57. If 20 masons, in 12 days of 8 hours each, build a wall 360 ft. long, in how many days of 10 hours each ought 16 masons to build 500 ft. of the same wall?

58. In the last question let the days be, in both cases, of 10 hours; but let the first wall be 6 ft. high and 2 ft. thick, and the second 5 ft. high and 3 ft. thick. Required the time taken by the party of 16 masons to build this wall, the other conditions remaining the same?

59. If the interest (or compensation allowed for the loan) of 100£ for a year be 5£, what interest must be allowed for the loan of £320 15s. 6d. for the same time and at the same rate?

60. If the interest of 100£ for 1 year be 3£ 5 fl., what is the interest of 450£ 7 fl. 5 c. for 5 years, at the same rate?

61. If 100£ in 1 year produces 5£ of interest, what is that sum which in 7 years produces 262£ 5 fl. of interest, at the same rate?

62. The amount of 100£, with 4£ of interest added at the end of a year is 104£. What sum with its interest at the rate of 4£ per 100£ (or 4 per cent.) will come to 200£ at the end of a year?

63. What must be paid for 1000£ 3 per cent. consols, at 92£ 3 fl. 7 c. 5 m. (£92.375) for every 100£ of stock?

64. A and B have a joint capital of 900£, of which 500£ belong to A and 400£ to B: they gain 100£: what share of this gain ought each to have?

65. If a wine merchant pays 70£ for a pipe of sherry, at what rate per gallon must he sell it in order to have a profit of 10£ by the transaction?

66. The same things being supposed, at what price per gallon must the sherry be sold to afford a profit of 10£ per cent.?

67. How many gallons of port, at 4 fl. 2 c. 5 m. (*425£) per gal., are worth 500 yd. of broad cloth at 6 fl. 1 c. 2.5 m. (6125£) per yd.?

68. How many American dollars at 4s. 21d. each ought to be received in exchange for 100£ ?

69. If 5 apples are worth 3 oranges and 4 apples are worth 7 pears, how many oranges are worth 360 pears?

70.

A grocer mixed 4 cwt. of sugar at 56s. per cwt., 7 cwt. at 43s. per cwt and 5 cwt. at 37s. per cwt.; what are 2 cwt. of this mixture worth?

Questions like those from that numbered 60. to the end of the examples in Proportion, are, in most treatises of Commercial Arithmetic, given as examples under other rules-all of which, however, depend on Proportion.

Thus the solution of Example 60. is made to depend on the Rule of Simple Interest; that of 62. on Discount; of 63. on the Sale of Stocks; of 64. on Partnership or Fellowship; of 65. and 66. on Profit and Loss; of 67. and 69. on Barter; of 68. on Exchange; and of 70. on Alligation.

The solutions of these questions are subjoined; with obser

vations, where they seem necessary; and, in some instances, special rules and examples.

171. Example 60., considered as a question in Compound Proportion, is resolved thus

In questions of this kind the money lent is called the Principal; the compensation, at the rate of so much yearly for 100%., is called the Rate per cent. per annum; and the sum made up of the principal and interest together is called the Amount.

The interest of a sum of money, for a given time, and at a given rate per cent., is found by the following rule (which is merely a translation into words of the preceding solution):

Multiply the principal by the rate per cent. and time (in years, or parts of a year), and divide the product by 100.

To find the amount, add the interest to the principal.

Commission, or the allowance given by a merchant to an agent who buys or sells his goods, is allowed at so much per cent., and computed like simple interest.

Insurance, or money paid by one party to another who engages, in consideration of such payment, to make good to the former any damage which his property may accidentally suffer by fire, shipwreck, &c., is computed at so much per cent. on the value of the property insured, in the same manner as simple interest.

Examples. Find the simple interest of —

1. 346£ 3 fl. 5 c. at 4 per cent. per annum, for 1 yr.
2. 715£ 6 fl. 2 c. 5 m. for 1 yr. at 4 per cent. per an.

3. 476£ 8 fl. 5 c. for 3 yr. at 5 per cent. per an.

4. 57£ 4 fl. 8 m. for 5 yr. at 3 per cent. per an.

5. 154£ 7 fl. for 4 yrs. at 41 per cent. per an.

6. What is the commission on the sale of 1000£ worth of goods at the rate of £ per cent.?

7. What is the commission on the sale of 475£ 8 fl. 5 c. worth of goods at the rate of per cent.?

8. What is the insurance on a ship, value 5000£, for 3 months, at 14£ per cent.?

172. If the interest of money is added, as it becomes due, to the principal, and the amount is continually made a new principal, the result at the expiration of a given time is the amount,

at Compound Interest, of the original principal for that time, and the difference between the amount and the said principal is the compound interest of the principal for that time.

The following rule for the calculation of compound interest is an immediate consequence of this definition :

Find the amount of the given principal for the time of the first payment by the rule of Simple Interest; consider this amount as the principal for the second payment; find the amount of the second principal as before; and thus continue making always the last amount the principal of the next payment. The last result is the compound interest required, together with the given principal.

This rule may be given in a different form.

Let the rate per cent. for each time of payment be r; then 100+r is the amount of 100£ for the time of the first payment, and 100£ 100 +r£::1£ 100+r

100

Again,

r

£=1+ =the amount of 1£ for the time of the first payment.

100

100£ 100 +r£ :: 1+

(1 + 100)2

2=the amount of 1£ for the time of the second payment.

In the same manner it is found that

4 the amount of 1£ for the time of the

t

fourth payment; and, generally, (1+) the amount of l£ for the

number of payments denoted by t.

Now the amount of different principal sums (the rate and time being the same) are proportional to these sums: the amount of p£ is consequently equal to p times the amount of 1£. The amount of p£ for t times of payment

(1+). Whence if the

100

at r£ per cent. compound interest is therefore Р amount of 1 for the time of the first payment is raised to a power indicated by the number of payments, and this result is multiplied by the principal, the last result is the compound interest together with the given principal.

Ex. Required the compound interest of 250£ for 3 yr. at 5 per cent. per an.? The first proportion is 100£: 250£::5£: interest for 1 year. Omitting this proportion, the calculation may be made as follows:

250£x=250£ × 12£ 10s. or 12.5£=interest for 1 yr.
250£ + 12·5£=262.5£=principal for the second payment.
262.5£5=262·5 × =13·125£=interest for second year.
262.5+13·125£=275.625£=principal for the third payment.

275.625€ 50275.625 × 13-78125£=interest for third year. ... 275.625£ + 13·78125£=289-40625£= amount at end of third year. And 289-40625£ −250£=39•40625£=compound interest required. 39-40625£=39£ 4 fl. 6.25 m, = £39 8s. 1d.

Otherwise by the second rule,

r

18

い

100) = 1·051; t=3; p
p(1.

Τ t

+

250 x 1.053

100/

p=250£; r=5; (1+100)

250 × 1.053 250 × 1.157625=289-40625£, as before.

Examples. Find the compound interest of

1. 50£ for 5 yr. at 5 per cent.

5.

6.

100£ for 3 yr. at 5 per cent., the interest payable half yearly.

100£ for 11 yr. at 5 per cent., the interest payable quarterly. 7. 478£ 1 fl. 5 c. 2 m. for 2 yr. at 3 per cent.

8. 296£ 8 fl. 6 c. for 31 yr. at 4 per cent., the interest payable half yearly.

173. In Question 62., the amount of an unknown principal for a certain time at a known rate of interest is given to find the principal or the interest thereon.

The unknown principal is called the Present Worth of the proposed amount, and the difference between the principal and amount, the Discount.

Quest. 62. is resolved by Proportion as follows:

104£ 100£ :: 200£ :

200 x 100 2500
£
104
13

£=192€.

Whence the present worth of 200£ payable in a year, at 4 per cent. discount, is 192£.

The discount=200£-192£ or 7£.

Examples. What is the present worth of

1. 50£ payable in 3 yr. at 5 per cent. discount?

2. 400£ payable in 8 months, discount being allowed at the rate of 4 per cent. per annum?

3. 256£ 9 fl. 6 c. payable in 11 yr. discount being allowed at the rate of 31 per cent. per annum?

4. What is the discount on a bill for £217 4s. 6d. due 5 months hence, allowing discount at 5 per cent.?

174. Sale of Stock.-The national debt of Great Britain, under the name of Funds or Stocks, bears different rates of interest. The public creditor who owns, for example, 1000£ of a 3 per cent. stock, is entitled yearly to receive 30£ of interest, which is payable in two half-yearly dividends at the

Bank of England. But he cannot, at any time he pleases, demand payment of his 1000£ of principal. He may, however, sell his 1000£ of stock. The market value of 100£ is variable. Supposing it (as now) 91£, the value in money of 1000£ 3 per cent. consols is given by the proportion 100£ : 91.5£:: 1000£ 915£, the value required. In like manner

the answer to Quest. 63. is given by the proportion 100£: 92.375£1000£ : 923·75£=923£ 7 Al. 5 c.

Examples.

1. What must be paid for 750£, three per cent. consols, at 973 per cent.? 2. What must be paid for 500£ bank stock at 2153£ per cent.?

175. Partnership or Fellowship.- Question 64. is resolved as a question of distributive proportion thus

If in the preceding question A's share of the joint capital were employed for 3 months and B's for 4 months, to apportion the profits it is assumed that,

500£ for 3 months=1500£ for 1 month, and

400£ for 4 months 1600£ for 1 month.

=

Then 1500+ 1600£ : 1500£ : 100£ : 150o=48}}£=A's share.
1500+1600£ : 1600£: 100£: 1600=51}£=B's share.

Whence to resolve questions in partnership, time being taken into consideration, multiply the contribution of each partner by the time of its continuance in the common stock, and divide the gain into shares proportional to these products.

Examples.

1. Two persons, A and B, make a joint stock. A puts in 496£ for 2 months, and B 620£ for 3 months. They gain 456£. What are their respective shares of this gain?

2. Three persons, A, B, C, hold a pasture in common, for which they are to pay 30£ a year. Into this pasture A put 7 oxen for 3 months, B 9 oxen for 5 months, and C 4 oxen for 12 months. How much must each pay of the rent?

176. Profit and Loss.

Solutions of Quest. 65. and 66.—

70£ price of wine + 10£ profit=80£=price at which, by the question, the pipe of sherry ought to be sold. The price of 1 gal. at this rate for 1 pipe or 126 gal. is required.

.. 126 gal. : gal. 80£ £=12s. 8d.=price of gal,

80 T26