When numbers are under the vinculum, or are included within any of the other signs of aggregation, they are commonly said to be in parenthesis.

107. The Terms of an expression are the parts connected by the signs

or

Thus, the expression 4 + 5 contains two terms. (4 + 5) × 2 + (6 — 3) X 4 also contains but two terms; for the expression included in each parenthesis is treated as a single number.

1. 482

4+4 or 8; but (48) ÷ 2 = 12 ÷ 2, or 6.

Hence, to find the value of expressions containing parenthetical forms:

1st. Simplify the expressions within the parenthesis by performing the operations indicated by the signs. 2d. Perform the multiplication or division indicated. 3d. Combine the terms as indicated by the signs and

When expressions contain two or more parentheses, brackets, etc., perform the operations indicated by the inner ones first.

4. (45) X7-3 5. (62) X4 +3

6. (37) X (3 + 2)

7. (86) 2 + 5

9. (57) X 4+ (6 +8)=7 10. 8124 (4 +62) 11. (63) × 5+ (10 − 6) 2 ÷ 12. 1284-4+6 ÷ 2

13. 10 + 3 × [7 + 3 × (8 − 2) + 5]
14. [(6515) X 216 + 4] ÷ 5

15. 7 (184—36) ÷ 4 — (9 × 3 + 4) + 24 X 3}

RELATION OF TERMS IN DIVISION.

108. The value of the quotient depends upon the dividend and the divisor. If either is changed, the quotient will be changed. If both are changed, the quotient may or may not be changed.

Take the equation, 32 ÷ 8 = 4. If we multiply the dividend by 2, we have 6488, which multiplies the quotient by 2. If we divide the divisor by 2, we have 32 ÷ 4 = 8, which also multiplies the quotient by 2.

If we divide the dividend by 2, we have 1682, which divides the quotient by 2. If we multiply the divisor by 2, we have 32 ÷ 16 = 2, which also divides the quotient by 2.

If we multiply both dividend and divisor by 2, we have 64 ÷ 16 = 4,

which does not change the quotient.

If we divide both dividend and divisor

by 2, we have 1644, which also does not change the quotient. These facts may be shown to be true of any equation. Hence, the following:

109. PRINCIPLES.-1. Multiplying the dividend or dividing the divisor by any number, multiplies the quotient by that number.

2. Dividing the dividend or multiplying the divisor by any number, divides the quotient by that number.

3. Multiplying or dividing both dividend and divisor by the same number, does not change the quotient.

ARITHMETICAL ANALYSIS.

110. Analysis is the process of solving problems by a comparison of their parts to each other directly; or indirectly, through their relation to the unit.

1. If 5 apples cost 7 cents, what will 10 apples cost?.

ANALYSIS.-If 5 apples cost 7 cents, 10 apples, which are 2 times 5 apples, will cost 2 times 7 cents, or 14 cents.

2. If 6 combs cost 9 cents, what will 12 combs cost?

3. If 4 lemons cost 12 cents, what will 16 lemons cost?

4. How much will 15 oranges cost, if 3 oranges cost 8 cents?

5. What must be paid for 32 barrels of flour, if 8 barrels cost $40.

6. If 7 tons of coal cost $22, what will 21 tons cost?

7. If 12 yards of muslin cost 66 cents, how much would 4 yards cost?

ANALYSIS.—If 12 yards of muslin cost 66 cents, 4 yards, which is of 12 yards, would cost of 66 cents, or 22 cents.

8. If 18 pigs cost $30, how much must be paid for 9 pigs? 9. If 14 pencils cost 24 cents, what will 7 pencils cost?

10. How much must be paid for 9 sheep, if 36 sheep are worth $140?

11. In 15 quires of paper there are 360 sheets. How many sheets are there in 5 quires?

12. If 48 bushels of corn cost $24, what must be paid for 12 bushels?

13. If 5 pounds of rice cost 35 cents, what will 9 pounds cost?

ANALYSIS.—If 5 pounds of rice cost 35 cents, 1 pound will cost of 35 cents, or 7 cents; and since 1 pound costs 7 cents, 9 pounds will cost 9 times 7 cents, or 63 cents.

14. James walked 36 miles in 9 hours. How far, at this rate, could he walk in 11 hours?

15. How many tons of hay will a drover feed in 12 weeks, at the rate of 10 tons in 5 weeks?

16. If 8 men can dig 32 rods of ditch in a day, how much can 15 men dig in the same time?

17. If 7 bushels of wheat are worth $5.60, how much are 20 bushels worth?

18. A farmer raised 360 bushels of wheat on 12 acres of land. How much wheat, at this rate, could he raise on 15 acres?

REVIEW.

111. 1. A man gave $2150 for a house, $750 for a lot, and had $1700 left. How much money had he before purchasing?

2. Mr. M. had $6000. He gave to each of his four children $1250. How much money had he left?

3. My farm contains 175 acres.

I sell it to receive $14175?

At what price per acre must

4. John went to a store and bought 28 pounds of sugar at 5 cents a pound; 3 pounds of butter at 16 cents a pound; 5 dozen eggs at 17 cents a dozen; and 7 pounds of oatmeal at 5 cents a pound. He gave the clerk a 10-dollar bill. What change was given back?

5. A dealer bought 17 mowers at $25 each, and gave in exchange 12 tons of old metal at $5.25 a ton, and the balance in cash. How much cash did he pay?

6. A company invested a million dollars in bonds of $1000 each. How many bonds did the sum secure?

7. Mr. R's barn cost $2350; his house cost 3 times as much as his barn; and his farm cost as much as both. What was the cost of the farm?

8. A dealer bought 176 barrels of oil at $3.20 per barrel. He lost 21 barrels by a fire, and sold the remainder at $4.45 a barrel. Did he gain, or lose, and how much?

9. A clerk receives a salary of $60 a month, and his expenses are $45 a month. In how many years, at this rate, can he pay for a house which cost him $1800?

10. Mr. Mancor has $10000 which he wishes to invest in real estate. If he should buy a house for $4500, how many acres of land could he buy with the balance at $110 an acre?

11. A farmer wished to pay a mortgage of $300. He sold 180 bushels of wheat at $.90 per bushel, and enough clover seed at $3 per bushel to lift the mortgage. How many bushels of clover seed did he sell?

12. A drover had 360 sheep. He sold to one man 120 sheep, and to another 75, and then bought enough to make his number 500. How many did he buy?

13. Mr. B. sold 3450 bushels of wheat, then bought 1200 bushels, and then had 1750 bushels. How many bushels had he at first?

14. A ship whose rate of sailing is 112 miles a day is already 540 miles on its course, when another vessel whose rate is 148 miles a day leaves the same port for the point as the other. In how many days will the former vessel be overtaken by the latter?

15. Two men start from the same place at the same time and travel in opposite directions, the one at the rate of 23 miles a day, and the other 26 miles. How far will they be apart at the end of 6 days?

16. The distance from New York City to London by postal route is 3740 miles, and the mail time is about 6 days. What is the rate of sailing per hour, in whole numbers?

17. Mr. H. sold from his farm 265 bushels of wheat at $.80 per bushel; 400 bushels of corn at $.35 per bushel, 130 bushels of oats at $.25 per bushel, and 7 tons of hay at $8.50 per ton. How much did he realize from the farm?

18. I sold 75 acres of land at $90 per acre, and invested the money in land at $50 per acre. How many acres did I purchase? 19. If Mr. Row's field of 16 acres should yield 27 bushels of wheat to the acre, what would the grain be worth at $.98 per bushel?

20. A teacher's salary is $100 per month. His expenses per week average $19.35. How much can he save in a term of 9 months?

PROPERTIES OF NUMBERS.

112. Numbers are either odd or even, prime or composite.

113. An Odd Number is a number that cannot be exactly divided by 2.

The unit figure of an odd number is always, 1, 3, 5, 7, or 9; as 5, 21, 33.

114. An Even Number is a number that can be exactly divided by 2.

The unit figure of an even number is always 0, 2, 4, 6, or 8; as 4, 20, 36.

115. A Prime Number is a number that can be exactly divided only by itself and 1.

Thus, 1, 3, 5, 7, 11, 13, 17, etc., are prime numbers.

116. A Composite Number is a number that can be exactly divided by others besides itself and 1.

Thus, 4, 6, 8, 10, 12, 14, 15, etc., are composite numbers.

DRILL EXERCISES.

117. 1. Tell which of the following numbers are odd or even: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 27, 31, 36, 39, 40, 42, 45, 43, 46, 48, 50.

2. Tell which of the following numbers are prime or composite: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30.

3. Name the prime numbers from 31 to 67. From 67 to 101. 4. Name the composite numbers from 20 to 50. From 50 to 75. From 75 to 100.

5. Select all of the prime numbers from the following: 8, 10, 11, 17, 15, 18, 19, 23, 41, 45, 20, 21, 27, 29, 38, 42, 36, 35, 31, 43, 45, 47, 63, 61, 67.

6. Select all of the composite numbers from the following: 13, 19, 20, 21, 24, 25, 29, 26, 27, 32, 34, 37, 39, 58, 71, 73, 78, 80, 47, 45, 84, 87, 93, 99.

DIVISIBILITY OF NUMBERS.

118. A number is divisible by another when the latter will divide the former without a remainder.

119. Any number is divisible

1. By 2, if it is an even number.

2. By 3, if the sum of the digits is divisible by 3.

Thus, 231 is divisible by 3, for 2 + 3 + 1, or 6, is divisible by 3.

3. By 4, if the two right hand figures are ciphers or express a number divisible by 4.

Thus, 300, 516, 928, etc., are divisible by 4.