A map has scale 1 : 50 000. Two towns are 7 km apart. How far apart are they on the map?
- Change the real distance to centimetres: 7 km = 7000 m = 700 000 cm.
- Map length = real length ÷ 50 000 = 700 000 ÷ 50 000.
- 700 000 ÷ 50 000 = 14.
Answer: 14 cm
Units, scale and ratio
Maths · Ratio, proportion and rates of change. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.
The idea
Many questions need a change of units before any other working. For compound units such as km/h or g/cm3, change one unit at a time. Remember that area and volume conversions use the length factor squared or cubed.
A scale such as 1 : 25 000 means 1 cm on the map stands for 25 000 cm in real life. Areas change by the scale factor squared.
A ratio compares parts. Simplify it like a fraction, after putting both amounts in the same units. To share an amount, add the parts of the ratio, find the value of one part, then multiply.
Rules to remember
- 1 km = 1000 m, 1 m = 100 cm, 1 kg = 1000 g, 1 litre = 1000 ml = 1000 cm3.
- 1 m2 = 1002 = 10 000 cm2; 1 m3 = 1003 = 1 000 000 cm3.
- m/s to km/h: multiply by 3.6. km/h to m/s: divide by 3.6.
- 1 g/cm3 = 1000 kg/m3.
- Scale 1 : n: real length = map length × n; real area = model area × n2.
- Sharing in the ratio a : b: one part = total ÷ (a + b); the first share is a/(a + b) of the total.
- x : y = a : b means x/y = a/b, so bx = ay.
Common mistake
In the ratio 3 : 5 the first share is 3/8 of the total, not 3/5. The denominator is the total number of parts.
Worked example
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