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MULTIPLICATION CONTRACTED. Any number, from 12 to 20, can be multiplied at one operation, or one line of product.
Rule. Multiply the multiplicand by the unit figure of the multiplier, and add to the product of each multiplication that figure which stands next on the right hand of that which you multiplied ; and when you have gone through the whole, add to the last figure in the multiplicand, what you have to carry, (which is the last figure of the product.)
EXAMPLES. 1. Multiply 4241 by 14. Thus, 4241
product 59374 59374 2. Compute the product of 16 times 2414, with ten figures, in the whole operation. 3. Multiply 24351 by 19.
Ans. 462669. To multiply by any number of nines in one line, as 9, 99, 999, &c.
RULE.-Annex as many ciphers to the multiplicand as there are nines in the multiplier, and from this number subtract the given multiplicand, and the remainder will be the answer required.
EXAMPLES. 2. What is the product of 48652 multiplied by 99999?
Aus. 8465115348 2. Multiply 72031 by 999 ?
Ans. 7131069. The circumference of the earth being 360°, and as it performs one entire revolution in 24 hours, it is evident, that the motion of the earth, on its surface, from west to east, is
150 of motion in 1 hour of time; consequently,
l' of motion in 4 seconds of time. So having the longitude of two places given, we can find the difference of time: the place easterly having the time of day earlier than the place westerly.
Rule.-Multiply the difference of longitude by 4, and the prodact is the difference of time in minutes.
1. Boston being 6° 40' E. longitude from the city of Washington, when it is 12 o'clock at the city of Washington, what is the hour at Boston ? Ans. 26 minutes 40 seconds past 12 o'clock.
2. Required the difference of time between New-York and Philadelphia; the difference of longitude is 1 degree, 7.25 minutes.
Ans. 478 minutes.
my horse ?
MISCELLANEOUS EXAMPLES. 1. What decimal is that, which being multiplied by 15, the pro duct will be .75 ?
Ans. 05. 2. A man exchanged 70 bushels of rye, at $.92 cents per bushel
, for 40 bushels of wheat at $1.375 per bushel, and received the balance in oats, at $.40 per bushel; how many bushels of oats did he receive ?
Ans. 23.5. 3. What premium must I pay for the insurance of my house against loss by fire, at the rate of į or .005 percent., if my house be valued at $2475 ?
Ans. $12.375. 4. If my horse and saddle are worth $84, and my horsę be worth 6 times as much as my saddle, pray what is the value of
Ans. $72. 5. If the floor of a square room contain 36 square yards, how many feet does it measure on each side ?
Ans. 18 feet. 6. How many solid inches in a brick which is 8 inches long, 4 inches wide, and 2 inches thick?
Ans. 64. 7. How many bricks in a cubic foot ?
Ans. 27. 8. How many bricks will it take to build a wall 40 feet long, 12 feet high, and 1 foot thick?
Ans. 12960. 9. If a circle be 14 feet in diameter, what is its circumference?
Ans. 43.9724–44. 10. If the distance through the centre of the earth, from side to side, be 7911 miles, how many miles around it?tra 24863+\?
11. What is the number of square miles on the surface of the earth, supposing its diameter 7911 miles, and its circumference 24853 miles?
Ans. 196612083. 12. How many square inches of leather will cover a ball 3.5 inches in diameter?
NOTE.-The area of a globe or ball is 4 times as much as the area of a circle of the same diameter, and may be found therefore, by multiplying the whole circumference by the whole diameter.
Ans. by rule 1st. 38.5, and by rule 2d. 38.48,5 13. What are the solid contents of a round stick, 20 feet long, and 7 inches through; that is, the ends being 7 inches in diameter.
Ans. by rule 1st. 5; cubic feet=64} square feet. NOTE.—The mean diameter of a cask (of a common curvature) may be found by adding two-thirds, or if the staves be but little curving, six-tenihs of the difference between the head and bung diameters, to the head diameter.-The cask will then be reduced to a cylinder.
ALLIGATION. ALLIGATION is the method of mixing two or more articles, of different qualities, so that the composition may be of a mean, or middle quality.
Alligation consists of two kinds, viz. Alligation Medial, and Alligation Alternate.
ALLIGATION MEDIAL, Is when the quantities and prices of several articles are given, to find the mean price of the mixture, composed of those articles, or things given.
RULE.Find the value of all the articles, and then divide this value by the sum of the articles, which will give the price required.
1. If 6 gallons of brandy, worth $1.25 a gallon, be mixed with 9 gallons worth 80 cents per gallon, and 5 gallons of whiskey, worth 40 cents a gallon, what is 1 gallon of this mixture worth?
Ans. $.835, or 831 cents. 2. If a bushel of Indian corn, at 75 cents a bushel, be mixed with 5 bushels of rye, at 80 cents per bushel, and 15 bushels of oats, at 30 cents per bushel, what will be the value of a bushel of the mixture ?
Ans. 442cents. 3. A wine merchant mixed 12 gallons of wine, at 75 cents per gallon, with 24 gallons at 90 cents,
end 16 gallons, at $1.10 cents; what is a gallon of this composition worth?
Ans. $.92611 Questions.-Alligation Medial is whal? Recite the rule, &c.
ALLIGATION ALTERNATE, Teaches the nethod of finding the quantity of each article given, at their respective rates, that must be taken to make a compound at any proposed rate.
1. Write the given prices of the articles under each other, and the mean rate on the left hand of those.
2. Connect with a continued line each price that is less than the mean rate with one or more that is larger, and each price larger than the mean rate with one or more that is less.
3. Write the difference between the mean rate (or price), and the price of each article opposite to the price with which it is connected.
4. Then the sum of the differences, standing against any price, will express the relative quantity to be taken of that price.
1. A grocer would mix sugars, at 9 cents, 11 cents, and 13 cents per pound; what quantity of each kind must he take, that the mix. ture may be worth 12 cents a pound?
16. cis. lb. 9
1 at 9 9 Mean rate 12 cents
1 at 11=11 134
3+1=4 at 13=52
12 proof. 2. A merchant would mix wines, at $1.20, $1.50, $2, and $2.50 per gallon, so that the mixture should be worth $1.75 per gallon; what quantity of each kind must he take?
3. A grocer would mix rum at 80 cents, and at 70 cents a gallon, with water, that the mixture may be worth 75 cents a gallon; what quantity of each sort must be take?
Ans. 80 gal at 80 cts., 5 gal. 'at 70 cts. and 5 ga). of water.
When the rates of all the articles, the quantity of one of them, and the moon rate of the whole mixture are given, to find the several quantitios of the rest, in proportion to the quantity given.
Take the difference between each price and the mean rate, and place them alternately, as in case 1. Then, as the difference standing against that article whose quantity is given, is to that
quantity, so is each of the other diferencos, severally, to the several quantities required.
1. A merchant has 20 pounds of tea, at $1.04 per pound, which he would mix with some at 98 cents, some at 92 cents, and some at 80 cents per pound; how much of each kind must he take to mix with the 20 pounds, that he may sell the mixture at 96 cents per pound?
104. 4 stands against the given quantity.
cts. 92 8
16 : 80 at 98 809 2 As 4 : 20 :: 8 : 40 at 92 Ans.
2 : 10 at 80 2. Bought a pipe of brandy, containing 120 gallons, at $1.30 a. gallon; how much water must be mixed with it to reduce the first price to $1.10'a gallon?
Ans. 21, gals.
PERMUTATION. PERMUTATION is the method of finding how many different changes can be made of any proposed number things.
RULE.---Multiply all the terms of the natural series of numbers continually together, from one up to the given number, and the last product will be the changes required.
1. Five gentlemen agreed to remain together as long as they could arrange themselves differently at dinner; how many days did they remain together ?
Ans. 120 days. 2. Christ church, in Boston, has 8 bells; how many changes may be rung on them?
Ans. 40320 changes. 3. Seven gentlemen, who were travelling, met together by chance, at a certain inn upon the road, where they were so well pleased with their host and each other's company, that (in a frolic) they offered him 30 dollars, to stay at that place so long as they, together with him, could sit every day at dinner in a different order. The host, thinking that they could not sit in many different positions, imagined that he should make a good bargain, and readily entered into an agreement with them, and so made himself the eighth. I demand how many different positions they sat in; and how long they stayed at the inn?
Ans. They sat in 40320 positions; and the
Questions.--Permutation teaches what? What is the rule?
COMBINATION. COMBINATION teaches how many different ways a less number of things may be combined out of a larger; thus, out of the letters a, b, c, d, are 6 different combinations of 2, viz. ab, ac, ad, dc, db, bc. Thus, 4x3=12
1x2= 2 RULE.-1. Take a series of as many terms decreasing by 1, from the number, out of which the election is to be made; and find the product of all its terms for a dividend.
2. Take a series beginning with 1, and increasing by 1, up to the number to be combined, and find the product of all its terms for a divisor.
3. Divide the dividend by the divisor, and the quotient will be the number sought.
EXAMPLES. 1. How many combinations may be made of 5 letters in 10?
Ans. 252. Questions.-Combination teaches what? Recite the rule.