MULTIPLICATION OF COMPOUND NUMBERS.

ART. 104. The process of taking a number consisting of different denominations, a certain number of times, is termed Multiplication of Compound Numbers.

1. A farmer takes to mill 5 bags of wheat, each containing 2 bu. 3 pk. 4 qt.: how much in all?

Next, multiply the 3 pk. by 5, making 15 pk., to which add the 2 pk. carried, making 17 pk., which, reduced, give 4 bu. and 1 pk.; write the 1 (pk.) in the column of pk., and carry the 4 bu.

Then, multiply the 2 bu. by 5, add to the product the 4 bu. carried, and we have 14 bu. to be written in the column of bu.

In Compound, as in Simple Multiplication, multiply the lowest denomination first, so as to carry from a lower to a higher order.

2. Multiply 2 bu. 5 qt. 1 pt. by 6. 3. Multiply 2 bu. 2 pk. 2 qt. by 9.

Ans. 13 bu. 1 qt.

Ans. 23 bu. 2 qt.

Rule for Multiplication.-1. Write the multiplier under the lowest denomination of the multiplicand.

2. Multiply the lowest denomination first, and divide the product by the number of units of this denomination, which make a unit of the next higher; write the remainder under the denomination multiplied, and carry the quotient to the product of the next higher denomination.

REVIEW.-103. How find the time between any two dates? 104. What is Multiplication of Compound Numbers? Why multiply the lowest denomination first? Repeat the Rule.

3. Proceed in bike manner with all the denominations, writing the entire product at the last.

PROOF.--The same as in Simple Multiplication.

REM.-1. In Simple Multiplication, we carry one for every ten, because ten units of a lower order make one unit of the next higher. In Compound Multiplication, we carry one for the number of units in each lower order which make a unit of the next higher.

2. The multiplier is always an abstract number, and shows how many times the multiplicand is to be taken.

4. If 4 bu. 3 pk. 3qt. 1 pt. of wheat make 1 bl. of flour, how much will make 12 bl. ? Ans. 58 bu. 1 pk. 2 qt.

5. What is the weight of 6 silver spoons, each weighing 2oz. 11 pwt. 6gr.? Ans. 1lb. 3oz. 7 pwt. 12 gr.

6. What is the weight of 10 bars of silver, each 10 oz. 10 pwt. 10 gr.? Ans. 8lb. 9oz. 4pwt. 4 gr.

7. I put up 8 packages of medicine, of 43 20 15 gr. each what did all weigh? Ans. 2lb 83 73 19.

8.

Find the weight of 9hhd. of sugar, of 8cwt. 2qr. 12 lb. each. Ans. 3 T. 17 cwt. 2 qr. 8 lb.

9. How much hay in 7 loads, each weighing 10 cwt. 3 qr. 14lb.? Ans. 3T. 16 cwt. 23 lb.

10. If a ship sail 208 mi. 4 fur. 16 rd. a day, how far will she sail in 15 days?

Ans. 3128 mi.

2fur.

11. If a man travel 30 mi. 4fur. 10 rd. a day, how far will he travel in 12 da. ?

12. Multiply 130 A. 3R. 30 P. by 4.

Ans. 366 mi. 3 fur.

Ans. 523 A 3R.

13. Multiply 23 cu. yd. 9 cu. ft. 228 cu. in. by 12.

14. How many yards in 6 pieces of muslin, of 26 yd. Ans. 159 yd. 3qr.

2qr. 2na. each ?

15. Multiply 62 gal. 1 qt. 1 pt. by 8.

16. How many gallons in 5 casks,

123 gal. 2qt. 1 pt.?

Ans. 499 gal.

each containing Ans. 618 gal. 1 pt.

REVIEW.-104. REM. 1. In Compound Multiplication, for what number do you carry one? 2. Is the multiplier concrete or abstract?

17. In a solar year are 365 da. 5 hr. 48 min. 48 sec.: how long has B lived, who is 12 years old?

Ans. 4382 da.

21 hr. 45 min. 36 sec.

18. Multiply 4° 11' 15" by 8.

Ans. 1s 3° 30'.

19. How many buttons in 3 great gross?

Ans. 5184. 20. How much water in 48 casks, each containing 62 gal. 1qt. 1pt. 1gi.? Here, 486 X 8. (See Art. 47.)

21. Multiply 2bu. 3pk. 5qt. by 24. Ans. 69 bu. 3 pk.

22. 3 mi. 5 fur. 16 rd. X 60.

23. 6 A. 3R. 30 P. X 56.

Ans. 220 mi. 4fur.

Ans. 388 A. 2R.

24. 8cwt. 2qr. 14lb. 12 oz. 13 dr. × 22.

Ans. 9T. 10 cwt. 1qr. 9oz. 14 dr.

25. 3 gal. 2qt. 1pt. 1gi. X 112. (1128 × 2 × 7.) Ans. 6hhd. 31 gal. 2qt.

NOTE. When the multiplier exceeds 12, and is not a composite number, it is most convenient to multiply each denomination, and reduce it separately, and write only the result.

26. Multiply 16 cwt. 2qr. 24lb. by 119.

27. 37 yd. 3qr. 2na.X89.

28. 47 gal. 3qt. 1 pt. X59.

Ans. 99 T. 12 cwt. 6 lb.

Ans. 3370 yd. 3qr. 2na.

Ans. 44hhd. 52 gal. 2 qt. 1 pt.

29. A travels 27 mi. 3 fur. 35 rd. in 1 day: how far will he travel in 1 mon. of 31 days?

Ans. 852 mi. 5 rd.

30. In 17 piles, each containing 7C. 98 cu. ft.: what quantity of wood? Ans. 132 C. 2 cu. ft.

3d Bk.

8

DIVISION OF COMPOUND NUMBERS.

ART. 105. The process of dividing numbers consisting of different denominations, is termed Division of Compound Numbers.

The Divisor may be either a Simple, or Compound Number. This gives rise to two cases:

FIRST CASE. To find how often one Compound Number is contained in another Compound Number.

SECOND CASE. To divide a compound number into a given number of equal parts. (See example 1, below.)

NOTE. Examples of the First Case are solved by reducing both divisor and dividend to the same denomination, and then dividing. They are treated of under Reduction. See examples 6, 8, page 100. Examples of the Second Case, are usually considered under the head of Compound Division.

ART. 106. 1. Divide 14 bu. 1 pk. 7 qt. 1 pt. of wheat equally among 3 persons.

SOLUTION.-Divide the highest denomination first, as in Simple Numbers, that if there be a remainder, it may be reduced to the next lower order, and added to it.

OPERATION.

bu. pk. qt. pt. 3)14 1 7 7 1

Ans. 4. 3

2 1

First, 3 in 14 is contained 4 times, and 2 bushels left; write the 4 in the order of bushels, and reduce the remaining 2 bushels to pecks, to which add the 1 in the order of pecks, and the sum is 9 pecks, which, divided by 3, gives a quotient of 3 pecks, to be written in the order of pecks.

Next, divide 7 quarts by 3, and the quotient is 2 quarts, with 1 quart remainder; write the 2 quarts in the order of quarts; reduce the 1 quart to pints, and add it to the 1 in the order of pints; the sum is 3 pints, and this, divided by 3, gives a quotient of 1 pint, which write in the order of pints.

Rule for Division.-1. Write the quantity to be divided in the order of its denominations, beginning with the highest; write the divisor on the left.

2. Begin with the highest denomination, divide each number separately, and write the quotient beneath.

3. If a remainder occurs after any division, reduce it to the next lower denomination, and, before dividing, add to it the number of its denomination.

PROOF.-The Same as in Simple Division.

REM.-In Simple Numbers when a remainder occurs in dividing any order except the lowest, it is prefixed to the figure in the next lower order, which is equivalent to multiplying it by 10, and adding to the product the figure in the next lower order.

Hence, in both Simple and Compound Division, each remainder is multiplied by that number of units of the next lower order which make a unit of the same order as the remainder.

Each partial quotient is of the same denomination as that part of the dividend from which it is derived.

4. Divide 67 bu. 3 pk. 4qt. 1 pt. by 5.

Ans. 13 bu. 2 pk. 2qt. 1pt.

5. Eight silver tankards of the same size, weigh 14lb. 8 oz. 16pwt. 16gr.: what is the weight of each?

Ans. 1 lb. 10 oz. 2pwt. 2 gr.

6. What will 1 dollar weigh; the weight of 10 dollars being 8oz. 12 pwt. 12 gr.? Ans. 17 pwt. 6gr.

7. Eleven bl. of sugar weigh 35 cwt. 1qr. 17 lb. 3oz. 7 dr. find the weight of one. Ans. 3 cwt. 22 lb. 5 dr. 8rd. in 7 hours: at what Ans. 5 mi. 5 fur. 24rd.

8. I traveled 39 mi. 7 fur. hour did I travel?

rate per

9. Divide 62 yd. 3na. by 5.

Ans. 12yd. 1qr. 3na.

REVIEW.-105. What is Division of Compound Numbers? What may the divisor be? What is the first Case? What the second? NOTE. How are examples of the first Case solved?

106. In dividing a Compound Number, why divide the highest denomination first? When a remainder occurs, how proceed, Rule?