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2. The larger of two numbers is 1685, and the smaller is 768; what is their difference ?

3. The difference between two numbers is 146 and the less is 467; what is the larger ?

4. If 1406 be the difference between two numbers, and the greater number is 4879, what is the less number?

5. The two factors of a number are 27 and 36; what is the number?

Ans. 972. 6. The product of A's and B’s ages is 3200 years, and B's age is 50 years; what is the age of A?

7. The product of two numbers is 515522, and the larger number is 1601; what is the smaller ?

Ans. 322. 8. If 168465 pounds of beef be divided equally between 11231 soldiers, how many pounds will each receive?

Ans. 15 pounds. 9. If a quantity of beef was divided equally between 11231 soldiers, and each received for his share 15 pounds, what quantity was divided ?

10. The product of two numbers is 890368, and the larger number is 3478; what is the smaller ?

Ans. 256. 11. 168465 pounds of beef having been equally divided between a number of soldiers, each received as his share 15 pounds; what was the number of soldiers ?

12. If the integral part of a quotient be 23, the remainder 692, and the dividend 39792, what is the divisor ?

Ans. 1700. 13. John Reed says his number of sheep is such that if he should separate the flock so as to put 115 in each of 5 fields, there would be 4 sheep remaining. How many sheep has he?

Ans. 579 sheep. 14. A farmer having 579 sheep, on putting an equal number of them into each of five fields, had 4 remaining. How many did he put into each of the fields ?

ARITHMETICAL ANALYSIS.

74. The process of solving questions, by considering their elements, or by comparing their numbers, and reasoning from them to a required number, is called Analysis. Hence,

75. Arithmetical Analysis is the process of solving & question, by considering its elementary parts, and obtaining a required result by a course of reasoning.

In analyzing a simple question, the reasoning is usually from a given number to a unit of the same name, and then from this anit to the required number.

Exercises.

1. If 7 acres of land produce 525 bushels of wheat, how many bushels will 11 acres produce ?

ANALYSIS. If 7 acres of land produce 525 bushels of wheat, 1 acre will produce one seventh of 525 bushels, or 75 bushels, and 11 acres will produce 11 times 75 bushels, or 825 bushels. Therefore, etc.

2. If 11 acres of land produce 825 bushels of corn, how many bushels will 7 acres produce ?

3. When 120 barrels of flour cost 1560 dollars, what will 139 barrels cost?

Ans. 1807 dollars. 4. If 139 barrels of flour cost 1807 dollars, what will 120 barrels cost? 5. If a man can travel 2440 miles in 4 weeks, how

many miles can he travel in 9 weeks ?

Ans. 5490 miles. 6. If 20 men can do a piece of work in 11 days, how many days will it take 17 men to do it ?

ANALYSIS. If 20 men can do a piece of work in 11 days, it will take 1 man 20 times 11 days, or 220 days, to do it, and 17 men 1 seventeenth of 220 days, or 1214 days. Therefore, etc.

What is Arithmetical Analysis ?

7. When 13 men can hoe a field in 21 days, in what time can 39 men hoe it?

Ans. 7 days. 8. If 19 cords of wood will pay for 57 yards of cloth at 2 dollars a yard, what is the wood a cord ? Ans. 6 dollars.

9. When 540 tons of coal will pay for 30 tons of iron, at 90 dollars a ton, what is the coal a ton ?

Ans. 5 dollars. 10. If 14 horses can be bought for 2240 dollars, how many can be bought for 3040 dollars ?

ANALYSIS. If 14 horses can be bought for 2240 dollars, 1 horse can be bought for 1 fourteenth of 2240 dollars, or 160 dollars; and if i horse can be bought for 160 dollars, as many horses can be bought for 3040 dollars as 160 dollars are contained times in 3040 dollars, or 19. horses. Therefore, etc.

11. If 19 horses can be bought for 3040 dollars, how many can be bought for 2240 dollars ?

12. When 23 men earn 1380 dollars in a month, how many men will earn 1980 dollars in the same time? Ans. 33 men.

13. If 31 horses will consume 93 tons of hay in a certain time, how many horses will consume 87 tons in the same time?

Ans. 29 horses. 14. How many hogsheads of molasses can be bought for 17670 dollars, if 16 hogsheads can be bought for 912 dollars ?

Ans. 310 hogsheads. 15.* Daniel White bought an equal number of cows and oxen for 2880 dollars, each cow at 45 dollars, and each ox at 75 dollars; how many of each did he buy?

ANALYSIS. Since 1 cow cost 45 dollars and 1 ox75 dollars, 1 cow and 1 ox together were bought for 45 dollars plus 75 dollars, or 120 dollars; and since the number of each was equal, he bought as many of each as 120 dollars are contained times in 2880 dollars, 24. There

fore, etc.

REVIEW QUESTIONS. Which are the Fundamental Operations of Arithmetic ? (66) What are called Symbols of Operation ? (67) What is an Arithmetical Formula ? (69) How many of them are called Fundamental ? 170)

* Optional.

16.* I have 11900 dollars in National currency consisting of an equal number of 5 dollar, 10 dollar, and 20 dollar bills; how many have I of each?

Ans. 340. 17. Frank Homer bought a farm for 11200 dollars, giving 60 dollars an acre for one half of it and 80 dollars an acre for the other half; how many acres in each half ? Ans. 80 acres.

18. A farmer sold an equal number of colts at 90 dollars each, and horses at 175 dollars each, for 3445 dollars; how much did he get for each kind ?

Ans. Colts, 1170 dollars; horses, 2275 dollars. 19. If A and B together have 1625 dollars, and B 825 dollars more than A, how much is the money of each?

SOLUTION. If A and B together have 1625 dollars, and B 825 dollars more than A, 1625 dollars less 825 doliars, or 800 dollars, must be twice A's money.

If 800 dollars be twice A's money, one half of 800 dollars, or 400 dollars, is A's money; and, since B has 825 dollars more than A, B's money is 400 dollars plus 825 dollars, or 1225 dollars.

Therefore, etc.

20. At a certain election A and B were candidates; the whole number of votes cast for them was 5963, of which A had 321 more than B; how many votes did each receive?

Ans. A 3142; B 2821. 21. Two men having met on a journey, found that they had travelled 1200 miles, and that one had travelled 360 miles more than the other; what distance had each travelled ?

22. Albert Smith gave his three sons 1350 dollars, Samuel receiving 50 dollars more than Edmund, and Ernest 150 more than Samuel; how much did each receive?

Ans. Edmund 366f dollars, Samuel 416f dollars, and Ernest 566; dollars.

REVIEW QUESTIONS. How is the sum of two or more numbers found ? (37) How is the difference of two numbers found ? (42)

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77. Coins are pieces of metal converted into money by legal stamping.

The Coins of the United States are of gold, silver, nickel, and bronze, as follows:

GOLD, – Double eagle, eagle, half-eagle, three dollar, quartereagle, dollar ; SILVER, - dollar, half-dollar, quarter-dollar, dime; NICKEL, — five-cent ; BRONZE, cent.

What is United States Money ? Repeat the Table. What are Coins ? Name the gold coins. The silver. The nickel coin. The bronze.

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