9. Find the sum of the n first terms of the progression of uneven numbers 1, 3, 5, 7, 9 ... Ans. S=n2.

10. One hundred stones being placed on the ground in a straight line, at the distance of 2 yards from each other, how far will a person travel who shall bring them one by one to a basket, placed at 2 yards from the first stone? Ans. 11 miles, 840 yards.

Geometrical Proportion and Progression.

144. Ratio is the quotient arising from dividing one quantity by another quantity of the same kind. Thus, if the numbers 3 and 6 have the same unit, the ratio of 3 to 6. will be expressed by

6

2.

3

And in general, if A and B represent quantities of the same kind, the ratio of A to B will be expressed by

B
Α

145. If there be four numbers

2, 4, 8, 16,

having such values that the second divided by the first is equal to the fourth divided by the third, the numbers are

QUEST. 144. What is ratio? What is the ratio of 3 to 6? Of 4 to 12?

said to be in proportion. And in general, if there be four quantities, A, B, C, and D, having such values that

then A is said to have the same ratio to B that C has to D; or, the ratio of A to B is equal to the ratio of C to D. When four quantities have this relation to each other, they are said to be in proportion. Hence, proportion is an equality of ratios.

To express that the ratio of A to B is equal to the ratio of C to D, we write the quantities thus:

A B C D;

and read, A is to B as C to D.

The quantities which are compared together are called the terms of the proportion. The first and last terms are called the two extremes, and the second and third terms, the Thus, A and D are the extremes, and B and

two means.

C the means.

146. Of four proportional quantities, the first and third are called the antecedents, and the second and fourth the consequents; and the last is said to be a fourth proportional to the other three taken in order. Thus, in the last proportion A and C are the antecedents, and B and D the consequents.

QUEST.-145. What is proportion? How do you express that four numbers are in proportion? What are the numbers called? What are the first and fourth called? What the second and third?-146. In four proportional quantities, what are the first and third called? What the second and fourth?

147. Three quantities are in proportion when the first has the same ratio to the second that the second has to the third; and then the middle term is said to be a mean proportional between the other two. For example,

3 66 12;

and 6 is a mean proportional between 3 and 12.

148. Quantities are said to be in proportion by inversion, or inversely, when the consequents are made the antecedents and the antecedents the consequents.

Thus, if we have the proportion

3 6 8 16,

the inverse proportion would be

6 3 16 8.

149. Quantities are said to be in proportion by alternation, or alternately, when antecedent is compared with antecedent and consequent with consequent.

Thus, if we have the proportion

QUEST.-147. When are three quantities proportional? What is the middle one called?-148. When are quantities said to be in proportion by inversion, or inversely?-149. When are quantities in proportion by alternation?

150. Quantities are said to be in proportion by composition, when the sum of the antecedent and consequent is compared either with antecedent or consequent.

Thus, if we have the proportion

151. 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

152. 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

6×9=54, and 5x9=45.

QUEST.-150. When are quantities in proportion by composition? -151. When are quantities in proportion by division?-152. What are equi-multiples of two or more quantities?

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

153. Two quantities, A and B, are said to be reciprocally proportional, or inversely proportional, when one increases in the same ratio as the other diminishes. When this relation exists, either of them is equal to a constant quantity divided by the other.

Thus, if we had any two numbers, as 2 and 4, so related to each other that if we divided one by any number we must multiply the other by the same number, one would increase just as fast as the other would diminish, and their product would be constant.

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 between the numbers

2 10 12 : 60,

which gives

2 × 60=10×12=120.

QUEST.-153. When are two quantities said to be reciprocally proportional?-154. If four quantities are proportional, what is the product of the two means equal to ?