(e) It follows, from (a) and (b), that the greater the dividend, the greater is the quotient; and the less the dividend, the less the quotient.

(f) Also, from (c) and (d), that the greater the divisor, the less is the quotient; and the less the divisor, the greater the quotient.

84. From the illustrations in Art. 83 we see that any change in the dividend causes a SIMILAR change in the quotient, and that any change in the divisor causes an OPPOSITE change in the quotient. Hence,

(a) Multiplying both dividend and divisor by the same number does not affect the quotient; thus,

12÷3 - 4

2 2

2464, Quotient unchanged.

(b) Dividing both dividend and divisor by the same number does not affect the quotient; thus,

20 응 10 2

5) 20

4

5) 10

22, Quotient unchanged.

(c) It follows from (a) and (b), that the operations of multiplying and dividing by the same number cancel (i. e. destroy) each other; e. g.,

If a number be multiplied by any number, and the product be divided by the multiplier, the quotient will be the multiplicand; thus,

8 X 7 56, and 56÷7

8, the multiplicand.

Also, if a number be divided by any number, and the quotient be multiplied by the divisor, the product will be the dividend; thus,

1535, and 5 × 3

15, the dividend.

83. What follows from (a) and (b)? From (c) and (d)? 84. Any change in the dividend, how does it affect the quotient? Any change in the divisor, how? First inference? Second? Third? Illustrate.

85. These general principles may be more briefly

stated as follows:

1st. Multiplying the dividend multiplies the quotient; and dividing the dividend divides the quotient (Art. 83, a and b).

2d. Multiplying the divisor divides the quotient; and dividing the divisor multiplies the quotient (Art. 83, c and d).

3d. Multiplying both dividend and divisor by the same number; or dividing both by the same number does not affect the quotient (Art. 84, a and b).

EXAMPLES IN THE FOREGOING PRINCIPLES.

1. How many bushels of corn at $1 per bushel must be given for 6 barrels of flour at $7 per barrel?

2. How many barrels of apples at $2 per barrel must be given for 8 cords of wood at $6 per cord?

3. A speculator bought 640 acres of land at $3 per acre, and sold the whole for $3200; how much did he gain by the transactions? How much per acre?

4. Bought 320 acres of land for $1760, and 320 acres more at $7 per acre, and sold the whole at $6 per acre; did I gain or lose? How much? Ans. Lost $160.

5. The expenses of a boy at school for a year are $126 for board, $24 for tuition, $15 for books, $35 for clothes, $10 for railroad and coach fare, and $9 for other purposes; what will be the expenses of 250 boys at the same rate?

6. If 3 men build 24 rods of wall in 4 days, in how many days will 5 men build 70 rods?

Ans. 7. 7. The product of 4 factors is 1155; three of the factors are 3, 5, and 7; what is the fourth? Ans. 11.

8. How many miles per hour must a ship sail to cross the Atlantic, 2880 miles, in 12 days of 24 hours each?

9. The first of 3 numbers is 6, the second is 5 times the first, and the third is 4 times the sum of the other two; what is the difference between the first and third?

85. A more brief statement of these principles: First? Second? Third?

10. Sold two cows at $30 apiece, 3 tons of hay at $20 per ton, 50 bushels of corn for $50, and 10 cords of wood at $7 per cord, and received in payment $200 in money, a plow worth $15, 50 pounds of sugar worth $5, and the balance in broadcloth at $4 yer yard; how many yards did I receive? Ans. 5,

11. In how many days of 24 hours each will a ship cross the Atlantic, 2880 miles, if she sails 10 miles per hour?

12. If I receive $60 and spend $40, per month, in how many years of 12 months each shall I save $2160? Ans. 9. 13. What is the value of 27 hogsheads of molasses at $32 per hogshead?

14. What is the value of 87 yards of cloth at $4 per yard? 15. Bought 87 acres of land at $50 per acre, and paid $3150 in cash, and the balance in labor at $240 a year; how many years of labor did it take?

16. Bought 42 yards of cloth at 15 cents per yard, and paid for it in corn at 90 cents per bushel; how many bushels did it take?

17. If I take 13729 from the sum of 8762 and 14967, divide the remainder by 50, and multiply the quotient by 19, what is the product? Ans. 3800.

REDUCTION.

86. All numbers are simple or compound.

A SIMPLE NUMBER consists of but one kind or denomination; as 2, $4, 8 books, 5 men, 6 days, 10 miles.

A COMPOUND NUMBER is composed of two or more denominations; as 4 days and 7 hours; 3 bushels, 2 pecks, and 5 quarts; 5 rods, 4 feet, and 6 inches.

All abstract numbers (Art. 2) are simple.

86. What is a Simple Number? A Compound Number? An Abstrac Num ber, is it simple or compound?

A concrete number, whether simple or compound, is often called a Denominate Number.

NOTE 1. All operations in the preceding pages are upon simple numbers.

NOTE 2. The several parts of a compound number, though of different denominations, are yet of the same general nature; thus, 2 weeks, 3 days, and 6 hours are SIMILAR quantities, and constitute a compound number; but 2 weeks, 3 miles, and quarts are UNLIKE IN THEIR NATURE, and do NOT constitute a compound number.

87. REDUCTION is changing a number of one denomination to one of another denomination, without changing its value.

It is of two kinds, viz. Reduction Descending and Reduction Ascending.

REDUCTION DESCENDING consists in changing a number from a higher to a lower denomination.

REDUCTION ASCENDING is changing a number from a lower to a higher denomination.

ENGLISH MONEY.

88. ENGLISH MONEY is the Currency of Great Britain.

89. REDUCTION DESCENDING is performed by multiplication; thus, to reduce 15£ to shillings, we multiply 15 by 20, because there will be 20 times as many shillings as pounds. So to reduce 15£ and 12s. to shillings, we multiply 15 by 20, and to the product add the 12s.

86. A Concrete Number, what is it called? 87. What is Reduction? How many kinds of Reduction? What are they called? What is Reduction Descending? Reduction Ascending? 88. What is English Money? Repeat the table. 89. How is Reduction Descending performed?

In a similar manner all such examples are reduced. Hence, 90. To reduce the higher denominations of a compound number to a lower denomination:

RULE. Multiply the highest denomination given by the number it takes of the next lower denomination to make one of this higher, and to the product add the number of the lower denomination; multiply this sum by the number it takes of the NEXT lower denomination to make one of THIS; add as before, and so proceed till the number is brought to the denomination required.

Ex. 1. Reduce 11£ 17s. 9d. 3qr. to farthings.

NOTE. Since there are no pence in the 3d example, there is nothing to add to the product obtained by multiplying by 12.

4. Reduce 27£ 15s. 6d. 2qr. to farthings.

5. Reduce 32£ 8d. 3qr. to farthings.

91. REDUCTION ASCENDING is performed by division; thus, to reduce 4299 farthings to pence, we divide the 4299 bý 4, because there will be only one fourth as many pence as farthings. Performing the division we obtain 1074d. and a remainder of 3qr. If we wish to reduce the 1074d. to shillings, we divide by 12, because there will be only one twelfth as many shillings as pence, and obtain 89s. and a remainder of 6d. Again,

90. Repeat the rule. Explain the process in Ex. 1. How are the 237 shillings obtained? How the 2853 pence? The 11415 farthings? 91. How is Reduction Ascending performed?