## 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: