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123. When the divisor is any number greater than 12.

Rule. Divide the highest denomination, and place the quotient on the right hand, exactly like common Long Division. Reduce the remainder (if any) to the next lower denomination, taking in the figures which are of the same denomination. Divide this number, the quotient will be of the same denomination with it. Reduce the remainder, take in, divide, &c. until the whole is finished.

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OPERATION. - L. s. d. L. s. d. 215)12345 6 73 (57 8 4; 1075 1595 - 1505 90 JEarplanation. 2O Having divided the pounds, the quotient Tsog is 57, and the remainder 90; this latter I o multiply by 20, and take in the 6; I then 1720 divide the result, viz. 1806, and the quotient 86 8 is shillings; the remainder 86 I multiply 12 by 12, and take in the 7; the result, viz. - 1039, I divide, and the quotient 4 is pence;. 1039 the remainder 179 I multiply by 4, and take 86O in the 3; the result I divide, and the quo. Tog tient 3 is farthings. 4 719 645

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89. Divide 391. 1s. 5d. by 31. Quot. 11. 5s. 2d.;. 90. Divide 412l. 1s. 9d. by 111. Quot. 31. 14s. 3d. 91. Divide 97.531. 14s. 8d., by 2345. Quot. 4l. 3s. 2d.; 92. Divide 177kild. Ifir. 6gal. 1gt. by 65. Quot. 2kild. Ifir. 4gal. lgt. 124. ProMiscuous ExAMPLEs for PRACTICE. 93. If I pay 91.9s, for 8 lb. of tea, what is the price per lb. ? 4ns. 11. 3s. 7d.4. 94. Paid 131. 18s. 8d. being the week's wages of 11 carpenters; what sum did each receive 2 Ans. 11. 5s. 4d. 95. Bought 12 pigs for 10l. 12s. 6d. what is the value of each? 4ns. 17s. 8d.:#. 96. Fourteen gentlemen hire a yacht or pleasure-boat, the expences of which will amount to 40l. 9s. 4d.;; what will each have to pay Ans. 21. 17s. 9d.4. 97. If 17 gallons of brandy cost 18l. 18s. 3d. how much is that per gallon 2 Ans. 1.l. 2s. 3d. 98. A club of 39 persons divide a lottery-prize of 20000l. equally; how much does each receive Ans. 512l. 16s. 4d.4, 27 rem, 99. A farm of 173 acres was reaped by 72 persons; how many acres is that apiece? Ans. 2a. 1r. 24p. 82 rem. 100. Suppose 7 puncheons of rum are just sufficient to serve 123 sailors during a voyage, how much may each man drink? 4ns. 4gal. 34t. 15 rem. WOL. I. I

IOI. If five thousand fathoms of rope be made up into coils of 117 fathoms each, how many coils will there be? Ans. 42 coils and S6 fathoms over.

102. How much can a person, whose income is a thousand a year, afford to spend per day? Ans. 21. 14s. 9d.;, 50 rem.

103. Ho a greyhound in going over a mile of ground make 1537 leaps, what is the length of each leap? Ans. 3f. 5in. 343 rem.

104. If a pipe of wine cost 120l. how much will be the charge per dozen, supposing a dozen equal to 3 gallons? Ans. 21. 17s. ld.4, 36 rem.

105. A silversmith, out of 23lb. 9oz. 6dwt. of silver, made 9 dozen of spoons; required the weight of each 2 Ans. 202. 12dwt. 20gr.

I25. The following questions require both multiplication and division. Rule. To multiply by + you must divide by 2; to multiply by 4, divide by 4; and to multiply by 4, divide by 2, and that quotient by 2, and add both quotients together. 106. What is the value of 34 lb. of tea, at 12s. 9d per lb. ? Ans. 21.4s. 7d.4. Multiply the top line by 3, divide it by 2, and add both results together. 107. What will 8+ cwt. of cheese cost, at 4l. 4s. 6d. per cwt. 2 Ans. 34l. 17s. 1d.4. Multiply by 8, divide (the top line) by 4, and add both results together. 108. What will 123 dozen of wine cost, at 21. 10s. per dozen : Ans. 311. 17s. 6d. Multiply by 12, divide (the top line) by 2, and this last result by 2, then add all the three results together. 109. Required the value of 25+ yards of cloth, at 3s. 4d.) per yard 2 Ans. 41.6s. Od.#. 110. What must be given for 1173 lb. of tea, at 12s. 6d. per lb. ? Ans. 731. 5s. 7d.). * 111. If a gallon of brandy cost 11. 8s. 6d. what cost 294 gallons 2 Ans. 34l. 7s. 4d.g. 112. What will 214 yards of lace cost, at ll. 1s. 6d. per yard 2 Ans. 231. 7's. 7d.;.

113. What will 373 lb. of nutmegs cost, at ll. 4s. 8d. per lb. ? Ans. 46l. 5s.

114. What will 873 gallons of oil cost, at 8s. 6d. per gallon 2 Ans. 37 l. 5s. 10d;.

115. Required the value of a parcel, containing 567, hundred of Whitechapel needles, at 1s. 8d. per hundred. Ans. 47 l. 5s. 5d.

P R O PORTION.

126. *Proportion, called also the Golden Rule, and the Rule of Three, teaches from three numbers given (whereof two are of the same kind) to find a fourth ; it consists of two branches, viz. The Rule of Three Direct, and The Rule of Three Inverse.

127. DIRECT PROPORTION, or, THE RULE OF THREE DIRECT,

teaches from three numbers given to find a fourth, which (when the three numbers are properly arranged) will be as great when compared with the second, as the third is when compared with the first; so that, if the third be greater than the first, the fourth will also be greater than the second; and if the third be less than the first, the fourth will, in like manner, be less than the second. 128. 1 Rule I. Examine the question carefully, and when you

* The comparison of one number to another is called their ratio; and when of four given numbers the first has the same ratio to the second which the third has to the fourth, these four numbers are said to be proportionals. Hence it appears, that ratio is the comparison of two numbers, but proportion is the equality of two ratios: we cannot then with propriety talk of the proportion of one number to another, nor confound the terms, as some authors have done. The name Proportion comes from the Latin pro and portio. * The fundamental principle of the rules of Proportion is this, namely, If four numbers are proportionals, the product of the two extreme terms is equal to the product of the two means. Thus, since 2 is as great when compared with 3, as 4 is when compared with 6, 2 has the same ratio to 3 that 4 has to 6; and consequently these four numbers are proportionals, that is, 2 : 4 : : 3 : 6. Now the product of the extremes equals the product of the means, namely, 2 × 6=4x 3; and since these products are equal, we are at liberty to substitute one product for the other. And further, if any product be divided by one of its factors, the quotient will evidently be the other. These particulars being premised, the rule will be easily accounted for as follows. Let the three terms 2:4:: 3 be given to find the fourth; now 4 and 3

have discovered the three numbers or terms contained in it, you will find that two of them are of the same kind, and that the remaining term is of the same kind with the fourth term, or answer required. Also, of the two that are alike, one will be a term of supposition, and the other a term of demand.

II. State the question, that is, place the three given terms in a row from left to right; let the odd term (which is of the same kind with the answer) stand in the middle, and the two remaining terms (which are both of one kind) in the first and third places, observing to put the term of supposition in the first (or left hand) place, and the term of demand in the third; and place two dots vertically between the first and second terms, and four dots in form of a square between the second and third.

III. Reduce the first and third terms to the same denomimation, (if they are not so already,) and if either or both of them consist of different denominations, both must be reduced to the lowest mentioned in either. Likewise the second term must be reduced to the lowest denomination mentioned in it.

IV. Multiply the seeond and third terms together, and divide the product by the first; the quotient will be the fourth term, or answer, in the same denomination into which the second term was reduced.

are the two means, and their product, viz. 4 x 3, equals the product of the extremes, and may be therefore taken for it: but we have one of the extremes, viz. 2, given; wherefore if the said product be divided by 2, the quotient will

4 × 3

be the other extreme, that is, = 6, the other extreme or fourth term

required : and the same may be shewn in every other case. "Wherefore, in the Direct Rule of Proportion, if the second and third terms be multiplied together, and the product divided by the first term, the quotient will be the answer; which is the rule. To make the rule perfectly clear, one or two more particulars will require an explanation. If four numbers are proportionals, that is, if the first be to the second as the third to the fourth, then will the first be to the third as the second to the fourth ; and this accounts for the usual method of stating questions belonging to this rule: and the reason why the first and third terms must be reduced to the same denomination is, that numbers cannot be compared together except they are of the same denomination, as is evident; and hence it will readily appear, that the fourth term or answer will be in the same denomination with that to which the second was reduced.

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