the proportion by composition would be,

and,

2 + 42: 8 + 16: 8;

2 + 4:4: : 8 + 16 : 16.

192. Quantities are said to be in proportion by division, when the difference of the antecedent and consequent is compared either with antecedent or consequent.

Thus, if we have the proportion,

39 12: 36,

193. Equi-multiples of two or more quantities are the products which arise from multiplying the quantities by the same number.

Thus, if we have any two numbers, as 6 and 5, and multiply them both by any number, as 9, the equi-multiples will be 54 and 45; for,

Also, m × A, and m × B, are equi-multiples of A and B, the common multiplier being m.

194. Two quantities A and B, which may change their values, are reciprocally or inversely proportional, when one is proportional to unity divided by the other, and then their product remains constant.

192. When are quantities in proportion by division?

193. What are equi-multiples of two or more quantities?

194. When are two quantities said to be reciprocally proportional ?

We express this reciprocal or inverse relation thus,

1

Αα

Β ́

in which A is said to be inversely proportional to B.

and by clearing the equation of fractions, we have,

BC AD.

That is: Of four proportional quantities, the product of the two extremes is equal to the product of the two means. This general principle is apparent in the proportion be tween the numbers,

which gives,

2 10 12 : 60,

2 × 60 10 x 12 120.

196. If four quantities, A, B, C, D, are so related to each other, that

That is: If the product of two quantities is equal to the product of two other quantities, two of them may be made the extremes, and the other two the means of a proportion.

195. If four quantities are proportional, what is the product of the two means equal to?

196. If the product of two quantities is equal to the product of two other quantities, may the four be placed in a proportion? How?

That is: If three quantities are proportional, the square of the middle term is equal to the product of the two extremes Thus, if we have the proportion,

That is: If four quantities are proportional, they will be in proportion by alternation.

197. If three quantities are proportional, what is the product of the extremes equal to ?

198. If four quantities are proportional, will they be in proportion by alternation?

A B C D, and A: B:: E: F

That is: If there are two sets of proportions having an an tecedent and consequent in the one, equal to an antecedent and consequent of the other, the remaining terms will be proportional.

If we have the two proportions,

26824, and 2: 6:: 10: 30,

we have, by dividing 1 by each member of the equation,

A

=

C

B D'

and consequently, B: A:: D: C.

199. If you have two sets of proportions having an antecedent and consequent in each, equal; what will follow?

200. If four quantities are in proportion, will they be in proportion when taken inversely?

That is: Four proportional quantities will be in proportion, when taken inversely.

To give an example in numbers, take the proportion,

A B C D, gives, A x D = BX C.

:

To each member of the last equation add B x D. We shall then have,

(A + B) × D = (C + D) × B;

and by separating the factors, we obtain,

A + B : B : : C + D : D.

If, instead of adding, we subtract B x D from both members, we have,

That is: If four quantities are proportional, they will be in proportion by composition or division.

Thus, if we have the proportion,

9: 27: 16: 48,

201. If four quantities are in proportion, will they be in proportion by composition? Will they be in proportion by division? What is the difference between composition and division?