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four ; eleven. Those in italics are the only ones really essential. Exemplification of the Proof by Addition with large numbers. 35241 First factor,
324 Second factor.
11418084 Product of 35241 by 324.
70482 First factor added to itself X10 by position 105723 Sum of 1st factor and 4th line x100 by pos.=300 140964 Sum of 1st line and 5th line
With larger figures in the multiplier. 32541 First factor.
978 Second factor.
31825098 Product of 32541 by 978.
65082 First factor added to itself. 130164 The last line added to itself.
prod. by 2.
260328 The last line added to itself. .
prod. by 8 227787 Sum of 1st, 4th, and 5th lines x 10 by position= 70 292869 Sum of 4th and 7th lines x 100 by position =900
37. Multiply 213-54 severally by 2-34, by 32'4, and by 423, and prove by the long method, or by addition.
38. Multiply 3521'4 severally by 451, by 32'5, and by 3-53, and prove.
39. Multiply 765-324 severally by 56-2, by 7*24, and by 258, and prove.
40. Multiply 9815462 severally by 374, by 865, and by 914, and prove.
41. Multiply 521432 severally by 1324, by 2413, and by 4132, and prove.
[Though the student may be allowed at first to practise by
the long method, yet he ought not to pass on to Division till he can use the short method with ease and rapidity.]
Practical Exercises. 1. If 25 men can do a piece of work in 25 days, how long will it take 1 man to do it?
2. If 16 men can do a piece of work in 14 days, how long will it take 1 man to do it ?
3. Two men set out from the same place, travelling in opposite directions; one at the rate of 42 miles, the other at the rate of 36 miles a day. How far would they be apart at the end of 5 days?
4. Two men set out from the same place, going in the same direction; the one in railroad cars at the rate of 300 miles a day, the other in a wagon, at the rate of 38 miles a day. How far would they be apart at the end of 3 days ?
5. Two men set out at the same time, but in contrary directions, to travel round a large circular course; the one at the rate of 3, the other at the rate of 5 miles an hour, and after 3 hours' travel they meet each other. How many
miles was the circumference of the course ?
6. A carpenter was employed on a building for 25 days, at $1.25 per day. He received at different times $20. How much remained due ?
[The following bills of parcels should be transferred to the slate, and the multiplication be performed horizontally. Where the price is given in cents, as 100 make a dollar, the whole number of cents divided by 100 will give the amount in dollars and cents.]
Boston, Oct. 4, 1853. 7. Mr. James Scott,
Bought of Wm. Smith,
5 lbs. of tea, at 65 cents.
3 gallons of lamp oil, at 94 cents
Boston, Sept. 15, 1854. 8. Miss Jane Roberts,
Bought of John Smith, 9 yards mousseline de laine, at 45 cents 8 do. do.
at 34 cents 32 do. cotton cloth,
at 7 cents 6 do. linen,
at 65 cents 9 do. calico,
at 15 cents 6 do. do.
at 13 cents 5 do. gingham,
at 24 cents 2 pairs of gloves,
at 58 cents
$17.40 John Smith.
Pittsford, Vt., Oct. 19, 1854. 9. Mr. John Fox,
Bought of Henry Sawyer,
X-inch do. at $7
Charged in account,
Henry Sawyer. 10. Mr. John Brown, 1854.
In account with Clement & Norton, Dr. Jan. 18. To 68 gallons of molasses, at 31 cents
To 425 lbs. brown sugar, at 6 cents Feb. 21. To 83 lbs. old hyson tea, at 54 cents
To 75 lbs. coffee, at 7 cents
By 18 bushels rye, at 75 cents
By 32 bushels buckwheat, at 45 cents Mar. 4. By cash to balance
11. Mr. Jacob Jones,
Bought of Henry Wheaton,
Albany, N. Y., Jan. 4, 1849.
12. Make a bill, like that in Ex. 10, of the following statement: On the 28th of June, 1849, William Jenkins bought of S. Talbot & Co., 34 gallons of molasses, at 36 cents per gallon; 26 bushels of salt, at 85 cents per bushel ; 14 pounds of tea, at 42 cents per pound; and paid 36 bushels of corn, at 58 cents per bushel; and 45 bushels of oats, at 42 cents per bushel. The balance, $0.44, was paid in money. By whom was it paid ?
13. Make a bill and receipt of the following statement : Mr. A. Williams bought of Samuel Roberts the following articles : A quarter of lamb, weight 7 pounds, at 8 cents a pound; a fillet of veal, weight 9 pounds, at 7 cents a pound; à quarter of mutton, 16 pounds, at 6 cents a pound; a pig, weight 12 pounds, at 10 cents a pound; 2 bunches of celery, 8 cents a bunch; and a bushel of turnips, for 35 cents.
Amount, $3-86. 14. William Hudson sold the following articles to Robert Benson. Make a bill and receipt for them. 325 bushels of corn, at 65 cents a bushel ; 73 bushels of wheat, at $1,25 ; 150 bushels of oats, at 40 cents; and 115 bushels of rye, at 72 cents.
Amount, $445-30. 15. Make a bill, like that in Ex. 10, of the following statement: Samuel Brown bought of William Roberts, of Philadelphia, Dec. 31st, 1853, 24 lbs. of tea, at 45 cents per pound; 16 bushels of salt, at 45 cents per bushel ; 25 yards of cotton cloth, at 13 cents per yard ; and 54 lbs. coffee, at 10 cents per pound. He paid a hog, weighing 425 pounds, at 8 cents per pound. The balance, $735, was paid in money. By whom was it paid ?
Or Multiplication by Two or more Equal Factors. INVOLUTION teaches the method of finding the powers of numbers.
1. A square is a figure with four equal sides, and four equal angles.* Squares are employed for the measurement of surfaces, or of any thing of which only two dimensions (length and breadth) are considered. In measuring surfaces, the square is the form to which all others are reduced. Thus, painters' work is estimated by the number of square feet covered by the paint; a sail, by the number of square yards it contains ; a field, by its contents in square rods. A great mistake is frequently made by using the terms square miles, square yards, &c., for miles square, yards square, &c. For, although 1 square mile is the same as 1 mile square, yet, with all other numbers the result is very different. Thus, 16 square miles are only equal to 4 miles square, as is evident from an inspection of the figure, in which each of the sixteen small squares may represent
one square mile, or square yard, or square foot, and the whole sixteen square miles form but four miles square. A slight examination of the figure will show that a square surface, or superficies, is measured by multiplying the length by the breadth.
2. If a number be once multiplied by
itself, the product is called the square, or second power, of that number. Thus, 4X4=16; 16 is the square, or second power of 4.
3. A cube is a solid body, with six equal square sides, and consequently of three dimensions,-length, breadth, and depth,
* An angle is the space comprised between two straight lines which meet in a point, as at A and B; or the quantity by which two straight lines departing from a point diverge al from each other. The lites containing the angle are called its sides, or legs. The size of an angle has no reference to the length of its sides. Thus, the angle at A is much greater than the angle at B.