Lesson Applications of Proportions

Proportions can be applied in several forms and has applications in several places. We would discuss some forms of applications and then application of proportions in similar triangles.

Direct Proportion/ Direct Variation
If two quantities changes or vary in such a manner that their ratio always remains constant, the quantities are said to be in proportion. In other words, the two quantities are related in such a manner that the positive change in one quantity leads to proportionately same positive change in the other quantity. We represent this proportionality using 'α' (). Thus x α y is termed as 'x is directly proportional to y'.

If y α x then y = kx , where k is a non-zero constant called constant of proportionality. The equation is called the equation of direct proportionality.
Inverse proportion/ Inverse Variation
If two quantities changes or vary in such a manner that an increase or decrease in one quantity leads to the proportional decrease or increase respectively in the other quantity, they are said to be in inverse proportion.

If y α (1/x), then y = k/x where k is again a constant of proportionality and a non zero constant.

Mixed Variation
In this we have both types of variations, direct and indirect.
If x α (1/y) and x α z, then combining them we can say
x α (z/y) then x =((k*z)/y).

Some more results on proportions:
If a α b and b α c then a α c
If a α b and c α d then ac α bd
If a α b then ap α bp, where b is the constant of proportionality.
If a α b then a^n α b^ n.
If a α b and c α b then (a (+, -) c ) α b .

Application of proportion in similar triangles
Similar triangles have a property that the corresponding side are in proportion. If ΔABC is similar to ΔXYZ, then corresponding sides are
AB -> XY
BC -> YZ
CA -> ZX
These sides are in proportion would mean

If we know 4 of these sides, we can calculate the remaining two sides using the two equations which can be formed.