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many days to build a wall 200 feet in length as it will 20 feet in length; therefore we write 200 for the second term and 20 for the first; we also write 200 for the numerator of a fraction and 20 for the denominator.

Comparing 6 feet high, a term of supposition, with 8 feet high, its corresponding term of demand, we find that 8 feet is of 6 feet; hence it will take & as many days to build a wall 8 feet high as it will 6 feet high; consequently we write 8 for the second term and 6 for the first; we also write 8 for the numerator of a fraction, and 6 for the denominator.

Comparing 4 feet thick, a term of supposition, with 6 feet thick, its corresponding term of demand, we find that 6 feet is of 4 feet; hence it will take as many days to build a wall 6 feet thick as it will 4 feet thick; consequently we write 6 for the second term, and 4 for the first; we also write 6 for the numerator of a fraction and 4 for the denominator, and the terms are all arranged and the statements completed.

3. If $100 will gain $6 interest in 12 months, in what time will $750 gain $30 dollars interest? Ans. 8 months. 4. If 7 men can reap 84 acres of wheat in 24 days, how many men will be required to reap 100 acres in 10 days? Ans. 20 men.

5. If $200 will gain $6 interest in 6 months, how many dollars will $400 gain in 9 months? Ans. 18 dollars.

6. If $1200 will support a family of 24 persons 8 months, how many months will $900 support a family of 16 persons? Ans. 9 months. 7. If I pay $24 for the transportation of 96 barrels of flour 200 miles, what must I pay for the transportation of 480 barrels 40 miles? Ans. 24 dollars.

8. If 8 men can build a wall 75 rods in length in 12 days, how many men will be required to build a wall 300 rods in length in 6 days? Ans. 64 men.

9. If 12 pounds of wool will make 8 yards of cloth 6 quarters wide, how many pounds will be required to make 144 yards 4 quarters wide? Ans. 144 pounds.

10. If 5 men can make 1200 pairs of shoes in 48 days, how many men will be required to make 4800 pairs of the same kind of shoes in 16 days? Ans. 60 men.

11. If 15 girls can perform 180 questions in arithmetic in 75 minutes, how many questions can 25 girls perform in 35 minutes? Ans. 140 questions.

12. If a garrison of 1200 men consume 500 barrels of flour in 9 months, how many barrels will a garrison of 2500 men consume in 15 months?

13. If 12 men can build a wall 100 feet long, 4 feet high, and 3 feet thick, in 40 days, in what time will 6 men build one 20 feet long, 6 feet high, and 4 feet thick?

14. If 180 men, in 6 days of 10 hours each, can dig a trench 1200 yards long, 4 feet wide, and 3 feet deep, in how many days, of 12 hours each, will 90 men dig a trench 400 yards long, 6 feet wide, and 4 feet deep?

15. A garrison of 1800 men has provision sufficient to allow each man 32 ounces a day 35 days; suppose the garrison to be reinforced with 600 men, what must be the daily allowance for each man 45 days?

16. If 6 men can reap 30 acres of wheat in 5 days, what number of acres can 8 men reap in 10 days?

17. If 3 horses eat 30 bushels of oats in 50 days, what number of bushels will 30 horses eat in 25 days?

18. If it requires $600 to support a family of 9 persons 8 months, what number of dollars will be required to support a family of 6 persons 12 months?

19. If 12 ounces of wool be sufficient to make 1 yards of flannel 6 quarters wide, what number of pounds will be required to make 450 yards of flannel 4 quarters wide?

20. If 25 tailors can make 75 suits of clothes in 8 days, what number of tailors will be required to make 1200 suits of clothes in 6 days?

21. If 4 men can build a wall 30 rods in length in 6 days, what number of men will be required to build a wall 90 rods in length in 4 days?

22. If 14 men, working 9 hours each day, dig a ditch 420 feet long, 4 feet wide, and 3 feet deep, in 4 days, in what number of days will 35 men, working 12 hours each day, dig a ditch 840 feet long, 5 feet wide, and 4 feet deep?

23. A wall, to be built to the height of 27 feet, was raised to the height of 9 feet by 12 men in 6 days; how many men must be employed to finish the wall in 4 days at the same rate of working?

24. A wall 700 yards long was to be built in 29 days; at the end of 11 days 18 men had built 220 yards of it. What number of additional men was it then necessary to engage to work at the same rate, in order that the wall might be completed in the given time?

PROPORTION

INVOLVING A

CONSTANT QUANTITY.

Art. 135. We will explain what is to be understood by a constant quantity, by the following question.

1. A coal mine has the same quantity of water constantly running into it in the same time. At one time it was allowed to get full, and it required an engine of 12 horse-power 60 hours to empty it. At another time it was again allowed to get full, and it required an engine of 10 horse-power 80 hours to empty it. It is now full the third time; what must be the power of an engine that will empty it in 48 hours, and what must be the power of an engine that will discharge the water as fast as it runs into the mine after it is emptied ?

It is to be understood that the horse-power and the flow of water into the mine are quite constant, that is, exactly the same in all cases. It is also to be understood that the quantity of water in the mine is the same each time, when full.

As it required an engine of 12 horse-power 60 hours to discharge the water that was in the mine when full the first time, including also the water that runs into the mine during those 60 hours, it is plain that it will require only or of 12 horse-power to discharge the same quantity of water in 80 hours, and 12 X 29, the number of horse-power required to discharge the quantity of water in the mine when full, including also the water that runs in during 60 hours.

The second time it was allowed to get full, it required an engine of 10 horse-power 80 hours to empty it, including also the water that runs into the mine during 80 hours.

As it requires only 9 horse-power 80 hours to empty the mine when full, including also the water that runs into the mine during 60 hours, it is plain that the other 1 horsepower, employed 80 hours the second time it was allowed to get full, is the power required to discharge the water that runs into the mine during the remaining 20 hours. If it requires 1 horse-power 80 hours to discharge the water that runs into the mine during the remaining 20 hours, it will require or 4 times 1 horse-power to discharge the same quantity of water in 20 hours, which is 4 horse-power. Hence it is plain that an engine of 4 horse-power will discharge the water as fast as it runs into the mine during any length of time; which is one of the answers required.

We will now find what power will be required to empty the mine in 48 hours. It is plain that, whatever power is required to empty the mine in any given number of hours, there is always 4 horse-power that does nothing towards emptying the mine, because it only dicharges the water that runs into the mine during the given number of hours.

The first time that the mine was allowed to get full, it required an engine of 12 horse-power 60 hours to empty it. Deducting the 4 horse-power which is required to discharge the water as fast as it runs into the mine, there remains 8 horse-power, which was the number of horse-power required to empty the mine in 60 hours, not including the water that runs into the mine during those 60 hours.

=

To discharge the same quantity of water in 48 hours, it will require or of 8 horse-power, and 8 X 10, the number of horse-power that will be required to empty the mine in 48 hours, not including the water that will run into the mine during the 48 hours.

If to this we add 4, the number of horse-power which is required to discharge the water as fast as it runs into the mine, we have 14 for the power of an engine that will empty the mine in 48 hours, including also the water that will run into the mine during the 48 hours, which is the other answer required.

2. "If 12 horses eat 34 acres of grass in 4 weeks, and 21 horses eat 10 acres in 9 weeks, how many horses will be required to eat 24 acres in 18 weeks; the grass being at first equal on every acre, and growing constantly and uniformly ?" Ans. 36 horses.

3. If 14 cows, in 3 weeks, can eat the grass on 2 acres of land, also all that grows during the 3 weeks, and 16 cows, in 4 weeks, can eat all the grass on 3 acres, also all that grows during the 4 weeks; how many cows will be required to eat the grass on 6 acres, in 5 weeks, also all that grows during the 5 weeks; there being the same quantity on each acre, and growing constantly and uniformly? Ans. 26 cows.

4. Suppose 8 sheep, in 7 weeks, to eat all the grass on 400 square rods of land, also all that grows in the same time; and suppose 9 sheep, in 8 weeks, to eat all the grass on 500 square rods of land, also all that grows in the same time; how many sheep will be required to eat all the grass on 600 square rods of land, in 12 weeks, also all that grows in the same time; there being the same quantity of grass upon each square rod, and growing constantly and uniformly? Ans. 8 sheep.

PER CENTAGE AND PER CENT.

Art. 136. THE term per centage has but recently been introduced into text-books, and it is used to express a greater or less number of hundredths of any sum or quantity; thus, if a merchant invests his money and gains 25 hundredths of the sum invested, he gains a high per centage; if he gains only 5 hundredths of the sum invested, he gains a low per centage. Hence, any number of hundredths of any sum or quantity is called the per centage.

The term per cent. also signifies hundredths; thus, 1 per cent. of any sum or quantity is 1 hundredth of it, 2 per cent. is 2 hundredths of it, 6 per cent. is 6 hundredths of it; &c.

1. A man hired 20 dollars for one year, and agreed to pay 6 per cent. or 6 hundredths of the sum for its use; how much did he pay?

3. What is 4 per cent. of 5 dollars Of 10 dollars? Of 15 dollars? Of 20 dollars?

5. A man purchased a watch for 25 dollars, and sold it so as to gain 10 per cent.; how much did he gain by trading?

7. What is 1 per cent. of 75 dollars? 2 per cent.? 3 per cent.? 4 per cent.?

2. A girl found a lady's purse containing 50 dollars; the owner gave her 5 per cent. of the money. How much did the girl receive?

4. What is 6 per cent of 12 dollars? Of 25 dollars? Of 30 dollars? Of 40 dollars?

6. A market-man sold 50 dollars' worth of butter for a farmer, who paid him 5 per cent. for selling it; what did the market-man receive?

8. What is 5 per cent. of 100 dollars? 6 per cent.? 7 per cent.? 8 per cent.?

Art. 137. Since per centage and per cent. signify hundredths, we can express any per centage, or any number of per cent., by a decimal fraction; thus,

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