm = 1 km
- Write the conclusion. A show-that question needs a final sentence.2 cm on the map is 1 km in real life, as required.
Check. 1 cm = 0.5 km, so 2 cm = 1 km. ✓
Watch out. End with a sentence that says what you have shown.
Key words. A scale compares a length on a map or drawing with the real length. A scale of 1 : 50 000 means that 1 cm on the map is 50 000 cm in real life. A ratio compares two amounts in the same units.
Method. First find the scale of the map as a ratio 1 : n, then use it to change the length of the road.
- Change 9 km to centimetres. 100 000 cm = 1 km.9 × 100 000 = 900 000 cm
- Write the ratio map : real. Both are now in cm.6 : 900 000
- Divide both sides by 6. The first number must be 1.1 : 150 000
- Multiply the road on the map by the scale number. Map to real life.14.2 × 150 000 = 2 130 000 cm
- Change to kilometres. Divide by 100 000.2 130 000 ÷ 100 000 = 21.3 km
Check. Another way: 6 cm is 9 km, so 1 cm is 1.5 km and 14.2 × 1.5 = 21.3 km. ✓
Watch out. A map scale is the same everywhere on the map, so once you know it you can measure any distance.
Exercise 22B Maps and scale drawings in context
Calculator allowedKey words. A scale compares a length on a map or drawing with the real length. A scale of 1 : 50 000 means that 1 cm on the map is 50 000 cm in real life. A ratio compares two amounts in the same units.
Method. Multiply the length on the map by the scale number to get centimetres, then change to sensible units.
- Say what the scale means. A scale of 1 : 50 000 means that every 1 cm on the map stands for 50 000 cm in real life.1 cm on the map = 50 000 cm real
- Multiply the map length by 50 000. The real distance is 50 000 times as long as the length on the map.5 × 50 000 = 250 000 cm
- Change the centimetres into km. There are 100 cm in 1 m and 1000 m in 1 km, so there are 100 000 cm in 1 km. Divide by 100 000.250 000 cm ÷ 100 000 = 2.5 km
Check. Go backwards: 2.5 km = 250 000 cm and 250 000 ÷ 50 000 = 5 cm, the length we started with. ✓
Watch out. Multiply by the scale number to go from the map to real life. Divide to go the other way. Change the units at the end.
Key words. A scale compares a length on a map or drawing with the real length. A scale of 1 : 50 000 means that 1 cm on the map is 50 000 cm in real life. A ratio compares two amounts in the same units.
Method. Multiply the length on the map by the scale number to get centimetres, then change to sensible units.
- Say what the scale means. A scale of 1 : 50 000 means that every 1 cm on the map stands for 50 000 cm in real life.1 cm on the map = 50 000 cm real
- Multiply the map length by 50 000. The real distance is 50 000 times as long as the length on the map.6 × 50 000 = 300 000 cm
- Change the centimetres into km. There are 100 cm in 1 m and 1000 m in 1 km, so there are 100 000 cm in 1 km. Divide by 100 000.300 000 cm ÷ 100 000 = 3 km
Check. Go backwards: 3 km = 300 000 cm and 300 000 ÷ 50 000 = 6 cm, the length we started with. ✓
Watch out. Multiply by the scale number to go from the map to real life. Divide to go the other way. Change the units at the end.
Key words. A scale compares a length on a map or drawing with the real length. A scale of 1 : 50 000 means that 1 cm on the map is 50 000 cm in real life. A ratio compares two amounts in the same units.
Method. Find the real length of the route A to B to C, then take away the direct road.
- Real length of A to B. Map to real: 5 cm × 50 000 = 250 000 cm = 2.5 km.2.5 km
- Real length of B to C. It is also 5 cm on the map.2.5 km
- Add them. Route A to B to C.2.5 + 2.5 = 5 km
- Take away the direct road. The direct road A to C is 3 km.5 − 3 = 2 km
Check. In centimetres on the map: 5 + 5 − 6 = 4 cm and 4 cm is 2 km. ✓
Watch out. The extra distance is a difference, so subtract.
Key words. A scale compares a length on a map or drawing with the real length. A scale of 1 : 50 000 means that 1 cm on the map is 50 000 cm in real life. A ratio compares two amounts in the same units.
Method. Draw the triangle accurately at the stated scale. Measure the angle with a protractor. A calculator can check it.
- Convert the lengths to the scale. 1 cm stands for 1 m, so the foot-to-wall distance is 2.5 cm.Horizontal side 2.5 cm; the ladder is 6.5 cm long.
- Draw the base and the right angle. Draw a horizontal line 2.5 cm long, then draw a vertical line at the right-hand end with a set square.Base PQ = 2.5 cm, right angle at Q.
- Find the top of the ladder. Set the compasses to 6.5 cm, put the point at P and draw an arc that crosses the vertical line at R.PR = 6.5 cm
- Measure. Place the protractor with its centre on P and read the angle QPR.QR measures 6 cm. Angle QPR = 67.4° to 1 d.p., so about 67°
Check. Trigonometry: cos(angle) = 2.5 ÷ 6.5, so angle = cos−1(0.3846…) = 67.4°. ✓
Watch out. Drawing and measuring is accurate to about 1°, so answers 1° or 2° either side are accepted.
Key words. A scale compares a length on a map or drawing with the real length. A scale of 1 : 50 000 means that 1 cm on the map is 50 000 cm in real life. A ratio compares two amounts in the same units.
Method. Draw a right-angled triangle: the ground is the shadow (8 cm), the flagpole is upright, and the Sun’s ray makes 35° with the ground.
- Draw the shadow. 1 cm stands for 1 m, so draw a horizontal line 8 cm long. Mark its ends F (foot of pole) and S (end of shadow).FS = 8 cm
- Draw the upright pole. The pole is at 90° to the ground, so draw a vertical line upwards from F.Right angle at F
- Draw the Sun’s ray. At S draw a line making 35° with the ground. It crosses the pole at the top T.Angle at S = 35°
- Measure FT and use the scale. FT measures about 5.6 cm, so the height is 5.6 m.Height = 5.6 m ≈ 6 m
Check. tan 35° = height ÷ 8, so height = 8 × tan 35° = 5.60 m. ✓
Watch out. Choose a scale that makes the drawing big enough to measure accurately.
Key words. A scale compares a length on a map or drawing with the real length. A scale of 1 : 50 000 means that 1 cm on the map is 50 000 cm in real life. A ratio compares two amounts in the same units.
Method. Change the real length to centimetres, then divide by the scale number to get the length on the drawing.
- Change the real length into centimetres. 1 m = 100 cm, so multiply by 100.36 m × 100 = 3600 cm
- Say what the scale means. A scale of 1 : 200 means that every 200 cm in real life is only 1 cm on the drawing.200 cm real = 1 cm on the drawing
- Divide by 200. The drawing length is 200 times smaller than the real length.3600 ÷ 200 = 18 cm
Check. Go forwards: 18 × 200 = 3600 cm, which is the real length. ✓
Watch out. Going from real life to the map, you divide by the scale number (the map is smaller).
Key words. A scale compares a length on a map or drawing with the real length. A scale of 1 : 50 000 means that 1 cm on the map is 50 000 cm in real life. A ratio compares two amounts in the same units.
Method. Multiply the length on the map by the scale number to get centimetres, then change to sensible units.
- Say what the scale means. A scale of 1 : 50 000 means that every 1 cm on the map stands for 50 000 cm in real life.1 cm on the map = 50 000 cm real
- Multiply the map length by 50 000. The real distance is 50 000 times as long as the length on the map.1 × 50 000 = 50 000 cm
- Change the centimetres into km. There are 100 cm in 1 m and 1000 m in 1 km, so there are 100 000 cm in 1 km. Divide by 100 000.50 000 cm ÷ 100 000 = 0.5 km
Check. Go backwards: 0.5 km = 50 000 cm and 50 000 ÷ 50 000 = 1 cm, the length we started with. ✓
Watch out. Multiply by the scale number to go from the map to real life. Divide to go the other way. Change the units at the end.
Key words. A scale compares a length on a map or drawing with the real length. A scale of 1 : 50 000 means that 1 cm on the map is 50 000 cm in real life. A ratio compares two amounts in the same units.
Method. Write the map length and the real length in the same unit (cm), make a ratio, and divide so the first number is 1.
- Change the real length into cm. Both parts of a ratio must be in the same unit. 1 km = 1000 m = 100 000 cm, so multiply by 100 000.5 km × 100 000 = 500 000 cm
- Write the ratio map : real. The scale compares the map length with the real length, in that order.1 : 500 000
- The first number is already 1. So the scale is complete.Scale = 1 : 500 000
Check. 1 cm on the map stands for 500 000 cm = 5 km in real life. So 1 cm on the map should be 5 km. ✓
Watch out. Change to the same units first. A ratio like 2 : 4 km is wrong because the units do not match.
Key words. A scale compares a length on a map or drawing with the real length. A scale of 1 : 50 000 means that 1 cm on the map is 50 000 cm in real life. A ratio compares two amounts in the same units.
Method. Multiply the length on the map by the scale number to get centimetres, then change to sensible units.
- Say what the scale means. A scale of 1 : 500 000 means that every 1 cm on the map stands for 500 000 cm in real life.1 cm on the map = 500 000 cm real
- Multiply the map length by 500 000. The real distance is 500 000 times as long as the length on the map.7.4000000000000003552713678800500929355621337890625 × 500 000 = 3700000.00000000017763568394002504646778106689453125 cm
- Change the centimetres into km. There are 100 cm in 1 m and 1000 m in 1 km, so there are 100 000 cm in 1 km. Divide by 100 000.3700000.00000000017763568394002504646778106689453125 cm ÷ 100 000 = 37.0000000000000017763568394002504646778106689453125 km
Check. Go backwards: 37.0000000000000017763568394002504646778106689453125 km = 3700000.00000000017763568394002504646778106689453125 cm and 3700000.00000000017763568394002504646778106689453125 ÷ 500 000 = 7.4000000000000003552713678800500929355621337890625 cm, the length we started with. ✓
Watch out. Multiply by the scale number to go from the map to real life. Divide to go the other way. Change the units at the end.
Key words. A scale compares a length on a map or drawing with the real length. A scale of 1 : 50 000 means that 1 cm on the map is 50 000 cm in real life. A ratio compares two amounts in the same units.
Method. Use Pythagoras’ theorem on the map to find the straight-line distance in cm. Then use the scale.
- Draw the right-angled triangle. East and north are at right angles, so the straight line is the hypotenuse.Shorter sides 4.8 cm and 3.6 cm
- Use Pythagoras. Add the squares, then square root.4.82 + 3.62 = 23.04 + 12.96 = 36
√36 = 6 cm on the map - Multiply by the scale number. Map to real life.6 × 25 000 = 150 000 cm
- Change to km. Divide by 100 000.150 000 ÷ 100 000 = 1.5 km
Check. The numbers 4.8, 3.6, 6 are 1.2 times the 3, 4, 5 triangle (3.6 : 4.8 : 6). ✓
Watch out. Do the Pythagoras on the map, then scale up. Do not scale a squared number by 25 000.
Key words. A scale compares a length on a map or drawing with the real length. A scale of 1 : 50 000 means that 1 cm on the map is 50 000 cm in real life. A ratio compares two amounts in the same units.
Method. Find the real distance, then use time = distance ÷ speed.
- Real distance in cm. Map to real: multiply by 50 000.15.6 × 50 000 = 780 000 cm
- Change to km. Divide by 100 000.780 000 ÷ 100 000 = 7.8 km
- Time in hours. Time = distance ÷ speed.7.8 ÷ 12 = 0.65 hours
- Time in minutes. Multiply by 60.0.65 × 60 = 39 minutes
Check. 12 km/h is 1 km every 5 minutes, so 7.8 km takes 7.8 × 5 = 39 minutes. ✓
Watch out. Check the units match before you divide.
Key words. A scale compares a length on a map or drawing with the real length. A scale of 1 : 50 000 means that 1 cm on the map is 50 000 cm in real life. A ratio compares two amounts in the same units.
Method. Multiply each model length by the scale number, then change from cm to m.
- Real height of the tower in cm. Model to real: multiply by 300.18 × 300 = 5400 cm
- Tower in metres. Divide by 100.5400 ÷ 100 = 54 m
- Real height of the window in cm. Multiply by 300.0.5 × 300 = 150 cm
- Window in metres. Divide by 100.150 ÷ 100 = 1.5 m
Check. 1 cm on the model is 3 m in real life, so 18 cm is 54 m and 0.5 cm is 1.5 m. ✓
Watch out. The same scale applies to every length on the model, big or small.