19. A laborer dug 130r. 4yd. 21ft. of ditching at $2 per rod, for which he is to take $100 in cash, and wheat at 87 cents per bushel. To what quantity of wheat will he be en titled ? Ans. 259bu. 2pk. 4qt. 1.6'pt.

20. A grocer bought, at different times, 3cwt., 21gr. and 49lb. of soap, of which he has sold 2cwt. 1gr. Find the quantity remaining in lb. Ans. 2183lb.

21. A merchant had 3 pieces of cloth containing 29yd. 3qr. each, of which he has sold, to different persons, 51yd., 31qr., and 10yd. 11gr. Find the remainder in yards.

Ans. 72 yd.

22. A townsman who had a lot of ground containing 51A., sold to each of two persons 34R., at the rate of $100 per What is the remainder of it worth at the same rate?

acre.

Ans. $350.

23. A miller bought at one time 200bu. 3pk. of wheat, at another 313bu. 1pk., and at another 194bu. Having made 405bu. 1pk. of these purchases into flour, how much wheat has he still on hand? Ans. 302bu. 3pk. 24. A merchant had 5 pieces of cotton, containing 33yd. 3qr. each, which he sold in equal portions to ten customers. What quantity was bought by each customer?

Ans. 16yd. 31gr.

25. An agriculturist raised 500bu. 3pk. 4qt. of oats from one field, and he found the produce to be at the rate of 29bu. 1pk. 1qt. per acre. How many acres did the field contain? Ans. 17A. 16.8P.

26. A bought a tract of land containing 570A. 3R.; of which he sold to one person 90A. 1R. 20 P., and three times What is of the

that quantity to another, at $27 per acre. remainder worth at the same rate?

Ans. $3813.

27. How many yards of carpeting which is one yard in width, will be required to cover the floor of a room measuring 19ft. 6in. long, and 10ft. 6in. wide ?

Ans. 22yd. 3gr.

28. How many square yards of paper would be required to cover the walls of a room which is 10ft. 6in. high, and 56ft. in compass, deducting for doors, windows, &c. ? Ans. 59sq. yd. 8sq. ft.

29. How many cords of wood are there in a pile which is 19ft. 10in. long, 4ft. wide, and 3ft. 11in. high,—there being 128 cubic feet in a cord?

Ans. 2.427' cords. 3in. in length, 8ft. in

30. A certain reservoir is 10ft. breadth, and 4ft. 11in. in depth. How many barrels of water will it contain, allowing 282 cubic inches to a gallon, and 36 gallons to a barrel ? Ans. 68bar. 22gal. 1qt. 1.7'pt.

CHAPTER VII.

RATIO AND PROPORTION.

RATIO.

(135.) The RATIO of one quantity, called the antecedent, to another of the same kind, called the consequent, is the quotient of the former divided by the latter.

Thus the ratio of 12 to 4 is 3, since 12 is 3 times 4 ; and the ratio of 5 to 13 is, since 5 is five thirteenths of 13.

The antecedent and consequent together are called the terms of the ratio.

What is the ratio of 12 to 3?-which is the antecedent, and which the consequent? What is the ratio of 3 to 7 Of 15 to 3? Of 5 to 19 Of 20 to 4?

A colon (:) between two quantities denotes that the two quantities are taken as the antecedent and consequent of a ratio; thus 3: 5 denotes the ratio of 3 to 5.

Ratio of Monomials and Polynomials.

(136.) To find the ratio between two quantities, the antecedent and consequent must be taken in the same order of units.

The ratio of 2ft. to 5yd. is the ratio of 2ft. to 15ft., equal to 75.

What is the ratio of 3in. to 2ft.? (f 4yd. to 6ft.? Of 2hr. to Ida.?

Two quantities of different kinds can have no ratio to each other; as 2ft. and 5hr.

Direct and Inverse Ratio.

(137.) The direct ratio of the first of two quantities to the second, is the quotient of the first divided by the second; (135);

The inverse ratio of the first quantity to the second, is the direct ratio of the second to the first; thus the inverse ratio of 7 to 5 is 5.

What is the inverse ratio of 9 to 4?
What is the inverse ratio of 20 to 7?

Of 3 to 15?

Of 5 to 30?

Of 4 to 19?

Of 8 to 24?

The term ratio, when used alone, always means direct ratio.

Comparison of Fractions.

(138.) Two Fractions having a common denominator are to each other as their numerators; and two fractions having a common numerator are to each other inversely as their denominators.

Thus the ratio of to is the ratio of the numerator 2 to the

And the ratio of to 5 is

(135), which is

numerator 3.

= (135), which is the

inverse ratio of the denominator 8 to the denominator 9 (137).

(139.) One quantity varies directly as another when both increase or decrease together in the same ratio.

Thus the value of a given commodity varies directly as the quantity, since the value will be doubled, or trebled, &c., when the quantity is doubled, or trebled, &c.

(140.) One quantity varies inversely as another when one of them increases in the same ratio in which the other de

creases.

Thus the time in which a laborer will earn a given sum, varies inversely as his rate of wages, since one of these quantities will increase in the same ratio in which the other is diminished.

Will the value and the quantity of a piece of cloth vary directly or inversely with each other?—The time and the number of men required for a given work?—The number of men and the amount of provisions that will suffice them for a given time?—The weight of an article and the distance it may be carried for a given sum of money?—The length and the breadth of a garden containing a given area?—The weight of the five-cent loaf of bread and the price of flour?

PROPORTION.

(141.) PROPORTION consists in an equality of ratios.

Four quantities are in proportion when the ratio of the first to the second is equal to the ratio of the third to the fourth. Thus the numbers 6, 3, 8, 4 are in proportion, since

=

The first and third terms are the antecedents, the second and fourth the consequents; the first and fourth are the two extremes, the second and third the two means.

The fourth term is called a fourth proportional to the other three taken in order; thus 4 is a fourth proportional to 6, 3, and 8.

What is the fourth proportional to 10, 2, and 15, to which 15 has the same ratio that 10 has to 2? proportional to 16, 4, and 20? To 4, 8, and 10? To 2, 1, and 10? To, 1, and 2?

that is, the number

What is the fourth

To 5, 20, and 12?