Second, how much coffee is worth 112 of tea, or the 1120 of sugar. Third, how much biscuit is worth the 1120 of sugar, or 112 of tea, or 210 of coffee. If 40 lbs. of tea at 3s. 2d. per lb. are bartered away for Plantation coffee at 1s. 6d., find how much is received in exchange. 1. If 5 oxen are worth 24 sheep, and 4 sheep are worth £13, what are 55 oxen worth? 2. If 3 lbs. of tea be worth 8 lbs. of coffee, and 12 lbs. of coffee be worth 31 lbs. of sugar, what quantity of sugar may be had for a chest of tea weighing 28 lbs.? 3. If 1 lbs. of tea be worth 3 lbs. of coffee, and 6 lbs. of coffee worth 24 lbs. of sugar, and 71⁄2 lbs. of sugar worth 14 lbs. of soap, how many lbs. of soap are 3 lbs. of tea worth? 4. If 4 sheep are equal in value to 5 pigs, and 20 pigs are of the same value as 3 cows, while 40 cows are valued at the same price as 27 horses, how many horses can I exchange for a flock of 327 sheep? 5. If 4 books are worth 10s., and for 25s. I can buy 208 quires of white paper, while 32 quires of white paper are of equal value to 60 of brown, how much brown paper must I exchange for 18 books? 6. If 3 ducks are worth 4 chickens, and 3 geese are worth 10 ducks, find the value of a goose, a pair of chickens being worth 4s. 6d. 7. What is the value of 78 yds. of cloth, if 40 yards of cloth are worth 14 bush. of corn, and 26 bush. of corn are valued at 14 of £8, 10s. sperm 8. If 40 cwt. of coals at 29s. 6d. per ton be exchanged for candles at 1s. 6d. per lb., find how many lbs. of candles are given. 9. One grocer exchanges ginger at 1s. 4d. per lb. with another for 64 lbs. of nutmegs at 3s. 9d. per lb.; find what weight of ginger is returned. 10. A tradesman exchanges a hogshead of sherry wine (56 gals.) at 21s. a gal. for gin at 13s.; find the quantity of gin. 11. If 2 guineas make 3 Napoleons, and 15 rix dollars make 4 Napoleons, and 6 ducats make 7 rix dollars, how many ducats are there in £490? SQUARE ROOT. Involution means involving a number, or the multiplication of the same number by itself. If 4 be multiplied by 4, we have the square of 4, or 16: 4 squared is frequently written 42=16, the second power of 4. If 4 be multiplied by itself twice, as 4 × 4× 4, we have 64 or 4 cubed; frequently written 43=64, the third power of 4. 1 4 9 16 25 36 49 64 81 100 1 8 27 64 125 216 343 512 729 1000 Evolution is the rolling or evolving out of numbers. In involution, we, as it were, "fold up" the numbers, now we have to unfold them; or evolution is the reverse of involution, the number that has been cubed or squared has to be found. The number obtained by evolution is called the root, such as the square root, cube root, etc. The following signs and symbols are of frequent use: ✔ means the root or square root of 4; quantities written without an index, as 49, 100, mean the square root; 8 means the cube root. Fractions. The squares, cubes, etc., of fractions are obtained by multiplying both numerator and denominator, each by themselves as =;cubed ; () = 256 2 2 = 8 27; 4 1 The square root, cube root, etc., of fractions is obtained by extracting the square root, cube root, etc., of the numerator square root of is and denominator separately, as the 16 25 5 EXERCISES.-LIX. 1. Square the following numbers: 9, 12, 17, 20, 401, and 527. 2. Find the square of the following numbers: 1000, 999, 8471, and 5240. 3. Find the square of the numbers: %, 7, 24, 41, 231, 901. 4. Find the cubes of 9, 8, 7, 10, 40, 1000. To Extract the Square Root.-Find the square root of five millions three hundred and eight thousand four hundred and sixteen. Explanation of Square Root.—First, Begin at the righthand figure, and place a dot over each alternate figure, which will divide the number into pairs, 16, 84, 30, with an odd 5. Take the highest square number under 5, which is the square of 2 (4). The 2 is placed in the answer a, and the 4 under the 5; subtracting, I remains. Second, Bring down the next pair of figures, 30, and proceed to obtain the divisor b, by doubling the quotient 2, and placing the 4 thus obtained at b; 4 is the trial divisor, which goes into 13 three times; the 3 is placed both at a and b, and the 43 multiplied by the 3 and placed under 130; subtracting, 1 is left. Third, The next pair of figures (84) is brought down, and the answer a (23) doubled and placed at c; 46 is the new trial divisor, which will not go into 18, or, supposing there were another figure with it, as 460, will not go in 184, so place the nought at c and a, and bring down the next pair of figures (16). The trial divisor, 460, will go into 1841 about 4 times; put the 4 at a and c, and multiply 4604 by 4, this gives 18416 and nothing over. To find the square root of fractions. 46 Find the square root of, 13, 11, 26%. In each case the square roots of the numerator and denominator are taken, being careful, first, to reduce the mixed numbers to improper fractions. When the square root is obtained, if the answer be an improper fraction, it is wise to reduce it to a mixed quantity. Fourth Root.-The fourth root of a quantity is found by taking the square root of the square root. Since 256-4x4x4x4=42 x 42=162=44; ... the fourth root is the square root of the square root of a quantity. EXERCISES.-LX. 1. Find the square root of 1, 12. 2. Find the square root of the following number, viz., 844561. 3. Find the square root of 910116. 4. Find the square root of 3, 5, 7, 11. 5. Find the square root of 19, 23, 29. 6. Find the square root of 1, of 2, and of 5. 7. Extract the square root of 13 to five places of decimals. 8. Extract the square root of 459684. 9. Extract the square root of 6789265609. 10. Multiply the square root of 169 by 3 times the difference between 4 and 5. ANSWERS. 1. 420. 2. 529. 3. 6704. EXERCISES I. 4. 61,829. 5. 908,736. 7. 205,205,205. 10. 238,828. 11. 4821,732,943. 12. 206,003,976. EXERCISES II. five hundred and ninety; thirtyeight thousands one hundred and thirty-three. 5. One million one hundred and forty-seven thousands eight hundred. 6. Four hundred and nine thousands and forty. 7. Twenty-one millions and twenty-one; seventeen thousands and one. 8. Four millions seven hundred and one thousands seven hundred and one; eighty thousands and nine. 9. Four millions seven hundred thousands one hundred; eight 1. Ninety-seven; three hun-hundred and ten thousands and dreds and sixty-five; four thou-one. sands and eighty-one; four thousands and nine. 10. Ten millions; eleven millions one hundred and eleven thousands one hundred and eleven. 2. Seven hundred thousands eight hundreds and fifty-one ; 11. Two millions two hundred one million and one; one million and twenty-two thousands two one thousand and one; nine hundred and twenty-two; two thousands; fifty millions ninety-millions and two. one thousands and four. 12. One million and one; ten 3. Four thousand and fifty-one thousands and one; twenty-one millions six hundred and four millions twenty-one thousands thousands three hundreds; forty- and twenty-one. one millions forty-one thousands and forty-one; nine millions nine thousands nine hundreds. 4. Eight hundred and fortyfive thousands two hundred and twenty; seventy-three thousands 1. 2368. EXERCISES III. 2. 17377. |