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31. Wanting just an acre of land cut off from a piece which is 13 poles in breadth, what length must the piece be ?
Ans. 1! po. 4 yds. 2 ft. Olin. 32. At 76 91d per yard, what is the value of a piece of cloth containing 53 ells English 1 qu. Ans. 251.18s 1 d.
33. If the carriage of 5 cwt 14 lb for 96 miles de 1l 12s 6d; how far may I have 3 cwt 1 qr carried for the same money?
Ans. 151 m 3 fur 31. pol. 34. Bought a silver tankard, weighing 1 lb 7 oz 14 dwts ; what did it cost me at 6s 4d the ounce ? Ans. 61 48 94d.
35. What is the half year's rent of 647 acres of land, at 158 6d the acre ?
Ans. 2111 19s 3d. 36. A wall that is to be built to the height of 36 feet, was raised 9 feet high by 16 men in 6 days; then how many mer must be employed to finish the wall in 4 days, at the same rate of working ?
Ans 72 men. 37. What will be the charge of keeping 20 horses for a year, at the rate of 14;d per day for each horse?
Ans. 4411 Os 10d. 38. If 18 ells of stuff that is yard wide, cost 39s 6d; what will 50 ells, of the same goodness, cost, being yard wide ?
Ans. 71 6s 33;d. 39. How many yards of paper that is 30 inches wide, will hang a room that is 20 yards in circuit and 9 feet high.
Ans. 72 yards. 40 If a gentleman's estate be worth 3841 16s a year, and the land-tax be assessed at 2s 9d per pound, what is his net annual income?
Ans. 3311 18 9}d. 41. The circumference of the earth is about 24877 miles ; at what rate per hour is a person at the middle of its surface carried round, one whole rotation being made in 23 hours 56 minutes ?
Ans. 1039454 iniles. 42. If a person drink 20 bottles of wine per month, when it costs 88 a gallon ; how many bottles per month may he drink, without increasing the expence, when wine costs 10s the galloa ?
Ans. 16 bottles. 43 What cost 43 qrs 5 bushels of corn, at il 8s 6d the quarter.
Ans. 621 38 3;d. 44. How many yards of canvas that is ell wide will line 50 yards of say that is 3 quarters wide ?
Ans. 30 yds. 45. If an ounce of gold cost 4 guineas, what is the value of a grain ?
Ans. 27 d. 46. If 3 cwt of tea cost 401 12s; at how much a pound must it be retailed, to gain 101 by the whole ?
COMPOUND PROPORTION shows how to resolve such questions as require two or more statings by Simple Proportion ; and these may be either Direct or Inverse.
In these questions, there is always given an odd number of terms, either five or seven, or nine, &c. These are distin. guished into terms of supposition, and terms of demand, there being always one term more of the former than of the latter, which is of the same kind with the answer sought. The method is thus :
Set down in the middle place that term of supposition which is of the same kind with the answer sought.-Take one of the other terms of supposition, and one of the demanding terms which is of the same kind with it; then place one of them for a first term, and the other for a third, according to the directions given in the Rule of Three-Do the same with another term of supposition, and its corresponding demanding term ; and so on if there be more terms of each kind; setting the numbers under each other which fall all on the left-hand side of the middle term, and the same for the others on the right-hand side. Then, to work
By several Operations.—Take the two upper terms and the middle term, in the same order as they stand, for the first Rule-of-Three question to be worked, whence will be found a fourth term. Then take this fourth number, so found, for the middle term of a second Rule-of-Three question, and the next two under terms in the general stating, in the same order as they stand, finding a fourth term for them. And so on, as far as there are any numbers in the general stating, making always the fourth number, resulting from each simple stating, to be the second term in the next following one. So shall the last resulting number be the answer to the question.
By one Operation.-Multiply together all the terms standing under each other, on the left-hand side of the middle term ; and, in like manner, multiply together all those on the right-hand side of it. Then multiply the middle term by the latter product, and divide the result by the former prodact ; so shall the quotient be the answer sought.
1. How many men can complete a trench of 135 yards long in 8 days, when 16 men can dig 54 yards in 6 days ?
The same by two Operations.
2. If 1001 in one year gain 6l interest, wbat will be the interest of 7501 for 7 years ?
Ans. 2621 108. 3. If a family of 8 persons expend 1001 in 9 months ; how much will serve a family of 18 people 12 months ?
Aps. 3001. 4. If 273 be the wages of 4 men for 7 days ; what will be the wages of 14 men for 10 days ?
Ans. 61 158, 5. If a footman travel 130 miles in 3 days, when the days are 12 hours long ; in how many days, of 10 hours each, may he travel 360 miles ?
Ans. 98 days.
Ex. 6. If 120 bushels of corn can serve 14 horses 56 days ; how many days will 94 bushels serve 6 horses?
Ans. 10245 days. 7. If 3000 lb. of beef serve 340 men 15 days ; how many lbs will serve 120 men for 25 days ? Ans. 1764 lb 1111 oz.
8. If a barrel of beer be sufficient to last a family of 8 persons 12 days; how many barrels will be drank by 16 persons in the space of a year?
Ans. 60% barrels. 9. If 180 men, in 6 days of 10 hours each, can dig a trench 200 yards long, 3 wide, and 2 deep; in how many days, of 8 hours long, will 100 men dig a trench of 360 yards long, 4 wide, and 3 deep?
Ans. 15 days.
OF VULGAR FRACTIONS.
A FRACTION, or broken number, is an expression of a part, or sorce parts, of something considered as a whole.
It is denoted by two numbers, placed one below the other, th a line between them : 3 numerator
which is named 3-fourths. 4 denominator The denominator, or number placed below the line, shows how many equal parts the whole quantity is divided into; and it represents the Divisor in Division.—And the Numerator, or pomber set above the line, shows how many of these parts are expressed by the Fraction; being the remainder after division. Also, both these numbers are, in general,
— named the Terms of the Fraction.
Fractions are either Proper, Improper, Simple, Compound, or Mixed.
A Proper Fraction, is when the numerator is less than the denominator ; as, į, or ;, or , &c.
An Improper Fraction, is when the numerator is equal to, or exceeds, the denominator ; as, , or, or }, &c.
A Simple Fraction, is a single expression, denoting any number of parts of the integer ; as, f, or .
A Compound Fraction, is the fraction of a fraction, or several fractions connected with the word of between them; as, of, or off of 3, &c.
A Mixed Number, is composed of a whole number and a fraction together; as, 31, or 121, &c.
A whole or integer number may be expressed like a frac. tion, by writing 1 below it, as a denominator ; so 3 is t, or 4 is 4, &c.
A fraction denotes division; and its value is equal to the quotient obtained by dividing the numerator by the denominator ; soy is equal to 3, and į is equal to 4.
Hence then, if the rumerator be less than the denominator, the value of the fraction is less than 1. But if the numerator be the same as the denominator, the fraction is just equal to 1. And if the numerator be greater than the denominator, the the fraction is greater than 1.
REDUCTION OF VULGAR FRACTIONS.
Reduction of Vulgar Fractions, is the bringing them out of one form or denomination into another ; commonly to prepare them for the operations of Addition, Subtraction, &c. of which there are several cases,
To find the Greatest Common Measure of Two or more
The Common Measure of two or more numbers, is that number which will divide them both without remainder; 80, 3 is a common measure of 18 and 24 ; the quotient of the former being 6, and of the latter 8. And the greatest number that will do this, is the greatest common measure ; so 6 is the greatest common measure of 18 and 24 ; the quotient of the former being 3, and of the latter 4, which will not both divide further.
If there be two numbers only ; divide the greater by the less; then divide the divisor by the remainder; and so on, dividing always the last divisor by the last remainder, till nothing remains ; so shall the last divisor of all be the greatest common measure sought.
When there are more than two numbers, find the greatest common measure of two of them, as before; then do the same for that common measure and another of the numbers ;