# Proportion Properties

## 1st property

In a proportion, the sum of the first two terms is for the 2nd (or 1st) term, just as the sum of the last two terms is for the 4th (or 3rd).

Demonstration:Consider the proportions: and Adding 1 to each member of the first proportion, we get:  Doing the same in the second proportion, we have:  Example:

• Determine x and y in proportion , knowing that x + y = 84.
Solution: Like this: x + y = 84 => x = 84-y => x = 84-48 => x = 36.

Soon, x = 36 and y = 48.

## 2nd property

In a proportion, the difference of the first two terms is for the 2nd (or 1st) term, just as the difference of the last two terms is for the 4th (or 3rd).

Demonstration:Consider the proportions: and By subtracting 1 from each member of the first proportion, we get:  Doing the same in the second proportion, we have  (Multi 2 members by -1) Example:

• Knowing that x-y = 18, determine x and y in proportion .
Solution:

For the 2nd property, we have to: x-y = 18 => x = 18 + y => x = 18 + 12 => x = 30.Thus x = 30 and y = 12.

## 3rd property:

In a proportion, the sum of the antecedents is for the sum of the consequents, just as each antecedent is for its consequent.

Demonstration:Consider the ratio: By exchanging the means we have: Applying the 1st property, we get: By exchanging the means, we finally get: ## 4th property:

In a proportion, the difference of the antecedents is for the difference of the consequents, just as each antecedent is for its consequent.

Demonstration:Consider the ratio: By exchanging the means we have: Applying the 2nd property, we get: By exchanging the means, we finally get: Example:

• Knowing that a-b = -24, determine The and B in proportion .
Solution:
For the 4th property, we have to:  ## 5th property:

In a proportion, the product of the antecedent is to the product of the consequent, just as the square of each antecedent is to the square of its consequent.

Demonstration:Consider the ratio: Multiplying the two members by we have: Like this: Note: The 5th property may be extended for any number of reasons. Example: Next: Multiple Ratio