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Direct Proportion
Introduction
If 2 bottles of wine costs £8 then 4 bottles cost £16 and so on. In other words:
twice the number of bottles means twice the cost
three times the wine means three times the cost
half the number of bottles means half the cost
We say the cost of the wine is directly proportional to the number of bottles.
There are many situations where two related quantities vary in this way. For
example, if a car travels at a constant speed of 50mph then the distance travelled
is directly proportional to the time taken. The car travels twice as far in twice
the time. It travels 50 miles in 1 hour and 100 miles in two hours.
Notice that if one of the quantities is zero then so is the other:
no wine then no cost
A good (easy to understand) example of direct proportion is a map or a scale
drawing. On a plan of a house, a distance measured on the ground and the
corresponding distance on the plan are directly proportional. A room that is
twice the length on the plan will be twice the length on the ground.
You may wonder what direct proportion has to do with ratio. The point is that,
if two quantities are directly proportional, the ratio between the two quantities is
constant. For example, if 1cm on a plan corresponds to 100cm on the ground,
the constant ratio is 1cm:100cm or 1:100 .
Scale
The scale of a map or plan is the constant ratio between the distance measured
on the plan and the distance on the ground. In the example above the scale is
1:100 . In a scale drawing, the two units do not have to be the same. You can
have a scale of 1cm:1m . A typical scale for a walker’s map is 2cm:1km .
Different maps can have different scales but a single map must have a fixed
scale.
Remember: in questions with the same types of quantity but different units,
before you calculate the ratio, it is usual to convert to the same unit. If you are
asked to write a map’s scale in its simplest form (a pure ratio) you have to leave
out the units. Start by converting to cm and then leave out the unit.
2cm:1km = 2cm:1000m = 2cm:100,000cm = 1:50,000
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