5. Compound Multiplication. Compound Multiplication is the method of finding the amount of a given number, consisting of different denominations, by repeating it a proposed number of times. Rule. * Write the multiplier under the lowest denomination of the multiplicand. Multiply the several denominations successively by the multiplier, setting down the excess and carrying from each denomination to the next higher, as in Compound Addition. Proof. Examples 1. What will 5lb. of tea cost at 1 dol. 2 dimes, 7 cts. per pound ? $ d. cts. or $ cts, or cts. By this example it will be seen 1 2 7 1 27 127 that the operations by this rule in 5 5. 5 Federal Money are precisely the same as in Simple Multiplication, 6 3 5 6 35 635 cts. one doilar, 2 dimes, 7 cts. being just equal to 1 dol. 27 cts. or to 127 cts. Hence the 127 cents multiplied by 5 the answer is 635 cts.=6 dols. 35 cts.=6 dols. 3 dimes, 5 cts. and the given numbers may in all cases be expressed as a simple number in the lowest denomination mentioned, or as a compound number. 2. What is the cost of 6 lb. of tobacco, at 2s. 6d. 2qrs. per lb? Here 6 times 2 is 12, but 12qrs. are equal to 3d. there2 6 2 fore set down 0 and carry 3. Then 6 times 6 is 36 and 3 6 to carry is 39d.=3s. 3d. set down 3d. and say 6 times 2 is 12 and 3 to carry is 15s. which set down. 15s. 3d. 0 3. What will 3lb. of green tea 7. What will 5lb. of loat sugar cost, at 9s. 6d. per pound ? cost, at 1s. 3d. per pound ? Ans. £1 8s. 6d. Ans. 63. 3d. 4. What will 6lb. of nails cost 8. What will 8 bushels of corn at 9 cents per pound ? cost, at 5d. 7cts. or 57cts. per Ans. 5 dimes, 4cts, or 54cts. bushel ? 5. What will 9cwt. of cheese Aus. 84 5d. 6 cts. or 84 56cts. cost, at £1 11s. 5d. per cwt. ? 9. What will 9 yards of cloth cost at 5s. 4d. per yard ? Ans. £14 2s. 9d. Ans. 2 Ss. 6. What will 6 cows cost, at 10. What will 12 gallons of 4 6s. 8d. each? brandy cost, at 9s. 6d. per gal. A Ans. £26. Ans. £5 14s. * The product of a number consisting of different denominations by a simple number, is evidently expressed by the several products of the different parts multiplied by the simple number. Thus, £2 68. 4d. multiplied by 6, the several products will be £ 12 36s. 24d.=(by taking the shillings from the pence and the pounds from the shillings and placing them in the shillings and pounds respectively) to £13 18s. Od. wbich is agreeable to rule ; and the same will be true when the multiplicard is any compound number whatever. s. d. qrs. £ s. d. When the multiplier exceeds 12, and is a composite number, the component parts may be employed successively, as in Simple Multiplication, instead of multiplying by the whole number at once. Examples. 1. What will 16 cwt. of cheese 2. What will 28 yds. of broad cost, at €1 188 8d per cwt. ? cloth cost, at 19s 40 per yard ? Here because 16 is Ans. £27 1s. 4d. 1 18 8 produced by multiply 3. What will 96 quarters of 4 ing 4 by 4, multiply rye cost, at £i 3s 4d per quar7 14 8 the price by 4, and that ter? Ans. £112. 4 product again by 4. 4. What will 63 bushels of rye Ans. 30 188 cost, at 63 cents.per bushel ? 1 Ans. 3969cts, or $39 69cts. 2. When the multiplier cannot be produced by the multiplication of two small numbers, take two such nuinbers as come the nearest to it, and then find the value of the odd parts and add or subtract as the case requires. Examples. 1. What will 47 yards of cloth 2. What will 94 pair of silk cost, at 178 9d per yard ? stockings cost at 12s 2d per pair ? £ s. d. Multiplying by 5 Aus. £ 57 Ss. 8d. 0 17 9 and by 9 gives the 5 price of 45 yds. but 3. What will 31 bushels of this is 2 yds. short 4 8 9 of the given quan oats cost, at 25 cents per bush. ? Ans. 775cts, or $7 75 cts. Therefore multiply 17s 9d by. 39 18 9 2, and it gives £1 silver spoons, each weighing loz 4. What is the weight of 25 1 15 6 15s 6d for the price of 2 yds, which add-9pwts i4grs? Ans. 41,14 3 ed to the price of Ans. 21b. 10oz. Opwt. 10grs. 45 yds. gives the price of the whole. 3. When the multiplier exceeds 100, find the cost of 100, multiply it by the number of hundreds, and to this product add the cost of the odd parts and their sum will be the answer required. Examples: 1. What will 512 bushels of wheat 2. What will 235 bushels of cost, at 5s 100 per bushel? wheat cost, at 81 25 cents per 5 10 bushel ? Ans. 29375cts or $293 10 75cts. or 29 E. 83 70. 5cts. 2 18 4 price of 10 bushels. 10 3. What will 700 bushels of 29 3 4 price of 100 bushels. potatoes cost, at 1s Sd per bush. : 5 Ans. £43 15s. 145 16 8 price of 500 bushels. 4. What will 297 yards cost, 3 100 price of 12 bushels. at 17s 3d 2qrs per yard ? £149 6 9 price of 512 bushels. Ans. €256 155. 7{d. 9 tity 66 Duodecimals. DUODECIMALS are so called because the denominations decrease by 12 from the place of feet towards the right hand, as in the following TABLE. Rule. * Write the several terms of the multiplier under the corresponding terms of the multiplicand ; then multiply the whole multiplicand by the several terms of the multiplier successively, beginning at the right hand, and placing the first term of each of the partial products under its respective inultiplier, remembering to carry one for every 12 from a lower to the next higher denomination, and the sum of these partial products will be the answer, the left hand term being feet, and those towards the right primes, seconds, &c. This is a very useful rule in measuring wood, boards, &c. and for artificers in finding the contents of their work. · Examples 1. How many square feet in a 2. How much wood in a load floor 10 feet 4 long, and 7 feet 7ft. 6' long, 4ft. 8' wide and 4ft. 8' wille? 10f. 4' Ans. 140ft. or 1 cord 12ft. 7 8 Multiply the length by the width, and this product by the height. 3. How many square feet in a board 16ft 4in long, and 2ft sin Ans. 79f. 0'8" wide ? Ans. 43ft. 6in. 8n. high ? * The rule may be expressed in general terms thus. When feet are concerned, the product is of the same denomination as the term multiplying the feet; and whiện feet are not concerned, the name of the product will be expressed by the sum of the indices of the two factors, or of the strokes over them. Thus 4' * 2"=8"'. Here one of the factors is inches, the other seconds, and the indices or strokes over them amount to 3, hence the product, 8, is thirds. And in the same manner 8" *3=24" or, divide by 12,=2" The reason of the rule may be shown by the first example. The 4' are 4 twelfths of a foot and the 8' are 8 twelfths of a foot, and x= or 1 of or 32", which reduced gives 2 8"; putting down the 8" we reserve the 2' to be added of 10ft . by 8', or which product is to which 2 being added, we have for 6ft. 10. Next multiplying 4' or by 7 we have for 2ft. 4', which added to the product of 10 by 7 gives 72ft. 4', and these results added together give 79ft O'8" for the product. The same reasoning may be extended to cases where there is a greater Bumber of denominations. high? 4. How. niany feet in a stock 9. How many square yards in of 12 boards 14ft 6 long and ift the wainscoting of a room 18ft. 3' wide ? Ans. 217ft. 6in. long, 16ft. 6' wide and 9ft. 10' Ans. 324y. 4ft. 6'. Find the content of one board and multiply that by the number of 10. How much wood in a cuboards, as in Compound Multiplica- bick pile of wood measuring 8ft tion carrying for 12. on every side ? Ans. 4. 5. What is the content of a 11. How many square feet in ceiling 43ft. 3' long and 25ft. 6' a platform which is 37 feet, 11 broad? Ans. 1102ft. 10'6". inches long, and 23 feet 9 inches broad ? Ans. 900ft. 6', 3". 6. How much wood in a load 12. How much wood in a load, 6ft. 7' long, 3ft. 5' high, and 3ft. 8 wide ? 8 feet, 4 inches long, 3 feet 9 in ches wide, and 4 feet, 5 inches Ans. 82ft. 5' 8" 4". Ans. 138ft. O', S''. 7. What is the solid content of a wall 53ft. 6' long, 12ft. 3' high in a room which is 28 feet, 6 in 13. How many feet of ceiling and 2ft. thick ? ches long, and 23 feet, 5 inches Ans. 1310ft. 9'. broad? Ans. 667ft. 4', 6'. 8. How many cords in a pile 14. How many square feet are of 4 foot wood 24ft. long and 6ft. ! in a board which is 15 feet, 10 4' high? Ans. 4 cords. inches long and 9 inches wide ? Ans. 12ft. 10', 4", 6". high? QUESTIONS. 1. What is Compound Multiplica 8. What is the use of Compound tion ? Multiplication ? 2. How are the numbers to be 9. How do you prove Compound placed ? Multiplication? 3. How is the multiplication per- 10. Why are Duodecimals so callformed ? ed ? What is the Table ? 4. How, when the multiplier is a 11. How do you place the number composite number? for multiplication of Duodeci5. What is a composite number? mals? 6. What is to be done when the 12. Where do you begin to multiply? multiplier cannot be produced 13. How are the several products to by two small numbers ? be set down? 7. When the multiplier exceeds 14. What is the use of Duodecimals? 100, how do you proceed ? 6. Compound Division. COMPOUND Division is the method of finding how often one number is contained in another of different denominations. Rule. * Place the numbers as in Simple Division, and divide the several denominations of the dividend successively by the divisor. If there be a remainder after dividing any denomination, it must be reduced to the next lower, adding the number in the lower denomination. Divide the sum as usual ; and so on till the whole is finished. Proof. Examples. £ s. d. £ s. d. qrs. 35 ( 57 5 5 (1 12 8 2 Ans. 35 In 57 l find 35 75=775 is reduced to cts. once and 22 over. I and then the op 22 then reduce 22 to 31 ) 775 ( 25 Ans. eration becomes 20 shillings, adding the 62 the same as in 5s. and in the sum Simple Division. 35 ) 445 ( 12s. 445s. I find 35 12 155 35 times and 25 over. I 155 then reduce 25 to 95 pence, adding 5d. 2. If 9 yards of cloth cost £4 70 and in the sum 305 3s 7d 3qrs what is it per yard ? I find 35 8 times and 25 Again, I 12 reduce 25 to farthThis and other questions where the divisor is less than 10, may be ings, and divide by 35 ) 305 ( 8d. 35, and the quotient as conveniently solved by short di 280 is 2qrs. and 30 revision. When the number in the mains, which is highest denomination is less than 25 }of another the divisor, it must be reduced to 4 the next lower before dividing. farthing 3. If 126 lb. of nails cost $10 35 ) 100 ( 2qrs. 70 30 cts. cts. * The division of numbers of different denominations, or compound numbers, depends upon the same principles as Simple Division. This must be sufficiently obvious, when each of the several parts of the dividend can be divided without a remainder. And when there are remainders, the truth of the rule will |