is greater or less than the term sought; then the fourth term will be found by multiplying together the second and third, and dividing the product by the first." Or, Besides the general rule there are four others, either of which, when applicable, performs the work more concisely. They follow, Rule 1. By dividing the second term by the first, and multiplying the quotient by the third: Or Rule 2. By dividing the third term by the first, and multiplying the quotient by the second: Or Rule 3. By dividing the first term by the second, and the third by the quotient: Or Rule 4. By dividing the first term by the third, and the second by the quotient, the answer may be found. [The distinction between direct and inverse proportion is not necessary, the latter being the proportion where more requires less, and the reverse. If 10 dollars will furnish 5 men with provisions 5 days, it is obvious at once that the same money must furnish 8 men less time, and the reverse. RULE OF THREE IN VULGAR FRACTIONS. se.] Questions are stated as in whole numbers, after the fractions are prepared, and the operation is performed by inverting the first term, and then proceeding to multiply the numerators together, and also the denominators, as in the multiplication of vulgar fractions. Example. If of a dollar buy of a yard of cloth, what will of a dollar buy? 2×3×14 yd. Ans. SECTION XIX. COMPOUND PROPORTION. It frequently becomes necessary to involve a larger number of circumstances in a proportion, than those found in the preceding examples. 1. If $100 gain $6 in 12 months, what will $200 gain in 8 months? This may be resolved into two questions in simple proportion, and may be stated thus: If 12 months give $6, what will 8 months give? = $8. = $4. But it is perfectly easy to combine them both into one operation, by multiplying the numbers together which compose the terms, that are involved in the parts of the proportion. Thus : If 100 X 12: 6 :: 200 × 8?: As $100 for 12 months would be worth as much as $1200 for one month, these numbers may be multiplied together; and as $200 for eight months will be worth just as much as $1600 for one month, these numbers may be multiplied together. The proportion then may be expressed thus: If $1200 gain $6, what will 1600 gain? The answer must be just as much larger than $6, as $1600 is larger than $1200, $8. = 2. If 6 men in 12 days build 25 rods of wall, how much will 12 men build in 50 days? Here 6 men in 12 days must perform the same labor as one man in 6 times 12 days; and 12 men in 50 days must perform the work of one man in 12 times 50 days. Hence the operation may be thus expressed : 6 men If {Gmdays { 12 12 600 days' work, will build as much more than 50 rods, as 4163 rods. 6 men, rods, 12 men, 12 days 50: 50 days: 4163 rods, Ans. This is denominated Compound Proportion. RULE. "" Make the number which is of the same kind with the required answer, the third term; and of the remaining numbers, take any two that are of the same kind, and place one for a first term and the other for a second term, according to the direction in simple proportion; then any other two of the same kind, and so on till all are used; lastly, multiply the third term by the product of the second terms, and divide the result by the product of the first terms, and the quotient will be the fourth term, or answer required." (Lacroix.) SECTION XX. CONJOINED PROPORTION, The numbers of terms are frequently increased to a greater number than those mentioned in the last Section, and therefore such are denominated Conjoined Proportion. As the principles are the same as those already illustrated, it is necessary to furnish only a single example. If 6 shillings in N. E. are equal to 8 in N. York, and 16 in N. Y. are equal to 15 in Pennsylvania, and 30 in Pennsylvania are equal to 20 in Canada, how many in Canada are equal to 18 in New-England? 6 × 16 × 30=2880 : 8 × 15 × 20=2400 :: 18 : 15, Ans. The first numbers in each part of the question are antecedents, and the following are consequents. 6, 16, &c. are antecedents; 8, 15, &c. are consequents, as seen in the example above. Hence the following RULE. Multiply all the Antecedents for the first term, and all the Consequents for the second; the number that asks the question makes the third: then the product of the second and third terms, divided by the first, will give the desired answer. SECTION XXI. INTEREST, &c. When one person employs the property for his own benefit, which belongs to another, it is evidently right that a proper reward should be given. The reward or pay for the use of money, is called Interest, of which there are two kinds, Simple and Compound. Simple Interest is that which arises from the sum of money lent, which is called the principal. The number of cents paid for the use of a dollar for a year, is called the rate. The term amount means the sum of both principal and interest. When an agreement is made to consider interest as payable at the close of every year, and the borrower does not pay the interest, but keeps it in his own hands, it is considered as added to the principal, or money lent, and interest is allowed for it the same as for the money lent. This is called Compound Interest. The rate per cent of interest is different in different places, but it is commonly 6 per cent. This is established by the Congress of the U. States. Seven per cent is common in some of the states. In calculating Interest, 12 months are considered a year, and 30 days a month. What is the interest of 18 dollars and 24 cents, for 2 years and 6 months, at 6 per cent.? 18.24 1.0944 18.24 6 1.0944 1.0944 2 2.1888 Since the rate is 6 cents on a dollar, or .06 of a dollar, the principal multiplied by .06 and pointed according to the principles of decimal fractions, gives the interest for one year. Instead, however, of multiplying by .06, we may multiply by 6 merely, taking care to remove the decimal point two figures from its natural place towards the left. If we multiply the interest of one year by the number of years, we evidently get the interest for the given number of years. 2)1.0944 .5472 2.1888 2.7360 18.24 6 2)1.0944 2 2.1888 If we take half of one year's interest, it will obviously be 6 months' interest, which added to 2 years' interest makes the required interest, for 2 years and 6 months. The operation in all its parts, may be performed conveniently as seen here. .5472 2.7360 Ans. As the usual rate of interest is 6 cents for the use of a dollar 12 months, it must be part of 6 cents for one month, which is of a cent or 5 mills. For 2 months it is twice as much as for 1 month, equal to one cent. Hence it is easy to ascertain the interest for any number of months. If, as we find the true interest is 5 mills for the use of a dollar, one month or 30 days, of thirty days will be required to earn 1 mill. One fifth of 30 is 6. Hence when the number of days is known, it is easy to find the amount of interest; as a dollar must gain part as many mills as there are days in the given time. When the interest of one dollar is found, the interest of 2, 5, or 50, &c. may be readily known, for 2 must gain twice as much as one; 5, five times as many mills; 50, fifty times as many, &c. The foregoing illustrations will furnish the reasons for the following RULES. Ascertain the number of days, divide by 6 and the interest of 1 dollar will be obtained in mills. Multiply this by the number of dollars, and the answer is found. EXAMPLE.-What is the interest of 5 dollars for 90 days? 90615 mills, the gain of $1. And 15 m. X 5, the number of dollars, is = 75 mills=71⁄2 cents. 2. Or, ascertain the number of months and multiply the given sum by half of these. The answer is the number of cents. 3. Or, multiply the principal by the rate, and this is the interest for a year. Multiply the interest for one year by the number of years, and the answer is obtained. |