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5. If 20 cwt. of tobacco cost 9. If 24 lambs cost 30 dollars, 120 10s. what is it
cwt ? what are they a piere? Ans. 6 (s. 6d.
Ans. Si 25cts. 6. If 1 cwt. of tea cost £18 10. If 12 men draw a prize of 18s. what is it per pound ? 18000 dollars, what is each man's Ans. 35. 4 d. share ?
Ans. $1500. 7. If 4 men spend at a tavern
11. If 147 bushels of wheat £2 16s. 4d. what must each pay? Ans. 14s. 1d.
cost £47 12s. 6d. what is it per 8. If 12 silver cups weigh 131b. bushel ?
Ans. 6s. 52d. 1oz. 2pwt. 10grs. what is the 12. If 196lb. of cotton cost €6 weight of each cup ?
10s. 8d. what is it per pound? Ans. 11b. 1oz. 1pwt. 200 grs.
QUESTIONS. 1. What is Compound Division ? 4. What is to be done when there 2. How do you place the numbers ? is a remainder after dividing 3. How do you proceed in dividing? any denomination ?
5. What is the method of proof?
Miscellaneous Examples, 1. What is the difference between six dozen dozen and half a dozen dozen ?
Ans. 792. 2. What is the difference between half a solid foot and a solid half foot ?
Ans. 648 inches. 3. A note was on interest from March 20, 1819, till Jan. 26,1824; what was the length of time?
Ans. 4y. 10mo. 6ds. years. mo. days. Io operations of this kind, a month is considered 30 1824 0 26 days, and a year 12 months. This, though not per1819 2 20 fectly correct, will be found to be a good practical
method of ascertaining the time in computing interest. 4 10 6 4. How long from June 7, 1814, to August 3, 1823 ?
Ans. Gys. 1mo. 26ds. 5. Divide 5 guineas among 8 men-give a 8ů. more than B, and B 8d. more than C, &c. what does H receive ?
Ans. 15s. 2d. H's share.
appear equally plain, by, preparing the dividend in such a manner, before dividing, that the several parts may be divided without a remainder. If you would divide £3 13s. 8d. by 2, first make all the parts divisible by 2; thus £3 135. 8d.=£2 32s. 20d. These parts divided successively by 2, give £1 168. 10d. the same as by the rule.
In Compound Division there are usually given a variety of cases ; but it was thought better to give one general rule which would answer every purpose without unnecessarily incumbering the memory of the scholar. After becoming familiarly acquainted with the rule here given, the several contractions will readily suggest themselves in practice from what has been taught in Simple Division.
6. A horse is valued by A at 860, by B at 869 50, and by C at 872 25, what is the average judgment ? A. 1
$60 B. 1
69 50 The average in this and similar cases, is found C. 1
72 25 by dividing the sum of the several judgments by
the number of appraisers. 3 3) 201 75
$67 25 Ans. 7. M, N, O, and P, appraised a ship as follows, viz. M at 86700, N at $9000, O at $8750, and P at $7380; what is the average judgment?
Ans. $7957 50. 8. A and B wishing to swap horses, and disagreeing as to the conditions, referred the matter to 3 disinterested persons, X, Y and Z, whose judgments were as follows, viz. X said I should pay B $8; and Y said A should pay B 86; but Z said B should pay A $5; what is the average judgment ?
Ans. A must pay B 83. A B
In the exchange of articles, X 1. $0 $8 14 B where the judgment of the referees Y 1. 0 6 5 A is partly on one side of the equality Z 1. 5 0
between them, and partly on the
3)9(3 Ans. other, subtract one side from the Referees. 3 5 14
other, and divide the remainder by the number of referees for the average judgment.
9. C and D wishing to swap farms, referred the subjpct to O, P, Q and R, and agreed to abide their judgment, which was as follows, viz. O said C should pay D870; P said C should pay D 8100; and Q said C should pay D $55; but R said D should pay C 825; how was the matter settled ?
Ans. C pays D 850. 10. What is the weight of 4hhd. of sugar, each weighing 7cwt. 4qrs. 191b.
Ans. 31cwt. 2qrs. 20lb. 11. Three men and 2 boys hoed 30000 hills of corn, and each man hoed 2 hills while a boy hoed one ; how many hills were hoed by each man, and how many by each boy ?
Ans. Each man hoed 7500 and each boy 3750 hills. 3x2+28 Divisor. 12. If 8911.555 be divided among 5 men and 4 women, what is each man and woman's share ?
865.111=1 woman's share. Ans.
15. Two places differ in longitude 31° 37' 3'' what is their difference in reckoning time, allowing 15° to make an hour?
Ans. 2h. 6' 3". 14. How much wood in a load 6ft. 7' long, 3ft. 5' high and 3ft. 8' wide ?
Ans. 82ft. 5' 8" 4". 15. I bought a load of wood 8ft. long, 3ft. wide and 2ft. 8' high, how much did it cost at the rate of $1.75 per cord ?
Ans. 87] cents.
Fractions.* Fractions are parts of a unit.
Fractions are of two kinds, Vulgar and Decimal, which differ in the manner of expression and modes of operation.
A Vulgar Fraction is expressed by two numbers, written one over the other with a line between; as , ģ,
The number below the line is called the denominator, and expresses the number of parts into which the unit is divided.
The number above the line is called the numerator, and shows how many of those parts are contained in the fraction ; thus, the meaning of the expression, of a bushel, is, that a bushe) is divided into 8 equal parts, and that 3 of those parts are taken.
A Decimal Fraction is expressed by one number, which is distinguished from a whole number by a period at the left hand, called the separatrix, as .5, .45, the first denoting 5 tenth parts, and the second 45 hundredth parts.
A Decimal may be changed into a Vulgar Fraction by drawing a line under it, and writing under the line as many cyphers as there are figures in the decimal, with a 1 at the left hand. Thus .I is for .45 is 16 and .005 is roso
1. VULGAR FRACTIONS. Vulgar Fractions are those whose numerators and denominators are both expressed.
A proper fraction is one whose numerator is less than its denominator; as }, , *, &c.
An improper fraction is one whose numerator is greater than its denominator; as, , &c.
A compound fraction is a fraction of a fraction, as of #, &c. .
A mixed number consists of a whole number and a fraction, as 124, 65, &c.
A whole number is changed into an improper fraction by writing 1 under it, with a line between, as 4, 6, &c.
The common measure of two, or more numbers, is a number which will divide each of them without a reinainder.
The greatest common measure of two, or more numbers, is the greatest number which will divide those numbers severally without à remainder.
The common multiple of two, or more numbers, is a number which may be divided by each of those numbers without a remainder.
* Fractions comes from the Latin word, Frango, to break, because the unit is considered as broken into several equal parts. Vulgar and Decimal Fractions differ in this ; the denominator of the former may be any number whatever, but the denominator of the latter, when expressed, is always 10, 100, 1000, ar 1 with as many ciphers annexed as there are figures in the decimal.
The least common multiple is the least number, which can be so divided without a remainder.
A prime number is one which can be measured only by itself or by a unit.
A perfect number is one which is equal to the sum of all its aliquot parts.*
1. REDUCTION OF VULGAR FRACTIONS.
Reduction of Vulgar Fractions, is changing them from one form into another without altering their value.
Case I. To find the greatest common measure of two, or more numbers.
1. If two numbers only be given, divide the greater by the less, this divisor by the remainder, and so on till nothing remains, always dividing the last divisor by the last remainder; then will the last divisor be the common measure required.
2. If there be more than two numbers given, find the greatest common measure of two of them; then of that common measure and one of the others, and so on through all the numbers ; the greatest common measure last found will be the answer.
Examples. 1. What is the greatest com
2. What is the greatest common measure of 580, 320 and 45 ? mon measure of 918, 1998 and 320)580(1 20)45(2 322 ?
Ans. 18. 320
* The aliquot part of any pumber is such a part of it, as, being taken a cer. tain number of times, exactly makes that number. The smallest perfect number is 6. Its aliquot parts are 3, and 1, and 3+2+1=6. The next is 28, the next 496, and the next 8128. Only ten perfect numbers are yet known.
To find the least common multiple of two, or more, numbers.
Rule. 1. Arrange the numbers in a line, and divide by any number that will divide two, or more, of the given numbers without a remainder, and set the quotients together with the undivided numbers in a line below.
2. Divide the second line as before, and so on till there are no two numbers remaining that can be thus divided ; then will the continued product of the several divisors, and the figures in the last line, be the multiple required.
Examples. 1. What is the least common 3. What is the least number multiple of 3, 5, 8 and 10? which can be divided by 6, 10, 5)3 5 8 10 Five and 10
16 and 20 without a remainder ? divided by 5
Ans. 240. 2)3 1 8 2
are 1 and 2. 31 4 1 with which 3
4. Supposing 12 clocks to be and 8
set a-going together, the first of 5*2*3*4=120 Ans, bro't. down.
which strikes at the end of every
Again 8 and hour, the second at end of every 2 divided by 2, give 4 and 1 with second hour, the third at the end which 3 and 1 are brought down. of every third hour, and so on to Then the product of 5, 2, 3 and 4 is the 12th which strikes at the end the multiple required.
of every 12 hours; how long be2. What is the least common fore they will all strike togethmultiple of 3, 4, 8 and 12
Ans. 27720 hours. Ans. 24.
To reduce fractions to their lowest terms.
Rule. * Divide both the terms of the fraction by their greatest common measure, and the quotient will be the fraction required.
* Dividing both terms of a fraction by the same number does not at all alter its value. If the greatest common measure of a fraction be 1, the fraction is already in its lowest terms.