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RULE. --State the questions, and reduce the terms, the same as in the Rule of Three Direct; then multiply the first and second terms together, and divide the product by the third ; and the quotient will be the answer, in a denomination of the same name, to which the second term was reduced.
In this rule the fourth term or answer bears the same proportion to the second term, as the first term bears to the third.
If more require less, or less require more; the question belongs to Inverse Proportion.
More requir less, is when the third term is greater than the first, and requires the fourth term to be less than the second.
Less requiring more, is when the third term is less than the first, and requires the fourth term to be greater than the second.
NorE.-The principal difficulty which the student will experience in this rule, will be to distinguish Inverse or Indirect Proportion from Direct Proportion, but as the rule given for stating the questions is the same in both, the student can easily decide to which it belongs, by considering, whether more requires less, or less requires more.
EXAMPLES. 1. There was a certain dam erected across a river, in 60 days, by 12 workmeni; but the same being swept away, it is required to rebuild it in 30 days; how many men must be employed about it, working at the same rate ? Ans. 24 men.
d. men. di Dem.- From the condition of this question, 60 : 12 :: 30 it is plain, that it belongs to Inverse Proportion, 60
because it is evident, the less the time, the more
men must be employed to finish the work.310)7210 Here our third term is less than the first, and re
quires the fourth term or answer to be greater 24 Ans. than the second term; and it is evident from the
conditions of the question, that the fourth term or answer should be double the second term, because it must require double the number of men to perform the same work in 30 days, that it would in 60 days. And it will be seen, that the fourth term or answer, 24 men, bears the same proportion to the second term, 12 men, that the first term, 60 days, bears to the third term, 30 days; thus, 24 : 12 :: 60 : 30. To reduce this sum back to Compound Multiplication and Division, divide the first term by the third, and multiply the quotient by the second term; thus,
24 The answer as before. 2. If a footman perform a journey in 6 days, when the days are 16 hours long; how many days will he require to go the same journey, when the days are 12 hours long?
Ans. & lays.
h. d. h. DEM.-From this example the learner can 16 : 6 :: 12
plainly see, that the footman travelled 96 hours:
then when he travelled 16 hours in a day, it is evi. 16
dent, that it took him a less term of time in days, 36
than when he travelled only 12 hours in a day; 6
and it is plain, that every 12 hours are equal to a
day, when he travelled only that time in a day; 12)96
then as often as 12 is contained in 96, the whole
number of hours employed in travelling, so many Ans. 8
days he must have been emrioyed in travelling,
which proves to be 8 days. 3. If 3 teams can plough a field-in 12 days; in what time will 9 teams do it?
Ans. 4 days. 4. How many yards of shalloon that is 3 quarters wide will line 18yds. of cloth that is 5qrs. wide? Ans. 30yds.
5. Jf I lend a friend 200 dollars, for 90 days; how long ought he to lend me 450 dollars, to return the favour?
Ans. 40 days. 6. If a board be 8 inches wide; how much in length will make a square foot?
Ans.' 18 inches. 7. How many yards of paper, 3 quarters wide, will paper, a room that is 26 yards round, and 3 yards high?
Ans. 104 yards. 8. How much land, at $3,75 cents per acre, should be given in exchange for 300 acres, at $6,50 cents per acre ?
Ans. 520 acres. 9. How many men must be employed to do a piece of work in five days, which 8 men can do in fifteen days?
Ans. 24 men, 10. If 4 men will set 250 apple trees in six days; how many men will set the same number in two days?
Ans. 12 men. EXAMPLES Promiscuously placed, in Direct and Inverse Proportion. +
1. If any of a farm be worth 1250 dollars, what is the whole farm worth?
Ans. $2250. Shares. $ Shares. DEM.--Here it is plain, that the 5 : 1250 :: 9 farm is divided into 9 equal shares, or 9
9 parts; then it is evident, that as 5 5)11250 shares, are to the worth of 5 shares,
that is, $1250, so are 9 sh., the while Ans. $2250
farm, to the worth of the whole.
2. If a farm be worth 2250 dollars, what is of the farm worth?
Ans. $1250. 3. If a man receive $52,50cts. for the use of a certain sum, for the term of one year; what should he receive on the same sum, for 25 days?
Ans. $3,59cts. 5m. 4. If 100 dollars gain 7 dollars interest in 12 months ; what principal will gain the same interest in 4 months ?
Ans. $300. 5. If a man, worth, 4500 dollars, pay 12 dollars tax; how much should he pay, who is worth 1500 ?
Ans. $4. 6. If an ounce of silver be worth 63 cents, what is the price of 9 ingots, each weighing 41b. 3oz. ? Ans. $289,17.
7. If 60 bushels of corn, at 50 cents per bushel, will pay a debt, how many bushels, at 75 cents per bushel, will pay the same ?
Ans. 40 bushels. 8. A man is 1600 dollars in debt, and his whole estate amounts to but 992 dollars; how much can he pay on the dollar?
Ans. 62 cents. 9. If a staff, 4ft. cast a shade 5 feet; what is the height of that steeple whose shade, at the same time, measures 175 feet?
Ans. 140 feet.
QUESTIONS ON THE RULE OF THREE DIRECT. What is the Single Rule of Three? A. It is properly an application of Multiplication and Division. Why is it called the Rule of Three? A. Because three terms or numbers are given to find a fourth term or answer to the question. How may the first term be known? A. By being preceded, generally, by words like these, if, suppose, foc., and being of the same name or kind with the demanding term. How is the second term known? A. By being a term of supposition, and of the same name or quality with the answer. How is the third term known? A. By being the demanding term, and generally preceded by words like these, What cost? How fur? How much? What will ? fc.On what principle is the Rule of Three founded? A. An the obvious principle, that the magnitude 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. How is the Single Rule of Three divided ? A. Into two parts, the Rule of Three Direct, and the Rule of Three Inverse or Indirect. It is sometimes distinguished by the name of Proportion, Direct or Inverse. What is meant by Proportion ? A. That two numbers have the same relation to each other, that two other numbers have to each other ; thus 4 bears the same proportion to 8, that 6 bears to 12. What are the two first terms in the statement called ? A. Terras of supposition. What is the remaining term, (which is given,) called? A. The demanding terin. If the first and third terms are of different denominations, what must be done? Ą. They must be reduced to the same denomination. If the second term is of different denominations, what must be done ?. A. It must be reduced to the lowest denomination mentioned in that term. What must be done, after the terms are reduced ? A. The second and third terms must be multiplied together, and the product divided by the first term; then the quotient will be the answer in the same name of the middle term, if the question be in the Rule of Three Direct. How is the Rule of Three Direct distinguished from the Rule of Three Inverse ? A. If the third term be greater than the first, and require the fourth to be greater than the second; or if the third term be less than the first, and require the fourth term to be less than the second ; the question belongs to the Rule of Three Direct. How can the Rule of Three Direct be reduced back to multiplication and division ? A. By dividing the second term by the first, and multiplying that quotient by the third term; the quotient will be the answer, because we, in division, divide the price of a quantity, by the quantity, and the quotient is the price of a unit; and the price of a unit multiplied by the quantity, gives the price of the whole quantity. What rules strictly belong to the Rule of Three ? A. Interest, Practice, Exchange, Single and Double Fellowship, Tare and Tret, Barter, Loss and Gain, Alligation, Discount, Annuities, Position, and every other rule where proportion or ratio exists, may be considered as belonging to this rule. These rules have acquired different names principally on account of the business to which they are applied.
QUESTIONS ON THE RULE OF THREE INVERSE. How is the Rule of Three Inverse distinguished from the Rule of Three Direct ? A. When the third term is greater than the first, and requires an answer to be less than the second term; or when the third term is less than the first, and requires the answer or fourth term to be greater than the second, the question belongs to the Rule of Three Inverse. Are the questions stated the same in Inverse and Direct Proportion-? A. They are; and reduced in the same manner. What terms are multiplied together? A. The first and second ; and the product divided by the third. What proportion do the terms bear to each other, in Inverse Proportion ? A. The fourth term, or answer, bears the same proportion to the second, as the first bears to the third. In Direct Proportion, what proportion does the fourth term or answer bear to the third? AThe same as the second bears to the first
PRACTICE, Is a contraction of the Rulë of Thrée Direct, when the first term is a unit or one. It has acquired its name from its dail, use among mer chants, being an easy method of working, where the price of a unit or one is given, to find the price of a quantity. This rule was once o great use to tradesmen and merchants, when the price of one was given in sterling money, to find the price of a quantity ; but it is now rendered almost useless, as reckoning in federal inoney has almost be. come universal; and when the price of a unit or one is given in fede ral money it is much easier to work by multiplication.
Proved by the Single Rule of Three, Compou.. Multiplication, or by varying the parts.
CASE I.- When the price of one pound, one yard, foc. is given in farthings.
RULE.-Suppose the given number of yards or pounds to be so many pence, and take aliquot parts of a penny; thus, if it be at one farthing per yard, take one fourth; if at iwo farthings, take one half; if at three farthings, take one half
, and one fourth, and add the quotients; if there be 1 quarter of a yard, call it 1 farthing, if : of a yard, call it 2 farthings; if I of a yard, call it 3 farthings, in setting down the given sum.
EXAMPLES 1. What will 364 yards of tape come to, at two farthings per yard ?
Ans. 15s. 2d. grs. d.
DEM.-It is plain that at two farthings per ?=2)364 yard, one half the number of yards, would
be the value of the tape in pence; therefore
we suppose the number of yards to be so maAns. 155. 2d. niy pence, and take one half of them, and di
vide the quotient by 12, to bring the pence
into shillings. By Multiplication.
By the Rule of Threc Direct. yds.
git. qrs. yds. 364
1: 2 :: 364 2
12)182 Ans. 15s. 2d.
Ans. 15s. 2d. This operation, by Multiplication, and by the Rule of Three, is too plain to need demonstration.
2. What will 3961 yards come to, at three farthings per yard?
Ans. £l 4s. 9d. Iqr. qrs.
It will be seen, that there is a 2=)396 2
small loss in working according to 131)198 1 at 2 qrs.
this rule, when there is a remain99 0 at 1 qr.
der in the lowest denomination.
Where it is more convenient we 12)297 1 at 3qrs. 210)2 4 9d.
take parts of parts; thus, at lqr. it
is 1 of the price of the whole quanAnş. £1 4s. 9d.'1qr.
tity, at 2qrs. per yard. 3. What will 34692 yards come to, at one farthing per
Ans. £36 25. 9d.