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only? How do you cast the interest on any sum for one year ? A. Multiply the interest for one dollar by the number of dollars, and the product will be the interest for one year; or, multiply the given sum by the interest of one dollar, and the product will be the interest for one year. Why should that give the interest for one year ? A. Because it is plain, that the interest of one dollar repeated by the number of dollars, must give the interest on the whole number of dollars. Is this the most convenient way of casting interest ? A. It is not; the best method of casting interest is, to multiply the given principal. by the interest of one hundred dollars, and divide the product by one hundred. Why divide the product by 100 ? A. Because the product is one hundred times too much, the multiplier being the interest of one hundred dollars instead of the interest of one dollar. When you have computed the interest for one year, how do you compute it for a numher of years ? A. Multiply the interest of one year by the number of years, and the product will be the interest for the whole number of years. What general rule may be given to cast interest on a sum for any number of months ? A. First multiply by the per cent, and the product will be the interest for one year, then divide the interest of one year by 12, and the quotient will be the interest for one month, and the interest for one month multiplied by the number of months, gives the interest for the whole number of months; but the better way is to take aliquot parts of the interest of a year for the months. What general rule may be given to compute interest for any number of days ? A. Divide the interest for one year by 12, and the quotient will be the interest for one month; then divide the interest for one month by 30, and the quotient will be the interest for one day, and one day's interest multiplied by the number of days, will give the interest for the whole number of days; but the better way is to take aliquot parts of a month for the days. How is the amount obtained? A. By adding the principal to the interest. How do you cast interest on bonds in the state of New-York?. A. Compute interest to the time of the first payment, and from the amount subtract the payment, and the remainder will be a new principal, upon which cast interest till the time of the next payment, and from the amount again subtract the payment, and so on, till the time of settlement; but if the payment be less than the interest, at the time it is made, cast again on the same principal, and so continue to do, till the amount of payments exceeds the interest, and then subtract the sum of the payments from the amount of the note or bond at that time. On what principle is the computation of interest founded in the state of New York when partial payments are made ? A. On the principle that interest is due whenever a payment is made, and that the payment must first go to pay the interest, and then to discharge the principal. What is Commission or Factorage ? Ą. It a premium or an allowance of a certain per cent, to a factor engaged in buying and selling goods for his employer. How is the work performed ? A. The same as in Simple Interest. What is Brokerage ? A. It is a premium allowed, at a certain per cent, to persons assisting merchants or factors in sales and purchases. How is the work performed? A. The same as in Simple Interest. What is En.
e? A. It is premium of a certain per cent allowed to per
sons and offices indemnifying against hazard or losses of different kinds. What is Compound Interest ? A. It is reckoning interest upon interest; or at certain periods of time, the interest is incorporated with the principal, and draws interest. How do you proceed in the operation of the work? A. Compute interest on the given sum, the same as in Simple Interest, from time to time, as the interest becomes due, each time making the amount a new principal for the next period of time. Is compound interest allowed by law? A. It is not; yet it is usual to allow compound interest in purchasing annuities, pensions, or leases in reversion.
THE SINGLE RULE OF THREE. The Single Rule of Three is properly an application of Multiplication and Division; for on those two rules it principally depends. It is generally divided into two parts, viz. the Rule of Three Direct and the Rule of Three Inverse or Indirect. It is sometimes called, and very properly too, the Rule of Proportion, or the Golden Rule of Proportion, on account of its extensive usefulness, in the transac.. tion of business, and in the solution of almost every mathematical inquiry. The rule is founded on this obvious principle; that the magni-tude or result of any effect, varies constantly in proportion to the varying part of the cause; thus the quantity of articles purchased, is in proportion to the money laid out; and the space gone over by a uniform motion, is in proportion to the time. ,
This is the sign of proportion, : : : which is placed between nunbers thus, 4:8:: 5: 10, and read thus, as 4 is to 8, so is 5 to 10.
The Rule of Three is so called, because three terms or numbers are given to find a fourth, which in the Rule of Three Direct, shall bear the same proportion to the third, as the second bears to the first, thus, 4 is : to 8:: as 5 is : to 10.
It is obvious that 10, the fourth term, bears the same proportion to 5, the third term, that 8, the second term, bears to 4, the first term.
Two of the given numbers are called terms of supposition, and the other the term of demand. The terms of supposition may generally be known by being preceded, in most cases, by words like these, ij, suppose, foc. The term of demand is generally preceded by words like these, How far? What cost? What will? How many? Horo much? &.c.
RULE OF THREE DIRECT. The Rule of Three Direct is far more useful in business than the Rule of Three Inverse, and may be distinguished from it by the conditions of the question. When the third term is greater than the first, and requires the fourth term, or answer; to be greater than the second, it belongs to the Rule of Three Direct; or when the third term is less than the first, and requires the fourth term, or answer, to be less than the second, it belongs to this rule.
RULE FOR STATING. Write down for the first term, that term of supposition which is of the same name or kind with the demanding term. Place the remain
ing term of supposition in the second place, which must be of the same name or kind with the answer. Then set the demanding term in the third place, which must be of the same name or quality with the first
RULE FOR WORKING. If the first and third terms are of different denominations, reduce them to the lowest denomination mentioned in either. If the second term stands in different denominations, reduce it to the lowest denomination mentioned in that term: Then multiply the second and third terms together, and divide their product by the first term, and the quotient will be the fourth term or answer in the same denomination of the second term, in whatever denomination it stands or has been reduced;
then if the answer does not stand in the highest denomination, it should be brought to the highest by Reduction.
Note 1.--All the following rules strictly belong to the Rule of Three Direct, viz: the Double Rule of Three, Exchange, Interesi, Practice, Single and Double Fellowship, Tare and Tret, Barter, Loss and Gain, Alligation, Discount and Annuities.
These Rules have acquired different names from the business to which they are applied ; and in treating of them separately, it will be shown that they belong to this rule, and may, like Direct Proportion, or the Rule of Three Direct, be reduced back to the fundamental or Simple Rules.
The Double Rule of Three, or Compound Proportion, will in some cases embrace both the Rule of Three Direct and Inverse.
Interest, on account of its importance in business, and its evident relation to the Şimple Rules, has been placed before the Rule of Three, though in all cases may be worked by the Rule of Three.
NOTE 2.-As the chief difficulty which learners experience in this rule, is in stating questions, teachers should exercise their pupils in stating fifteen or twenty questions for a few days in succession, and the difficulty will then vanish. They should be in the daily habit of demonstrating to their pupils the rules, and explaining the principles upon which they are founded, tracing them back to the simple rules on which they immediately depend. Instructors by pursuing this course, will present to the young mind this important science free from all its obscurity, and students will soon learn that the whole science of Aritha metick, is nothing more than a proper application of the fundamental or Simple Rules.
EXAMPLES. 1. If 2 yds. of calico cost 4s. what will 4 cost? Ans. 8s. yds. s. yds.
Dem-When we multiply the second 2:4 :: 4
term by the third, we are multiplying the
price of a quantity, consequently the pro4
duct must be as many times too great as the 2) 16
first term exceeds a unit; therefore by divi
ding the product by the first term, the quoAns. 8 shillings. tient must be the answer, or fourth term
because the second term is double che price of a unit. N. B. The learner will perceive, that 8, the fourth term or answer; bears the same proportion to 4, the third term, that 4, the second term, bears to 2, the first term; because the fourth term is double the third, and the second term is double the first. The 1st and 4th terms are called extremes, and the second and third terms, the means; and the cele! brated property of proportional numbers is, that the product of the ex. tremes, is equal to the product of the means; thus, 2:4::4:8, then 2X8=4X4=16. The first term of two proportional numbers, is also called the antecedent, and the other, to which it bears proportion, is called its consequent; thus, in the example given, 2, the first term, is the antecedent, and 4, the second term, the consequent; that is, two yards have produced the 4 shillings; and the third teru, 4 yards, is also the antecedent which produced 8 shillings its consequent. Now it is evident, that the first antecedent bears the same proportion to its consequent, that the second antecedent bears to its consequent ; that is, 2 yards bear the same proportion to the cost, 4 shillings, that 4 yards bear to the cost, 8 shillings; because in cach case, the money bears the same proportion to the quantity of cloth purchased.
This rule may be proved by inverting the terms of the question. Or by multiplying the means, and if the product equal the product of the extremes, the work is right; thus, if 4 yds. cost 8s. what will 2 cost ?
Ans. 4s. Ayds. : 88. :: 2yds. This sum serves as a proof of 2
the first example; here we find that 2 yards cost 48. whịch agrees
with the first supposition. In order to render this excellent rule as plain as possible to the learner, the same example is again repeated, and reduced back to the simple rules on which it depends. If 2 yards of calico cost 4 shillings; what will 4 yards
Ans. 8s. 2yds. : 4s. - 4yds. Reduced to the simple rules by 4
dividing the second term by the 2)16
first, and multiplying the quotient
by the third term; thus, Ans. 8s.
PHAS. DEM. We have learned
2s. price of 1 yard. by division, that dividing the
4 price of a quantity by the quantity gives the price of a
Es. price of 4 yards. unit or 1; then when we divide 4 shillings, the price of 2 yards, hy 2, the quotient, 2 shillings, is the price of 1 yard. And we have found by multiplication, that multiplying the price of 1 yard, by the number of yards, gives the price of
the whole number, then when we multiply 28., the price of 1 yard, by 4, the number of yards, the product 8s. must be the price of 4 yıls Hence it is plain, that Direct Proportion, or the Rule of Three Direct, is nothing more than an application of Multiplication and Division, and from Multiplication and Division it might be reduced back to Addition and Subtraction, as those two rules have been reduced back, in their proper places. And it is the better way to divide the second term by the first, and multiply that quotient by the third term, whenever the first term will divide the second without a remainder.
2. If one yard of cloth cost four dollars; what cost eight yards?
Ans. $32. yd. yds. DEM.-It is plain, when the first term is a unit, 1: 4 :: 8
that the work is performed by multiplication, be8
cause it is only repeating the price of a unit or
one, by the whole number; and dividing that pro$32
duct by the first term, when it is a unit, can evi
dently make no difference. 3. If eight yards cost $32; what cost one yard ?
Ans. $4. yds. 8 yd. DEM.-It is plain, when the 3d term is a 8: 32 ::
unit, that the work is performed by division 1
merely, for multiplying by one can make no
difference; it is only bringing down the mid8)32
dle or second term for the convenience of
dividing by the first term, and it is evident, Ans. $4
if the price of a quantity be divided by the quantity, that the quotient must be the price
of a unit. 4. If 6 yards of cloth.cost $24; what cost 12 yards ?
Ans. $48. yds. $ yds. 6: 24 :: 12- Or thus, 6)24 the price of 6 yds. 12
4 the price of 1 yd.
12 Ans. $48 the price of 12 yds. A. $48 the price of 12 yds.
Dem.-It is evident from the reasoning under the first example, that when we multiply 24 dollars, the price of 6 yards, by 12, the product is 6 times too much; then when we divide the product by 6, the first term, the quotient must be the price of 12 yards. And it is also plain, when we divide $24, the price of 6 yards, by 6, that the quotient must be the price of one yard; and the price of one yard muk tiplied by 12, evidently gives the price of 12 yards.
5. If six pounds of tea cost $3,75cts.; what cost eighteen pounds?