15. What quantity of shalloon, that is be sufficient to line 7 yards of cloth, 1 15 yards. of a yard wide, will yards wide? Ans. 16. If 2 yards, 14 yard wide, be sufficient to make a coat, how much will it require of cloth that is of a yard wide to make the same kind of garment? Ans. 4 yd. 3 qr. 17. How many pieces of cloth, at $18 per piece, are equal in value to 224 pieces, at $12 per piece? Ans. $15015. 18. A merchant exchanged 73 cwt. of sugar, at 73 pence per pound, for tea at 94 shillings per pound; how many pounds of tea did he receive? Ans. 6034 lb. 19. If 8 men can perform a piece of work in 63 hours, in what time will 20 men do the same? Ans. 2 hours, 40 min utes. 20. How many yards of cloth, of a yard wide, will line 20 yards, of a yard wide? Ans. 12 yards. 21. How many pieces of cloth, at 18 shillings per yard, are equal in value to 350 pieces, at 124 shillings per yard? Ans. 2412 pieces. 22. Lent a friend $72 for 8 months; what sum must he lend me for 2 years, to balance the favor? Ans. $21.233.+ The following sums properly belong to Compound Proportion. They may be solved either by canceling, by analysis, or by the common rule of Compound Proportion. Ex. 23. If of a yard of cloth, which is of a yard wide, cost of a pound sterling, what is the value of of a yard, that is 12 yard wide? Analysis: x=3, the fraction of a square yard purchased, which cost of a pound sterling. Therefore, 215, the value of part of a square yard, and 3×32=16, the price of 1 yard. x=35, the quantity of which the price is required. Therefore, 604x35=2240-3 of 1 £. Ans. 13 s. 4 d. 2. 5. 7. 4. 8 The same canceled, 5. 8. 4. 3. 7° To understand why the fractions and are inverted, it must be remembered that a fraction is divided by a fraction, by inverting the divisor and then multiplying numerators and denominators together. (See Sec. 7th, Introduction to Fractions.) of a £.13 s. 4 d. Ans. The above solved, 2. 5. 7. 4. & 24. If 9 men spend 12 £. in 27 days, what sum will 25 men spend in 40 days? 486 Analysis: 12.25 £., and 25-9-15 £. the money one man spends in 27 days; and 25 £.-27-256, the money spent by one man daily. Therefore, 256 £.×25-625 £. the money 25 men spend daily; and 625 £.x40-25000 £. the sum of money required, which reduced gives 51 £. 8 s. 955 pence, Ans. 486 The same stated for canceling, Canceled, 20 25. 25. 40 486 25. 25. 40 2. 9. 27 and 25 × 25×20=12500; and 27×9 =243; and 12500÷243=51 £. 8 s. 955 d. Ans. 25. If 18 persons consume 27 lb. of tea in one month, how much will 8 persons consume in six months? Ans. 4 lb. 26. If the tuition of 2 boys for of a year be 564 £., how much will be the tuition of 3 boys for 5 years? Ans. 600 £. 27. If 90 cwt. be carried 30 miles for $29, how many cwt. be carried 45 miles for $5 ? Ans. 12 cwt. may 28. If 10 persons drink 15 gallons of wine in one week, how much will 16 persons drink in 43 weeks? Ans. 1073 gallons. 29. If cwt. be carried 600 miles for $12, how far may of a cwt. be carried for $305? Ans. 988 miles. QUESTIONS.-When is an arithmetical question solved analytically? What is the general principle by which sums may be solved analytically? How are sums in Simple or Compound Proportion solved without canceling? How are they solved by canceling? What is the note ? CONJOINED PROPORTION. Conjoined Proportion consists of a comparison instituted between a series of terms bearing a certain relation to each other, as the coins, weights, and measures of different countries. The principle involved in this rule is the same as in Single and Compound Proportion. No farther explanation is therefore needed. RULE. Above a horizontal line near the left, place the demanding term; then below the line, place the term of the same name as the demand, with the term which it equals in value directly above it. Next, seek another term of the same name as the one last placed, and set it also below the line, with the one it equals in value also above it. Thus proceed to arrange the terms, making each term standing below the line of the same name as the preceding term standing above it. The product of the numbers standing above the line divided by the product of those standing below it, will give the required number. The numbers may of course be canceled as far as practicable, before multiplying and dividing. Ex. 1. If 100 lb. English make 90 lb. Flemish, and 22 lb. Flemish make 28 lb. Bologna, how many pounds English are equal to 56 lb. Bologna ? The demand obviously lies on the 56 lb. Bologna; therefore, 2. If 40 lb. at New York make 48 lb. at Antwerp, and 30 lb. at Antwerp make 36 at Leghorn, how many pounds at New York are equal to 144 at Leghorn ? 3. If 70 braces at Venice make 84 braces at Leghorn, and 12 at Leghorn make 7 American yards, how many braces at Venice are equal to 96 American yards? Ans. 1374. 4. If 24 lb. at New London make 20 lb. at Amsterdam, and 50 lb. at Amsterdam make 60 lb. at Paris, how many Ib. at Paris are equal to 40 lb. New London? Ans. 40 lb. 5. If 50 lb. at New York make 45 lb. at Amsterdam, and 80 lb. at Amsterdam, 103 lb. at Dantzic, how many lb. at Dantzic are equal to 240 lb. New York? Ans. 278. 6. If 24 braces at Leghorn be equal to 15 vares at Lisbon, and 45 vares at Lisbon be equal to 90 braces at Lucca, how many braces at Lucca are equal to 120 braces at Leghorn? Ans. 150 braces. QUESTIONS.—In what does Conjoined Proportion consist? How does the principle involved, compare with Simple and Compound Proportion? What is the rule for Conjoined Proportion? DISCOUNT. Discount is an allowance made for the payment of money before it becomes due. The present worth of any sum of money, payable at some future time without interest, is that sum which, if put at interest, would in the given time and rate per cent. amount to the whole debt. Discount is not, therefore, a deduction of the given per cent. from a hundred cents or a hundred dollars. If I have a claim upon an individual for $100, payable a year hence, and propose to allow him 6 per cent. discount for present payment, I must receive more than $100 - $6=$94; since $94 put on interest at 6 per cent. will not amount to $100 in the given time. The interest on $94 one year at 6 per cent. is $5.64; and $94+$5.64 $99.64, which is 36 cents less than the required sum, or $100. If, however, a person owe me $106, payable in one year without interest, and I propose to allow him the same discount for immediate payment, he must obviously pay me $100, since $100 in one year at six per cent. will amount to precisely $106. Hence, we learn that the ratio which any sum due a year hence without interest bears to its present worth, is as 106 to 100; or, what is the same thing, as $1.06 to $1.00, whenever the discount is at 6 per cent. If the rate per cent. be any other than 6, or the time more or less than one year, the ratio varies accordingly. Therefore, as the amount of $1 for the given time and rate per cent. is to $1, so is the given sum to its present worth. Ex. 1. What is the present worth of $450, due 2 years hence, 6 per cent. discount being allowed? The interest of $1 for 2 years at 6 per cent. is 12 cents, and consequently the amount of $1, for the same time, is $1.12. Therefore, 1.12:1:: 450: the required sum. And, since nothing is effected by multiplying by 1, the required sum is obtained by dividing $450 by $1.12. Hence, $150.00÷ $1.12 $401.785.+ Ans. From the above we derive the following rule: RULE.-Divide the sum on which the discount is to be made, by the amount of one dollar for the given time and rate per cent. 2. What is the present worth of $700, due 3 years hence, at 5 per cent. discount? The amount of $1 for 3 years at 5 per cent., is $1.15. Therefore, $700.00÷1.15 $608.695. + Ans. 3. Sold goods to the amount of $1200, on 6 months' credit. What is the present worth, allowing 8 per cent. discount? Ans. $1153.846.+ 4. What is the present value of a legacy of $2000, due 2 years hence, discounting at 5 per cent. per annum ? Ans. $1818.18.+ 5. What is the difference between the interest and discount on $600 for 12 years, at 5 per cent.? Ans. Interest $360; discount, $225; difference, $135. Note. To obtain the discount, subtract the present value from the sum due. 6. What is the discount on $300 for 60 days, at 6 per cent. per annum? Ans. $2.97. 7. What is the present value of $750, due 3 years hence, discounting at 4 per cent. per annum ? Ans. $657.894.+ 8. What is the discount on $500 for 2 years, at 9 per cent. per annum? Ans. $76.272.+ 9. What is the present value of 350 £., due 4 years hence, discounting at 4 per cent. per annum? Ans. 301 £. 14 s. 5 d. 3 qr. 10. What is the present worth of 672 £., due 2 years hence, discounting at the rate of 6 per cent. per annum? Ans. 600 £. 11. Bought goods to the amount of $820, on 6 months' credit. What ought I to have paid, if I had advanced the money on |