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47. How much will a pile of wood 69 ft. long, 4 ft. wide, and 5 ft. high, cost at 3 cents per cubic foot ?
48. I bought 17 boxes of sugar, each containing 357 pounds, at $.13 per lb. What was the amount of my purchase ?
49. I bought a house at auction for $1975.37. I paid $547.26 for improvements and repairs, $137.29 for having it painted, and then sold it for a sum equal to three times as much as it had cost me in all. For how much did I sell it?
50. A man bought a bar of gold weighing 9 oz. 17 dwt. 13 gr., at $.04 per grain. What did it cost him ?
51. Definitions, Explanations, and Problems.
(a.) COMPOUND MULTIPLICATION is the multiplication of Compound Numbers. It does not differ in principle from Simple Multiplication.
1. What is the product of 18lb 53 63 multiplied by 6 ? to 3 3
Solution. -6 times 63 = 363, which, since 83 = 13, 18 5 6
must equal as many ounces as there are times 8 in 36, which are 4 times and 4 remainder. Hence 363 = 43
and 43. Writing the 43, we add the 43 to the product 110 10 4
of the ounces, thus: 6 times 53 = 303, and 43 are 343, which, since 123 = 1tb, must equal as many pounds as there are times 12 in 34, which are 2 times and 10 remainder. Hence 343 = 21 and 103. Writing the 103, we add the 21 to the product of the pounds, thus :
6 times 18th = 1081), and 21 are 11011), which, being the highest denomi. nation, we write.
The answer, then, is 1101 103 43.
23. IIow far will a man travel in 9 days, if he travels 33 m. 5 fur. 27 rd. 5 yd. 2 ft. 10 in. per day?
24. What is the weight of a dozen silver table-spoons, each weighing 1 oz. 5 dwt. 17 gr. ?
25. I bought in London 17 bags of coffee, each weighing 2 cwt.
(a.) DIVISION is a process by which we ascertain how many times one given number is contained in another; or by which we ascertain what number is contained a given number of times in another given number.
ILLUSTRATIONS. — “24 = how many times 3 ?” and, “How many apples at 3 cents each can be bought for 24 cents ?" are questions illustrating the first part of the definition (for they require us to find how many times 3 must be taken to equal 24); while "What is one-third of 24 ?” and “If 3 apples cost 24 cents, how many cents will 1 apple cost ?” are questions illustrating the second part of the definition (for they require us to find what number inust be taken 3 times to equal 24). Each question requires that 24 should be divided by 3.
(b.) The number to be divided is called the Dividend; the number by which we divide is called the Divisor; and the result is called the QUOTIENT.
(c.) The REMAINDER, if there be one, will always be of the same denomination as the dividend.
Note. — Division is the reverse of Multiplication; for in Multiplication the factors are given and the product is required; while in Division the product and one factor are given, and the other factor is required. The dividend corresponds to the product, and the divisor and quotient to the factors.
(d.) We begin the division with the left-hand or highest denomination of the dividend. We first find how many times the divisor is contained in the smallest part of the dividend which will contain it, writing the result as the first figure of the quotient. We then reduce the remainder, if there be one, to the next lower denomination, and add to it the next figure of the dividend. We find how many times the divisor is contained in this sum, writing the quotient, and reducing the remainder as before, and so proceed till the division is completed.
1. $1707 = how many times $6 ?
Solution. — $4767 contains $6 as many times as 6)4767 – 3 4767 contains 6.
794 We divide thus: 6 is contained in 47 hundreds, 7 hundreds times, with 5 hundreds remainder (for 7 hundreds times 6 = 42 hundreds, and 5 hundreds added are 47 hundreds). Writing 7 as the hundreds figure of the quotient, we reduce the 5 hundreds to tens, thus:
5 hundreds = 50 tens, and 6 tens added are 56 tens. 6 is contained in 56 tens 9 tens times, with 2 tens remainder (for 9 tens times 6 are 54 tens, and 2 tens added are 56 tens). Writing 9 as the tens figure of the quotient, we reduce the 2 tens to units, thus :
2 tens = 20 units, and 7 units are 27 units. 6 is contained in 27 units 4 units times, with 3 units remainder (for 4 units times 6 are 24 units, and 3 units added are 27 units). We write 4 as the units figure of the quotient. Hence the quotient is 794, and the remainder is 3.
Note. — The quotient really represents 794 times $6, or 794 units each equal to $6; and the remainder, 3, represents $3, and is an undivided part of the dividend.
2. What is f of $4767?
Solution. — * of $4767 may be found by dividing 4767 by 6, as in the example last explained. The quotient is 794, and the remainder is 3; thus showing that I of $4767 is $794, with a remainder of $3.
(e.) Methods of Proof. — 1st. Multiply the quotient by the divisor, and to the product add the remainder. The result should equal the dividend.
2d. Suhtract the remainder from the dividend, and divide the difference thus found by the quotient. The result should equal the divisor.
(f.) The remainder may be made a part of the quotient, by writing it over the divisor so as to form a fraction.
ILLUSTRATIONS. - In the first of the above questions, the quotient would become 7942, i. e. 7943 times $6; while in the second it would become $7943
(g.) Whenever there is a remainder, the quotient may be carried out to Decimal Fractions, by reducing the remainder to tenths, then to hundredths etc. dividing at each step as already explained.
ILLUSTRATION. — In the above example, reducing the 3 units remainder to tenths, gives 3 units = 30 tenths, and 6 is contained in 30 tenths 5 tenths times. The quotient now takes the form 794.5, instead of 7948.
53. Examples and Practical Problems.
16. 4 of 479.
31. How many barrels of flour at $8 per barrel can be bougb for $3576 ?
REASONING PROCESS. — If 1 barrel can be bought for $8, as many barrel can be bought for $3576 as there are times $8 in $3576, which may be found by dividing 3576 by 8.