RULE FOR EXTRACTING THE CUBE ROOT. I. Separate the given number into periods of three fig. ures each, placing a point over units, then over every third figure towards the left in the whole numbers, and over every third figure towards the right in decimals. II. Find the greatest cube in the first period on the left hand; then placing its root on the right of the number for the first figure of the root, subtract its cube from the period, and to the remainder bring down the next period for a dividend. III. Square the root already found, giving it its true local value; multiply this square by 3, and place the product on the left of the dividend for a divisor; find how many times it is contained in the dividend, and place the result in the root. IV. Multiply the root already found, regarding its local value by this last figure added to it, then multiply this product by 3, and place the result on the right of the dividend under the divisor; under this result write also the square of the last figure placed in the root. V. Finally, add these results to the divisor; multiply the sum by the last figure placed in the root, and subtract the product from the dividend. To the right of the remainder bring down the next period for a new dividend; find a new divisor, and proceed with the operation as above. PROOF-Multiply the root into itself twice, and if the last product is equal to the given number, the work is right. OBS. 1. When there is a remainder, periods of ciphers may be added, as in square root. 2. If the right hand period of decimals is deficient, this deficiency must be supplied by ciphers. The root must contain as many decimals as there are periods of decimals in the given number. 3. The cube root of a common fraction is found by extracting the root of its numerator and denominator. A mixed number should be reduced to an improper fraction. 3. What is the cube root of 1728 ? QUEST.-Obs. When there is a remainder, how proceed? When the right hand period of decimals is deficient, what must be done? How many decimals must the root contain? 4. What is the cube root of 13824? 5. If a box in the form of a cube, contains 373248 solid inches, what is the length of one side? 6. What is the side of a cubical vat, which contains 571787 solid feet? 7. What is the side of a cubical mound which contains 1953125 solid yards? 8. What is the cube root of 2? 9. What is the cube root of 2357947691? 10. What is the cube root of 12.167? 64 SECTION XV. EQUATION OF PAYMENTS. ART. 363. EQUATION OF PAYMENTS is the process of finding the equalized or average time when two or more payments due at different times, may be made at once, without loss to either party. OBS. The equalized or average time for the payment of several debts, due at different times, is often called the mean time. 364. From principles already explained, it is manifest, when the rate is fixed, the interest depends both upon the principal and the time. (Art. 241.) Thus, if a given principal produces a certain interest in a given time, Double that principal will produce twice that interest; Half that principal will produce half that interest; &c. In double that time the same principal will produce twice that interest; In half that time the same principal will produce half that interest; &c. QUEST.-363. What is Equation of Payments? Obs. What is the average time for the payment of several debts sometimes called? 364. When the rate is fixed, upon what does the interest depend? 365. Hence, it is evident that any given principal will produce the same interest in any given time, as One half that prin. will produce in double One third that prin. will 66 66 that time; "thrice that time; 66 half that time; 66 a third of that time, &c. For example, at any given per cent., The int. of $2 for 1 year, is the same as the int. of $1 for 2 years; The int. of $3 for 1 year, The int. of $5 for 1 mo. 366. The interest, therefore, of any given principal for 1 year, or 1 month, &c., is the same, as the interest of 1 dollar for as many years, or months, &c. as there are dollars in the given principal. 1. Suppose you owe a man $15 and are to pay him $5 in 8 months, and $10 in 2 months, at what time may both payments be made without loss to either party? In Analysis. Since the interest of $5 for 1 month is the same as the interest of $1 for 5 months, (Art. 365,) the interest of $5 for 8 months must be equal to the interest of $1 for 8 times 5 months. And 5 mo. X8=40 mo. like manner the interest of $10 for 1 month is equal to the interest of $1 for 10 months, and the interest of $10 for 2 months is equal to the interest of $1 for 2 times 10 months. And 10 mo. x2=20 months. Now 40 months added to 20 months make 60 months; that is, you are entitled to the use of $1 for 60 months. But $1 is of $15, consequently you are entitled to the use of $15, part of 60 months, and 60 months +15=4. Ans. 4 months. Proof. The interest of $5 at 6 per ct. for 8 mo. is $5 X.04=$.20 ' The interest of $10 66 "2 mo. is $10x.01.10 The interest of $15 at 6 per ct. for 4 mo. is 15X.02-$.30 367. Hence, we derive the following general RULE FOR EQUATION OF PAYMENTS. First multiply each debt by the time before it becomes due; then divide the sum of the products thus obtained by the sum of the debts, and the quotient will be the average time required. OBS. 1. If one of the debts is to be paid down, its product will be nothing; but in finding the sum of the debts, this payment must be added in with the others. 2. This rule is based upon the supposition that discount and interest paid in advance are equal. But this is not exactly true; (Art. 261. Obs. 1;) consequently, the rule, though in general use, is not strictly accurate. 2. If I owe a man $20, payable in 4 months, $40 payable in 6 months, and $60 in 3 months, at what time may I justly pay the whole at once? Operation. $20×4=$80, the same as $1 for 80 mo. (Art. 366.) $40×6=240, " 66 "$1 for 240 " $60×3=180, 66 $120debts.500 sum of products. 3. A merchant bought three lots of goods amounting to $300; for the first he gave $100, payable in 5 months; for the second $150, payable in 8 months; for the third $50, payable in 2 months: what is the average time of all the payments ? 4. A farmer has 3 notes; one of $50, due in 2 months; another of $100, due in 5 months; and the third of $150, due in 8 months: what is the average time of the whole? 5. A merchant buys goods amounting to $1200, and agrees to pay $400 down, $400 in 4 months, and $400 in 8 months; he finally concluded to give his note for the whole at what time must the note be made payable? 6. A man borrows $600, and agrees to pay $100 in 2 months, 200 in 5 months, and the balance in 8 months: when can he justly pay the whole at once? QUEST.-367. What is the rule for equation of payments? 7. A man buys a house for $1600, and agrees to pay $400 down, and the rest in 3 equal annual instalments: what is the average credit for the whole? 8. I have $1200 owing to me, of which is now due of it will be due in 4 months, and the remainder in 8 months. what is the average time of the whole? 9. A grocer bought goods amounting to $1500, for which he was to pay $250 down; $300 in 4 mo.; and $950 in 9 mo.: when may he pay the whole at once? 10. A young man bought a farm for $2000, and agrees to pay $500 down, and the balance in 5 equal annual instalments: what is the average time of the whole? PARTNERSHIP.. 368. PARTNERSHIP is the associating of two or more in lividuals together for the transaction of business. (Art. 299.) The persons thus associated are called partners; and the association is termed a company or firm. The money employed is called the capital or stock; and the profit or loss to be shared among the partners, the dividend. CASE I. Ex. 1. A and B formed a partnership; A furnished $300 capital, and B $500; they gained $200: what was each partner's share of the gain? Solution. Since the whole stock is $300+$500-$800, A's part of it was 300-3, and B's part was 500—5. Now since A put in of the stock, he must have of the gain; and $200×3=$75. For the same reason B must have of the gain; and $200×÷=$125. PROOF.-$75+125-$200, the whole gain. (Art. 284. Ax. 11.) Hence, QUEST.-368. What is partnership? What are the persons thus as sociated called? What is the association called? What is the money employed called? What the profit or loss? |