MISCELLANEOUS EXAMPLES. 101. 1. The United States export 105,000 sewing machines yearly. If each machine does the work of 12 women, what is the value of the labor thus contributed by the United States to other nations each year of 306 working days, if labor be estimated at $1 per day? 2. The Union Pacific Railway is 1777 miles in length, and was built at an average cost of $106,775 per mile; what was the total cost of construction? 3. The bills issued by the U. S. Treasury for National Bank circulation, are in denominations of $1, $2, $5, $10, $20, $50, $100, $500, and $1000. How much money has one possessing 73 bills of each denomination? 4. The gold coins of the U. S. are in denominations of $1, $2.50, $3, $5, $10, and $20. How much money in a bag containing 365 of each of these coins? 5. The U. S. notes-greenbacks-are of the following denominations, viz.: $1, $2, $5, $10, $20, $50, $100, $500, $1000, $5000, and $10000. How large a debt could be paid with 7 of each of the above-named greenbacks? 6. How many feet of wire will be required to fence a field 1116 ft. square, with six wires on each of the four sides? 67 yd. Moquette Carpet @ $3 per yd. 32 yd. Border No. 1 @ $3 per yd. 12. The Boston "boot-maker" will enable a workman to make 300 pairs of boots daily. How many pairs can he make in a year having 309 working days? 13. My grain sales for the year 1888 were as follows: 516 bu. White Wheat @ 85¢ per bu. 250 bu. Peas @ 95¢ per bu. 287 "Rye @ 924 per bu. 321 66 "Buckwheat @ 85¢ per bu. 14. In New York State a bushel of barley weighs 48 lb., of clover seed 60 lb., of flax seed 55 lb., of beans 62 lb., of buckwheat 48 lb., of rye 56 lb., of corn 58 lb., of oats 32 lb., of potatoes 60 lb., of timothy seed 44 lb., and of wheat 60 lb. What will be the total weight of 5 bushels of each of the products named? 15. In freighting, lime and flour are each estimated to weigh 200 lb. per barrel; pork and beef each 320 lb.; apples and potatoes 150 lb. each; cider, whisky, and vinegar each 350 lb. What will be the freight at 20¢ per hundred pounds, on a car containing 15 barrels of each of these products? 16. I bought 10 acres of land at $2250 per acre and laid it out in 75 city lots, expending $4725 for grading and streets, $CSO for sidewalks, and $87 for ornamental trees. I then sold 40 of my lots at $500 each, 20 at $450 each, and exchanged the remainder for a farm of 110 acres, the cash value of which was $65 per acre. How much was gained or lost? 17. A gardener rented 5 acres of land for $20 per acre and paid $63 for seeds, $20 for fertilizers, $246 for labor, and $52 for freight. He sold 2145 bushels of turnips for $429, 1710 bushels of beets for $513, 4350 bunches celery for $174, and 800 heads cabbage for $40. What was his gain? 18. A man earning $2.50 per day, works 306 days per year for five years. His annual expenses are, for board $156, for clothing $47, for charity $12, and he expends $2 per week for incidentals. If he deposit his surplus each year in a Savings Bank, what amount will he deposit during the time? 19. The U. S. coupon bonds are in denominations of $50, $100, $500, and $1000, and the registered bonds in denominations of $50, $100, $500, $1000, $5000, and $10000. Of the 4's of 1891, and the 4's of 1907, there are registered bonds of the denominations of $20,000 and $50,000. What would be the aggregate face value of twelve of each of the bonds above named ? 20. A man rented a farm of 132 acres of grain land, 67 acres of pasture land, and 45 acres of meadow land; paying for the grain land $7 per acre, for the pasture land $4 per acre, and for the meadow land $11 per acre. He produced 61 bushels of oats per acre on 45 acres, 32 bu. barley per acre on 30 acres, 75 bu. corn per acre on 15 acres, 150 bu. potatoes on 9 acres, 28 bu. buckwheat on 20 acres, and 24 bu. beans per acre on the remainder of the grain land. He re-let the pasture land for $200, and on the meadows cut 2 tons per acre of hay worth $13 per ton. If he paid $695 for labor and $467 for other expenses, did he gain or lose, estimating oats at $275, barley at $672, corn at $394, potatoes at $743, buckwheat at $420, and beans at $2 per bushel ? DIVISION. 102. Division is the process of finding how many times one number is contained in another of the same kind. 103. The Dividend is the number divided. 104. The Divisor is the number by which the dividend is divided. 105. The Remainder is the part remaining when the division is not exact. 106. The Sign of Division is the character ÷ ; it indicates that the number before it is to be divided by the number after it. Thus, 24 ÷ 3 = 8, is read 24 divided by 3 equals 8. We see by this operation that 3 is an exact divisor of 24, also that 3 and 8 are factors of 24. REMARK. From the above it is clear that the dividend in division corresponds to the product In multiplication, and the divisor and quotient to the multiplier and multiplicand, or the factors in multiplication. 107. General Principles.-1. Multiplying the dividend multiplies the quotient. Thus, 48 ÷ 6 = 8; (48 × 2 ) ÷ 6 = 16. 2. Dividing the divisor multiplies the quotient. Thus, 48÷6 = 8; 48÷ (6 ÷ 2 ) = 48 ÷ 3 = 16. 3. Dividing the dividend divides the quotient. Thus, 48 ÷ 6 = 8; (48 ÷ 2) ÷ 624 ÷ 6 = 4. 4. Multiplying the divisor divides the quotient. Thus, 48÷6 = 8; 48÷ (6 × 2) = 48÷124. 108. General Law.-I. Any change in the dividend produces a like change in the quotient. II. Any change in the divisor produces an opposite change in the quotient. III. A like change in both dividend and divisor will not change the quotient. 109. General Rules.-1. If the dividend and divisor be given, the quotient may be found by dividing the dividend by the divisor. 2. If the dividend and quotient be given, the divisor may be found by dividing the dividend by the quotient. 3. If the divisor and quotient be given, the dividend may be found by multiplying the divisor by the quotient. 4. If the divisor, quotient, and remainder be given, the dividend may be found by multiplying the divisor by the quotient and adding the remainder to the product. 110. To Prove Division.-Divide the dividend by the quotient, or multiply the divisor by the quotient. In divisions which are not exact, add the remainder to the product of the divisor and quotient; the sum thus obtained should be the dividend. 111. The Reciprocal of a number is one, or unity, divided by that number. A reciprocal will be produced by changing the relation of dividend and divisor; as, 2847, while 428; the resulting is the reciprocal of the first quotient 7. Operations in Division are of two kinds, Short Division and Long Division. 113. In Short Division, operations are restricted to those divisions in which the divisor consists of one figure, or is a number coming within one's thorough knowledge of the multiplication table. 114. When the Divisor consists of only one figure. EXAMPLE.-Divide 6482 by 2. OPERATION. 2) 6482 3241 EXPLANATION.-Write the divisor at the left of the dividend, separating them by a line, next draw a line below the dividend and then divide each figure of the dividend by the divisor, writing the quotient below the figure divided. Thus, 2 is contained in 6 thousands, 3 thousands times; write the 3 below the 6 in thousands' column, next, 2 is contained in 4 hundreds, 2 hundreds times; place the 2 below the 4 in hundreds' column; 2 is contained in 8 tens, 4 tens times; write the quotient in tens' column; 2 is contained in 2 units, 1 unit times, or once; write 1 in units' place, thus completing the division, and obtaining 3241 as a quotient 115. When the Divisor is a Number within one's thorough knowledge of the Multiplication Table. EXAMPLE.-Divide 31605 by 15. OPERATION. 15 ) 31605 2107 EXPLANATION.-Write the terms as before. Divide 31 by 15 and obtain 2, which write below the 1 as the first figure of the quotient; next, 15 is contained in 16, once; write 1 in hundreds' column; 15 in 10, 0, or no times; write the 0, or cipher, in tens' column; 15 in 105, 7 times; write the 7 as units of the quotient, thus completing the division, and obtaining the quotient 2107. 116. When any Figure or Figures of the Dividend will not Exactly Contain the Divisor. EXAMPLE.-Divide 394015 by 8. OPERATION. 8) 394015 49251 EXPLANATION -Write the terms as before. Since 3 hundreds of thousands is not divisible by the divisor 8, unite the 3 hundreds of thousands and the 9 tens of thousands, obtaining 39 tens of thousands; divide this by 8 and obtain for the first figure of the quotient 4 tens of thousands, with a remainder of 7 tens of thousands; write the 4 below the 9 as the tens of thousands of the quotient, and unite the 7 tens of thousands to the 4 thousands of the dividend and divide; 8 is contained in 74 thousands, or 7 tens of thousands+4 thousands, 9 thousands times with a remainder of 2 thousands; write the 9 in the column of thousands, and unite the 2 thousands to the next figure of the dividend; 8 is contained in 20 hundreds, 2 hundreds times with a remainder of 4 hundreds; write the 2 hundreds in the column of hundreds, and unite the 4 hundreds to the next figure of the dividend; 8 is contained in 41 tens, or 4 hundreds + 1 ten, 5 tens times, with a remainder of 1 ten; write the 5 in tens' column and unite the 1 ten to the last figure of the dividend; 8 is contained in 15 units, 1 unit times, or once, with a remainder of 7 units, or 7; write the remainder over the divisor in the form of a fraction and annex the result to the entire part of the quotient, thus obtaining 492513 as the complete quotient of 394015 divided by 8. Rule.-I. Write the divisor at the left of the dividend with a line separating them. II. Beginning at the left, divide each figure of the dividend by the divisor, and write the resulting quotient underneath the dividend. III. If after any division there be a remainder, regard this remainder as prefixed to the next figure of the dividend, and divide as before. IV. Should any partial dividend considered, be less than the divisor, place a cipher in the quotient and regard the undivided part as prefixed to the succeeding figure in the dividend and again divide. V. If the division is not exact, write the remainder over the divisor in fractional form, and annex the result to the integral part of the quotient. 118. When the Divisor is a Composite Number. When the divisor is a composite number the operation may be simplified by using the factors of the divisor. |