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The equality of ratios is called “proportion”.
If a : b :: c : d then a, b, c and d are said to be in proportion. Then a, b, c and d are called terms.
The first and fourth i.e. a and d are called extremas and second and third term i.e. a and d are called mean terms and d is called the fourth proportional.
If four quantities be in proportion, the product of the extremas is equal to the product of the means.
Proportional relationships tell us that two variables scale with each other in a predictable way:
If you drive twice as fast, you'll go twice as far in a given time: Speed and distance are directly proportional to each other.
If you drive twice as fast, it takes half the time to go a set distance: Speed and time are inversely proportional to each other.
To show proportionality we use a Greek alpha symbol in place of an =.
Sometimes we use a constant k to show that need to either multiply or divide something by our other variable.
If two quantities are linked in such a way that an increase in one quantity lead to a corresponding increase in the other and vice−versa, then such a variation is called direct variation.
If two things are directly proportional to each other, they will increase and decrease in the same proportion to each other - double one, double the other; increase one by 12%, the other increases by 12%.
We can find the exact link by looking at the numbers: If we know that the cost of a taxi journey is directly proportional to the distance travelled - which it probably isn't, but let's press on - we can use any pair of values to find the relationship between cost and distance:
If a 300 mile journey cost £1200, we could work out that a 30 mile taxi ride would cost £120, and a 3 mile journey would be £12 - meaning that the trip costs £4 a mile. We could then find the cost of any distance travelled, or how far we could get for a given price.
In this example, our constant k = 4, as the cost = 4 x the distance in miles.
Notes,worksheet and solved question for Maths class 8
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