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BU arhit first by one of the component parts of the divisor, and loch hotel. #tient by the other; and the last quotient will be the answer. 1. Divide £87 10s. by 21.
Ans. £4 3s, 4d. £ d.
DEMONSTRATION.--Seven and three 787 10 0 are the component parts of 21, because
three times 7 are 21; and it is the same 3) 12 10 0 as dividing directly by 21, as the folAns.
lowing example shows:
Here we find that the same 21)87 10 0 ( 4 3 4
result is produced, and the rea
son is obvious from the princi84
ples of multiplication, which are 3 exactly the reverse of those in division. 20 2. A man bought 36 bushels of apples for
£2 14s. ; what did he pay per bushel ? 63
Ans. Is. 6d. 7
3. Bought 28 cords of wood for £16 16s. ;
how much was paid per cord ? 12
Ans. 12s. 4. Bought 72 bushels of wheat for £41 88.; 84
what was paid per bushel ? Ans. 11s. 6d. 84
5. Sold 81-barrels of flour for £147 16s. 6d.;
Ans. £1 16s. 6d. 6. Bought 42 ploughs for £128 9s.; how much is that per plough?
Ans. £3 ls. 2d.
Of Weights, Measures, f:c. 1. If 24 pieces of cloth contain 426yds.; how many yards in one piece?
Ans. 17 yds. 3qrs. 2. If 6 chests of tea weigh 21cwt. Iqr. 261b.; what is the weight of 1 chest ?
Ans. 3cwt. 2qrs. Olb. 3. If 7 hogsheads of sugar weigh 69cwt. ; what is the weight of 1 hogshead ?
Ans. 9çwt. 3qrs. 121b. 4. If 11 pieces of cloth contain 163yds. 2qrs. 2na.; how much in each, suppose they contain equal quantities?
Ans. 14yds. 3qrs. 2na. 5. Divide 219 acres, 1 rood, 8 rods, into twelve equal parts.
Ans. 18 acres, 1 rood, 4 poles. 6. I 174yds. 1qr. 2na. be divided equally among 5 persons what will be the share of each ? Anš. 34 yds. 3qrs. 2na:
7. Divide eighteen gallons equally among one hundred and forty-four soldiers.
Ans. 1 pint apieoe.
PRACTICAL QUESTIONS In Compound Multiplication and Division.. 1. If one bushel of corn cost 45 cents; what will 8 bashels cost ?
Ans. $3,60 cents. 2. If 8 bushels of corn cost $3,60 cents; how much was that per bushel ?
Ans. $0,45 cents. 3. A man received $19,50 cents for 30 day's labour; how much was that per day?
Ans. $0,65 cents. 4: A gentleman wishes to put 130 bushels of apples into barrels, containing 3 þuşhels and 1 peck each; how many barrels does he need ?
Ans. 40. 5. The Prince of Wales receives a salary, of 150 thousand pounds a year; how much is that per day?
Ans. £410519s. 2d. 6. A piece of calico containing 29 yards cost $8,70 cents; what was that per yard ?
Ans. $0,30 cents. 7. A privateer took a prize of $30,000, of which the owner took one third, and the officers one fourth; the remainder is. equally divided among 125 seamen; how much must each seaman receive ?
Ans. $100. 8. One hundred and sixty-three men took a prize worth
11,16. cents, of which the captain had four shares, the first lieutenant three shares, and the second lieutenant two shares; what was the share of each officer and each private ?
Ans. Captain's share $42,12 cents; first Lieutenant's
share $31,59.cents; second Lieutenant's share $21,06
cents; and a private's share $10,53 cents. 9. Divide £136 14 shillings 6 pence among two men and three women, and give each man three times as much as a woman; what will each man and each woman receive :
Ans. £45 11s. 6d. 1 man's share. £15:35. 19d. 1 wo
2 men, and
d. 3 women.
9) 136 14 6 6
6 equal to 6 w'n. Ans. 15 3 10 one w's, share. add 3 women.
3 9 the number of 45 11 6 a man's share, or the shares in the whole.
2 shares of the 3 w. Divisor = 9 women.
91 3 O the sh. of the 2 men.
45 11 6 the sh. of the 3 w.
Proof. 136 14 6 DEM.-It is evident from the preceding work, that a woman must receive one ninth of the whole sum; then it is plain that we obtain a man's share by multiplying a woman's share by 3, because each man receives three times as much as a woman.
ro. Divide £60 85. 8d. among 14 men and 14 women, and give the women three times as much as the men.
Ans. £1 ls. 7d. m's. share. £3 4s. 9d. w's share, We have found, by division, where the price of a quantity is given, we obtain the price of one yard, one pound, &c. by dividing the price of the quantity by the quantity, and the quotient is the price of one yard, one pound, &c. And we have also found by multiplication, where the price of a unit or one is given, we obtain the price of a quantity by multiplying the price of one yard, one pound, &c. by the quantity, and the product is the price of the whole quantity.
Then from these two rules we may draw a very useful one in practical arithmetick, where the price of a quantity is given to find the price of any other quantity.
RULE.-Divide the price by the quantity of which it is the price, and the quotient will be the price of one; then multiply the price of one by the quantity of which you wish to obtain the price, and the product will be the price of the quantity required.
Note.—There is a double object in introducing this rule here: first, on account of its applying in one sum the principles of multiplication and division, and its importance in business; secondly, on account of its being a key to other rules which have been looked upon by the student as dark and intricate without relation to any thing preceding them ; but the student, from his knowledge of this rúle, will be able to say, when he advances in other rules, that they are easy, because they are only a different manner of working and applying rules which he already understands; and also the relation which they bear to what has preceded them.
EXAMPLES. 1. If 8 yards of cloth cost-24 shillings; what wiil 4 yards
Ans. 12 shillings.
DEM.--It is plain, when we di.
vide 24s. by 8, the number of yards 3s. the price of lyd. of which it is the price, the quotient 4
must be the price of 1 yard, because 12s. the price of 4yds.
one yard must cost the eighth part of 8 yards, which gives us 3 shillings for
1 yard; then to obtain the price of 4 yards, it is plain, the price of one must be repeated 4 times, which gives us 12 shillings for the price of 4 yards. 2. If 4 yards coșt 12 shillings; what will 8 yards cost?
Ans. 24s. 3. If 4 yards of broadcloth cost 20 dollars; what will 9 yards cost ?
Ans. $45. 4. Suppose 30 yards of Irish linen cost $19,50 cents, what will 8 yards cost?
Ans. $5,20cts. 5. If 3 cords of wood cost $4,35 cents; what will be the cost of 30 cords?
Ans. $43,50 cents. 6. If 30 cords.cost 43 dollars 50 cents; what will 3 cords cost?
Ans. $4,35 cents. 7. If 3 hogsheads of sugar weigh 26cwt. 1qr. 12lbs. ; what. is the weight of 9 hogsheads ?
Ans. 79cwt. Oqr. 8lb, 8. Suppose 30 bushels of rye cost $18; what will 11 bushels cost?
Ans. $6,60 cts. 9. If 3 bushels of apples cost 63.cents; what will 18 bushels
Ans. $3,78 cts. 10. Suppose 2 yards of broadcloth cost $10; what will 9 yards cost?
Ans. $36. 11. Divide 97deg. 55mi. 7fur. 35pol. 4ft. 2in. lb. c. by 6. Ans. 16deg. 20mi. 7fur. 12pol. 8ft
. 11in, 10b. c. 12. Divide 45deg: 10mi. 7fur. 37pol. 5yds, lft. 3in, and 2b. c, by 4.
Ans. 11deg. 20mi. Ofur. 39pol. 2yds. 2ft. 2in. 16. c. NOTE.-Let the student prove examples 11th and 12th. For the disposition of the fractions see note to example 3d, page 97, and demonstration to example 1st, page 97.
QUESTIONS ON THE COMPOUND RULES. What is compound addition ? A. Collecting in one sum two or more numbers of different denominations. How do you place your numbers for adding?_ A. Those of the same denomination exactly under each other. How do you proceed in adding? ' A Add up the right hand denomination, and divide the amount by as many of that denomination as will make one of the next greater, and set down
the remainder under the denomination added, and carry the quotient to the next greater, and so I proceed till I have come to the left hand denomination, which I add and set down, the same as in simple addition. In adding pounds, shillings, pence, and farthings; how do you carry, or by what numbers do you divide the several amounts to know what you must carry? A. Divide the amount in farthings by 4, and carry the quotient, because 4 farthings make a penny; and divide the pence by 12, and carry the quotient, because 12 penee make one shilling; and divide the amount in shillings by 20, because it takes 20 shillings to make one pound. What is Compound Subtraction? A. It is taking a less sum from a greater of different denominations. How do you place the two given numbers? A. The less sum under the greater, so that those of the same denomination may stand directly under each other. How do you proceed in subtracting? A. Take the figures in the subtrahend from those directly above them in the minuend, setting down the difference. When any denomination, in the subtrahend, exceeds that directly above it in the minuend, what do you do? A. Add as many to the upper denomination as will make a unit in the next higher, and from the amount subtract the figure or figures directly below in the subtrahend, and then add one to the next higher denomination in the subtrahend; or subtract from that number which is equal to a unit in the next higher, and to the difference add the figure or figures directly above in the minuend, and then add one to the next higher in the subtrahend. Why should this preserve the true difference between the subtrahend and minuend? A. Because it is adding equals to both the given sums, and adding equals to both, their difference must ever remain the same. If the pence in the subtrahend exceed the pence in the minucnd, what must you do? A, Subtract the pence in the subtrahend from 12, and to the difference add the pence in the minuend, and then add one to the shillings in the subtrahend. How do you prove compound subtraction ? A. By adding the difference to the subtrahend, and of the amount equals the minuend the work is right. Why should that prove it? A. Because the difference between two numbers added to the less must make it equal to the greater. If the difference be taken from the minuend, it must leave a number equal to the subtrahend, because if the difference between two numbers be taken from the greater, it must reduce it equal to the less. What is compound multiplication ? A. It is repeating a given number of different denominations a certain proposed number of times. How do you proceed in the work? A. Place the multiplier under the right hand denomination of the multiplicand; then multiply the right hand denomination of the multiplicand by the multiplier, and divide the product by as many of that denomination as will make a unit in the next higher, placing the remainder under the denomination multiplied, and carrying the quotient to the product of the next aç the left; and so I proceed tilĩ I come to the left hand denomination, and there set down the whole product the same as in simple work. Why do we divide the product in each place by the number which it takes to make a unit in the next higher, and carry the quotient ? A. For the same reason that we divide and carry in compound addition, this being only a short way of performing compound addition. If the price