(16) A wheel makes 514 revolutions in passing over 1mi. 467 yds. 1 ft., what is its circumference? Answer: 4yds. 1ft.

(17) If a person complete a journey of 422 mi. 3fur. 38 po. in 37 days, what distance does he travel each day? Answer: 11 mi. 3fur. 14 po.

(18) If 8 packages of cloth, each consisting of 4 parcels, each parcel of 10 pieces, and each piece of 26 yards, cost £6656., what is the price of a yard? Answer: 16 shillings.

(19) If the clothing of 754 soldiers come to £3178. 118. 7 d., how much is that for each man? Answer: £4. 4s. 3 d.

(20) A vintner bought 138 gals. at 10s. a gallon, of which he retained 18 gals. for his own use: at what rate per gallon must he sell the remainder, that he may have his own for nothing?

Answer: 11s. 6d.

65. The multipliers and divisors in the last two rules have always been regarded as abstract numbers: and though it may not be generally possible to determine the product of two concrete quantities, the quotient of one concrete magnitude by another of the same kind will be an abstract number.

Ex. Find how often £37. 12s. 8d., is contained in £263. 8s. 11 d.

Here the dividend = 252910 farthings:

and the divisor 36130 farthings:

=

whence the quotient is found to be 7 by common division: or £37. 12s. 8d. being repeated 7 times, amounts to £263. 8s. 111⁄2d.

Hence one concrete magnitude may be a measure or a multiple of another of the same kind.

CHAPTER III.

THE RULE OF THREE,

SOMETIMES CALLED THE GOLDEN RULE.

66. DEF. THE object of the Rule of Three is, by means of three quantities given, to determine a fourth, which shall be the same multiple, part or parts of one of them, that one of the remaining quantities is of the other; and it therefore follows, that the operation by which this may be accomplished, will depend upon those of Multiplication and Division already considered. Ex. 1. If 1lb. of any commodity cost 3s. 4d., it is required to find the price of 12lbs.

Here it is evident that the required price will be the same multiple of 3s. 44d., that 12lb. is of 1lb., which may therefore be found by Compound Multiplication; thus,

and the same result may be obtained by means of a statement and operation in the following form:

Ex. 2. If 11 bushels of wheat cost £4. 2s. 11 d., what sum must be paid for 45 bushels?

In this instance, we have, by Division,

and then by Multiplication, the required price is obtained thus:

£ 16. 19. 4 = price of 45 bushels :

and we shall arrive at the same conclusion by working the question in a form similar to that of the last example: as,

and it is easily shown that this sum consists of the same multiple and part of £4. 2s. 11d., that 45 does of 11.

67. Proper attention to the principles employed in these two Examples, will enable us to embody their substance in the following general rule.

RULE OF THREE.

For the Statement. Of the three quantities proposed, put down as the last, that which is of the same kind, or under the same circumstances as the one required; and the greater or less of the two others in the second place, according as the required one ought, from the nature of the case, to be greater or less than the last; and the remaining one in the first place.

For the Operation. Reduce, if necessary, the first and second terms to the same denomination, and the third to the lowest denomination contained in it: multiply together the second and third terms thus reduced, and the quotient arising from the division of the product by the first, will be the quantity required, expressed in the denomination to which the last term was reduced: which may be had in other terms by the proper divisions or multiplications.

It is sometimes necessary to consider what preparation may be required before the rule is applied: and it is evident that when the statement is made, the first, and the second or third terms, may be divided by any factor common to them both, without affecting the result, inasmuch as no alteration is produced from Multiplication and Division by the same number.

Examples for Practice.

(1) Required the price of 450lbs., at 4s. 8id. a lb. Answer: £105. 18s. 9d.

(2) Find the amount of a servant's wages for 215 days, at 2s. 4d, a day.

Answer: £25. 6s. 13d.

(3) If 25 cwt. 2qrs. cost £7. 6s. 74d., how much is that for 1 cwt.?

Answer: 5s. 9d.

(4) Required the price of 4 cwt. 1qr. 4lbs. 8 oz., when 1lb. costs 7s. 10d.

Answer: £189. 3s. 114d.

(5) If 6yds. 3 qrs. cost 5s. 3d., how much will 78yds. 2qrs. cost, at the same rate?

Answer: £2. 17s. 2d.

(6) If an artificer earn £19. 1s. in 20 days; in what time will he earn £23. 16s. 3d.?

if he perform a persone walk (216) miles in 7 days, of 16
a / Answer: 25 days. wing day.

hours each; in how many days of 12 hours each can
he do the same?

Answer: 9 days 4 hours.

(8) If 17 ells. 3 qrs., each ell containing 5qrs., be bought for £6. 17s. 6d.: how much must be paid for 18 yds.?

Answer: £5. 12s. 6d.

(9) If 12 quarts of wine cost £2. 5s., it is required to find the price of 5 pipes.

Answer: £472. 10s.

(10) How much wheat can be purchased for £55. Os. 3d., at the rate of 6s. 9d. a bushel?

Answer: 20qrs. 2 bush.

(11) If a farm of 375 acres, be let for £401. 11s. Sd. a year, what is that for each acre?

Answer: £1. 1s. 5d.

(12) If lodgings be let at 13s. 6d. a week, what will the demand amount to for 273 days?

Answer: £26. 6s. 6d.

(13) Required the price of 36 cwt. 1qr., when 2cwt. 2qrs. 10lbs. cost £4. 7s. 9d.?

Answer: £61. 9s. 1d.

(14) If a servant's wages be £30. Os. 8d. a year, what will be his demand for a service of 338 days? Answer: £27. 16s. 3d.

(15) If a person can walk 3mi. 6fur. 25po. in an hour, in what time will he complete a journey of 99 mi. 4 fur. 10po.?

Answer: 26 hours.

(16) What is the cost of 19bar. 24 gals. 3 qts. 1 pt. of beer, at 3d. a quart?

Answer: £41. 7s. Old.