68. Divide 71236yds. 3qr. 2n, by 1234.

yd. gr. n. yd. qr. n.

1234)71236 3 2(57 2 3 Quotient.

6170

9536

92. Divide 177kild. 1fir. 6gal. 1qt. by 65.

4gal. 1qt.

Quot. 2kild. Ifir.

124. PROMISCUOUS EXAMPLES FOR PRACTICE.

93. If I pay 9l. 9s. for 8 lb. of tea, what is the price per lb.? Ans. 11. 3s. 7d..

94. Paid 131. 18s. 8d. being the week's wages of 11 carpenters; what sum did each receive? Ans. 11. 5s. 4d.

95. Bought 12 pigs for 10l. 12s. 6d. what is the value of each? Ans. 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. 2l. 17s. 9d..

97. If 17 gallons of brandy cost 18l. 18s. 3d. how much is that per gallon? Ans. 11. 2s. 3d.

98. A club of 39 persons divide a lottery-prize of 20000l. equally; how much does each receive? Ans. 5121. 16s. 4d.,

27 rem.

99. A farm of 173 acres was reaped by 72 persons; how many acres is that apiece? Ans. 2a. 1r. 24p. 32 rem.

100. Suppose 7 puncheons of rum are just sufficient to serve 123 sailors during a voyage, how much may each man drink? Ans. 4gal. 3qt. 15 rem.

101. 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. 2l. 14s. 9d.4, 50 rem.

103. If 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 1201. how much will be the charge per dozen, supposing a dozen equal to 3 gallons? Ans. 21. 178. 1d., 36 rem.

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

125. The following questions require both multiplication and

RULE. To multiply by by, divide by 4; and to

division.

you must divide by 2; to multiply multiply by, 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.ż.

Multiply the top line by 3, divide it by 2, and add both results together.

107. What will 84 cwt. of cheese cost, at 41. 4s. 6d. per cwt.? Ans. 341. 17s. 1d..

Multiply by 8, divide (the top line) by 4, and add both results together.

108. What will 124 dozen of wine cost, at 27. 10s. per dozen ? Ans. 31l. 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

110. What must be given for 1174 lb. of tea, at 12s. 6d. per lb. ?

Ans. 731. 5s. 7d.4.

111. If a gallon of brandy cost Il. 3s. 6d. what cost 294 gallons ? Ans. 341. 7s. 4d..

112. What will 21 yards of lace cost, at 17. 1s. 6d. per yard? Ans. 231. 7s. 7d.4.

113. What will 374 lb. of nutmegs cost, at 11. 4s. 8d. per lb. ? Ans. 461. 58.

114. What will 87 gallons of oil cost, at Ss. 6d. per gallon? ns. 371. 58. 10d.

115. Required the value of a parcel, containing 567 hundred f Whitechapel needles, at Is. 8d. per hundred. Ans. 471, 5s. 5d.

PROPORTION.

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

:

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

eaches from three numbers given to find a fourth, which (when me three numbers are properly arranged) will be as great when ompared with the second, as the third is when compared with he first; so that, if the third be greater than the first, the fourth ill also be greater than the second; and if the third be less han the first, the fourth will, in like manner, be less than the cond.

128.1 RULE I. Examine the question carefully, and when you

* The comparison of one number to another is called their ratio; and when four given numbers the first has the same ratio to the second which the aird has to the fourth, these four numbers are said to be proportionals. Hence it appears, that ratio is the comparison of two numbers, but proporon is the equality of two ratios: we cannot then with propriety talk of the roportion of one number to another, nor confound the terms, as some authors ave done. The name Proportion comes from the Latin pro and portio. 1 The fundamental principle of the rules of Proportion is this, namely, If ur numbers are proportionals, the product of the two extreme terms is equal the product of the two means. Thus, since 2 is as great when comred with 3, as 4 is when compared with 6, 2 has the same ratio to 3 that has to 6; and consequently these four numbers are proportionals, that is, 4:36. Now the product of the extremes equals the product of the means, namely, 2 × 6=4 × 3; and since these products are equal, we are at berty to substitute one product for the other. And further, if any product e 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

101. 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. If 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 1201. how much will be the charge per dozen, supposing a dozen equal to 3 gallons? Ans. 21. 178. 1d., 36 rem.

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

125. The following questions require both multiplication and

RULE. To multiply by by, divide by 4; and to

division.

you must divide by 2; to multiply multiply by, 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 84 cwt. of cheese cost, at 41. 4s. 6d. per cwt.? Ans. 34l. 17s. 1d..

Multiply by 8, divide (the top line) by 4, and add both results together.

108. What will 124 dozen of wine cost, at 27. 10s. per dozen? Ans. 31l. 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

yard? Ans. 4l. 6s. Od.ş.

yards of cloth, at 3s. 4d. per

110. What must be given for 117 lb. of tea, at 12s. 6d. per lb. ?

Ans. 731. 5s. 7d..

111. If a gallon of brandy cost 11. 3s. 6d. what cost 294 gallons?

Ans. 341. 7s. 4d..

112. What will 21 yards of lace cost, at 11. 1s. 6d. per yard? Ans. 231. 7s. 7d.4.

113. What will 374 lb. of nutmegs cost, at 11. 4s. 8d. per lb. ? Ans. 461. 58.

114. What will 87 gallons of oil cost, at Ss. 6d. per gallon? Ans. 371. 58. 10d.

115. Required the value of a parcel, containing 5671⁄2 hundred of Whitechapel needles, at 1s. 8d. per hundred. Ans. 47l. 5s. 5d.

PROPORTION.

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. 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.

1 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, 2:4:36. Now the product of the extremes equals the product of the means, namely, 2 x 6=4 x 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 and the fourth: now 4 and 3