Art. 154. 14. Purchased a watch for $40, and sold it for $45; what per cent. did I gain? = $45— $40 $5, the gain, which is of the cost of the watch. If we change to hundredths, by annexing two decimal ciphers to the numerator and dividing this product by the denominator, we shall have 5.00÷40=.12 hundredths, the per cent. gained. Hence, when goods are purchased at one price and sold at another, we have the following rule for finding the per cent. which is gained or lost. RULE. Make the gain or loss the numerator of a vulgar fraction, and the first cost the denominator; this fraction will express what part the gain or loss is of the first cost; then change this fraction to hundredths; the number of hundredths will be the per cent. gained or lost. If there is a remainder, it will express the fraction of another hundredth, and must be written at the right of the number of hundredths. If the gain or loss is less than 1 per cent., it may be expressed by a fraction. 15. If $300 be paid for insuring a cargo of flour from Boston to Liverpool, valued at $15000, what per cent. is the insurance? Ans. .02 per cent. 16. If the price of flour be raised from $5.75 to $6.50 a barrel, on account of the scarcity in Europe, what per cent. is the rise? Ans. .13 per cent. 17. A bankrupt's debts amount to $25000; his property amounts to only $16250; what per cent. of his debts can he pay? Ans. .65 per cent. 18. A grocer bought a chest of tea at 35 cents a pound, and sold it at retail at 50 cents a pound; what per cent. did he gain? Ans. .42 per cent. 19. A speculator purchased a tract of new land, for which he paid $625; he afterwards sold it for $500; what per cent. did he lose? Ans. .20 per cent. 20. If I pay $18.75 for insuring $3000 on my house against fire annually, what per cent. do I pay? Ans. .00 of 1 per cent. 21. A merchant imported 5 pieces of broadcloth, which cost $4.624 a yard, and sold it at $5.25 a yard; what per cent. profit did he make? 22. A grocer purchased 1500 pounds of coffee, for which he paid $120, and sold it at 9 cents a pound; what per cent. did he gain? Art. 155. 23. A merchant sold a quantity of goods for $247.50, by which he gained 10 per cent. of the first cost; how much did the goods cost? 10 per cent. is $10 on $100, and $100+ 10 is $110; then $247.50 must be 188 of the first cost; hence, the first cost must be 18 of $247.50, and $247.50 × 118: first cost of the goods. = $225, the 24. A mechanic sold a new house for $8550, and gained 12 per cent. of what it cost him to build it; what did it cost to build it? Ans. $7600. 25. A merchant sold a piece of broadcloth containing 48 yards for $147.84, and lost 12 per cent.; what did the cloth cost him? 12 per cent. is $12 on $100, and $100-$12 is $88; then $147.84 must be of the cost; hence, the cost must be 10 of $147.84, and $147.84 X 10: = $168, the cost of the cloth. 26. An auctioneer sold 15 shares in the Western Railroad for $1275, by which the owner lost 15 per cent.; what was the cost of the 15 shares? Ans. $1500. 27. A merchant sold a quantity of goods for $22.50 more than he gave for them, and gained 7 per cent. of the first cost; what did the goods cost him, and for how much did he sell them? 7 per cent. is $72 on $100; hence, $22.50 must be 17580 of the first cost, and the first cost must be 1988o of $22.50, and $22.50 X 10000 $300, the cost of the goods. $300 +$22.50: =$322.50, the sum for which he sold them. = 28. A speculator sold a building-lot for $300 more than it cost him, and gained 20 per cent.; what did he pay for the lot, and for how much did he sell it? Ans. It cost $1500, and he sold it for $1800. 29. A trader sold a chaise for $15.50 less than it cost him, and lost 6 per cent.; what did the chaise cost, and for how much did he sell it? 30. When flour is sold at $4.62 a barrel, there is a loss of 12 per cent.; what will be the gain or loss per cent. when the same kind of flour is sold at $5.50 a barrel? 31. A merchant sold sugar at 8 dollars a cwt., and lost 10 per cent.; what will be the gain or loss per cent. when he sells the same kind of sugar at 9 dollars a cwt.? 32. A watchmaker sold a watch for $40, and lost 12 per cent.; for what number of dollars should he have sold the same watch to have made a profit of 12 per cent.? EQUALIZING OR AVERAGING THE TIME OF PAYMENTS. Art. 156. EQUALIZING or averaging the time of payments is finding a mean time for the payment of several debts due at different times, so that no loss shall be sustained by either debtor or creditor. Suppose A owes B $80 to be paid in 2 months, and $120 to be paid in 7 months; at what time may A pay B the whole sum of $200, and no loss of interest be sustained by either of them? $80 2 $160; hence, the int. of $80 for 2 mo. = the int. of $160 for 1 month. $120X7 $840; hence, the int. of $120 for 7 mo. = = the int. of $840 $1000 The interest on the two debts to the times of payment is equal to the interest of $1000 for 1 month, which is found by multiplying each debt by its time of payment, and adding the products. If the interest of $1 for 1000 months is $5, the interest of the sum of the two debts, which is $200, for go of 1000 months, which is 5 months, is also $5. Therefore, 5 months is the average or mean time for the payment of the sum of the two debts. PROOF. The interest of $80 for 2 months is $ .80. The interest of $120 for 7 months is $4.20. The interest of $200 for 5 months is $5.00. From the above illustration we obtain the following rule. RULE. Multiply each debt by the number of days, months, or years, between its date and the time it becomes due, and divide the sum of the products by the sum of the debts; the quotient will be the average or mean time for the payment of the sum of all the debts. 1. William owes 20 cents to 2. If I owe a man $200 to be be paid in 20 days, 30 cents to be paid in 2 months, $300 to be paid paid in 10 days, and 40 cents to be in 6 months, and $500 to be paid paid in 5 days; what is the aver-in 8 months, what is the average age or mean time for the payment or mean time for the payment of of the whole sum? the whole sum? 3. C owes D $1200, of which $200 is to be paid in 3 months, $400 in 6 months, and $600 in 9 months; what is the average or mean time for the payment of the whole Ans. 7 months. sum? 4. A merchant has due him $1200 to be paid in 30 days, $800 to be paid in 60 days, and $1000 to be paid in 90 days; what is the mean time for the payment of the whole sum? Ans. 58 days. If one of the payments is due on the day from which we begin to count the time, this payment must be added in finding the sum of the payments. 5. A owes B $1800, of which sum, $600 is due at the present time, $600 will be due in 4 months, and $600 in 8 months; what will be the average or mean time of payment? Ans. 4 months. 6. A merchant purchased goods to the amount of $2000, for which he agreed to pay $500 at the time he made the purchase, $500 in 30 days, $500 in 60 days, and $500 in 90 days; he afterwards proposed to pay the whole sum at one time. What will be the average or mean time for the payment of the whole amount? Ans. 45 days. This rule, which is universally adopted, is founded upon the supposition that the interest of the several debts which are due before the average or mean time of payment is equal to the interest of the several debts which are due after the average or mean time. The debtor, by keeping a sum of money after it is due, gains the interest of the sum for the time he keeps it; but the loss he sustains by paying a sum of money before it is due is equal to the discount of the sum for the time he pays it before it is due, which is less than the interest; therefore, the rule is not accurately true. The following rule will be found to be accurately true. Suppose A owes me $103, to be paid in 6 months, and $106 to be paid in 12 months; what will be the exact mean time for the payment of the whole sum? The present worth of $103 payable in 6 months is $100; the present worth of $106 payable in 12 months is $100. The sum of the present worth of the two debts is $200. The sum of the two debts is $209. In what time will $200 amount to $209, at 6 per cent. a year? = is $209-$200 $9. The interest of $200 for 1 year $12, and $9 ÷ $12—.75 of a year or 9 months, the exact mean time. Hence, the following rule. RULE. Find the present worth of each of the debts; then find the time at which the sum of these present worths will amount to the sum of the debts. The time thus found will be the true average or mean time. The time at which any given principal will amount to any required sum may be found by the following rule. RULE. Subtract the given principal from the required amount or sum, the remainder will be the interest of the given principal for the required time; then divide this interest by the interest of the given principal for one year, the quotient will be the time required. 7. A owes B $1976, of which $510 is to be paid in 4 months, $618 in 6 months, and the remainder in 12 months; what will be the true average or mean time for the payment of the whole amount at a single payment? Ans. 8 months. 8. A merchant purchased goods to the amount of $2033, of which $404 is to be paid in 60 days, $609 in 90 days, and the remainder in 120 days; what will be the true average or mean time for the payment of the whole amount? Art. 157. In finding the average or mean time for the payment of several sums due at different times, any convenient date may be taken from which to count the time. 9. Suppose a trader purchased goods as follows, on a credit of 60 days: May 1st, 1847, purchased to the amount of $ 50. June 1st, What was the average or mean time of payment, counting from the 1st of May? From 1st of May to 1st payt., 60 d's, $ 50 × 60= 3000. to 2d payt., 91 d's, $150 X 91 152: =30400. = 47050 400 =117 days. Hence, the average or mean time for the payment of the whole amount was 117 days, counting from the 1st of May, which was August 26th, 1847. 10. Purchased goods as follows, on a credit of 4 months: May 1st, 1846, purchased to the amount of $ 50. 66 12th, 66 $150. $120. $100. What was the average or mean time of payment, counting from 1st of May? The day from which you compute is not to be counted, but the day to which you compute is always to be counted. |