3. Multiply 87.32586 by 4.37. 4. Multiply 3.145 by 4.297.

Ans. 381.6140338.
Ans. 13.5169533.
Ans. 8s.

5. What is the value of .285714 of a guinea?
6. What is the value of .461607142857 of a ton?

Ans. 9cwt. Oqr. 26lb.

7. What is the value of .284931506 of a year?

Ans. 104da.

SECTION XXXIV.

DIVISION OF CIRCULATING DECIMALS.

1. Divide .54 by .15.

OPERATION.

.54 = §} = fr.

.15 = √ + % = £# = √5· £÷ ¬= £1 × 479 2703443.506493 Ans.

RULE.

Change both the divisor and the dividend into their equiva

lent vulgar fractions, and find their quotient as usual.

Change the vulgar fraction expressing the quotient into its equivalent decimal, and it

will be the quotient required.

2. Divide 345.8 by .6.

3. Divide 234.6 by .7.

Ans. 518.83.

Ans. 301.714285.

4. Divide .36 by .25. Ans. 1.4229249011857707509881.

SECTION XXXV.

MENTAL OPERATIONS IN FRACTIONS, &c.

If any number be divided into two equal parts, and into two unequal parts, the product of the two unequal parts together with the square of half the difference of the two unequal parts is equal to the square of one of the equal parts. Also,

The product of any two numbers is equal to the square of

half their sum, less the square of half their difference. See Euclid's Elements, Book Second, Proposition Fifth.

NOTE. A number is said to be squared when it is multiplied by itself; thus, the square of 5 is 5 x 5 = 25.

From the above proposition we deduce the following rules. To multiply any number containing a half by itself.

RULE 1. Multiply the whole number given in the question by the next larger whole number, and to the product add the square of the half = 1.

1. Multiply 5 by 51.

OPERATION.

5×6=30; × 1 = 1; 30 += 30 Ans.

NOTE. The whole number given is 5, and the next larger whole number is 6.

2. Multiply 7 by 71.
3. Multiply 34 by 31.
4. Multiply 9 by 91.
5. Multiply 11 by 11.
6. Multiply 20 by 204.
7. Multiply 301 by 301.

NOTE.

Ans. 561.

Ans. 121.

Ans. 904.

Ans. 132.

Ans. 4201.

Ans. 9301.

The same principle will hold good if we multiply any number by itself whose unit is a 5.

RULE 2. Take the next least number that ends in a cipher, and multiply it by the next larger number ending in a cipher, and add to the product the square of 5=25, and the result will be the product.

Ans. 625.

8. Multiply 25 by 25. The next less number ending in a cipher is 20, and the next larger is 30; 30 × 20 = 600; 5 × 5=25; 600 +- 25 = 625

To find the product of two mixed numbers, whose fractional part is a half, and whose difference is a unit.

RULE 3.

Multiply the larger number without the fraction by itself, and from the product subtract the fractional part multiplied by itself, and the result will be the product.

12. Multiply 6 by 71.

Ans. 48.

OPERATION.

7×7=49; × 1; 49-482 Ans.

NOTE. If the fractional parts of the numbers approach within a §, 1, or }, &c., of the larger number, the principle is the same.

17. What is the product of 43 multiplied by 54. Ans, 248.

OPERATION.

5×5=25; × }=}; 25 — }=248 Ans.

To find the product of two numbers, one of which is as much less than either 20, 30, 40, &c., as the other is more than either of these numbers.

RULE 4.- Multiply the 20, 30, or 40, as the case may be, by itself, and subtract from the product the square of half the difference of the two numbers to be multiplied, and the result will be the product.

22. Multiply 28 by 32.

Ans. 896.

We find that 28 is as much less than 30 as 32 is more than 30; we therefore multiply 30 by 30 900, and from this product we subtract the square of 2 = 4; 900—4=896 Ans.

=

23. What is the product of 75 by 85 ? 24. What is the product of 83 by 77 ?

25. What is the product of 97 by 103? 26. What is the product of 17 by 23 ?

Ans. 6375.

Ans. 6391.

Ans. 9991.

Ans. 391.

27. What cost 18cwt. of steel, at $22 per cwt.?

Ans. $396.

28. What cost 27 tons of hay, at $33 per ton?

Ans. $891.

29. What cost 64 gallons of oil, at $0.56 per gallon?

Ans. $35.84.

30. What cost 28 tons of hay, at $32 per ton?

Ans. $896.

31. What cost 49 tons of iron, at $ 51 per ton?

Ans. $2199.

SECTION XXXVI.

QUESTIONS TO BE PERFORMED BY ANALYSIS.

1. If a man travel 48 miles in 12 hours, how far will he travel in 17 hours? Ans. 68 miles.

[The following is the most obvious solution of this question. If he travel 48 miles in 12 hours, in 1 hour he will travel of 48 miles, which is 4 miles. Then if he travel 4 miles in 1 hour, he will in 17 hours travel 17 times as far, which is 68 miles, the answer.]

2. If 72 pounds of beef cost $6.48, what will 1 pound cost? what will 675 pounds cost? Ans. $60.75. 3. If of a dollar buy 1 pound of sugar, how much may be bought for $1? how much for $297 ? Ans. 239lb. 4. If a hogshead of wine cost $73.50, what is the value of 1 gallon? what cost 17hhd. 45 gallons? 5. Bought 11 bushels of rye for $9.00;

els?

Ans. $1302.00. what cost 25 bushAns. $20.455.

6. If a crew of 15 hands consume in 3 months 1620 pounds of beef, how much would be sufficient for 27 hands for the same time? Ans. 2916lb. 7. Bought 9 yards of flannel for $7.00; what would be the value of 371 yards? Ans. $29.31. 8. If a certain field will pasture 8 horses nine weeks, how long will it pasture 23 horses? Ans. 3 weeks.

9. If 7 dozen of hats cost $318.50, what will 1911 dozen cost? Ans. $874.958. 10. If 1 ton of hay cost $25.00, what will 17 tons 13cwt. 19 pounds cost? Ans. $441.46+.

Ans. 23.

Ans. 39

100.

11. What part of 9 is 25 ? 12. What part of 25 is 92 ? 13. If $25 will pay for 7 yards of broadcloth, what would be the price of 97 yards? Ans. $328.81.

14. If $47.25 will pay for the keeping of 7 horses 2 months, what would it cost to keep 43 horses for the same time? Ans. $290.25.

15. If 7 yards of cloth a yard wide is sufficient to make a cloak, how many yards would it take if the cloth was 12 yards wide? Ans. 4 yards. 16. If a barrel of beer will last 10 men a week, how long would it last 1 man? how long 37 men? Ans. 133 days. 17. If 5 calves are worth 9 sheep, how many calves will purchase 108 sheep?

Ans. 60 calves.

18. If 11 yards of cotton 3 quarters wide are sufficient to line a garment, how many yards would it require that were 5 quarters wide? Ans. 63 yards. 19. If 7 pairs of shoes will purchase 2 pairs of boots, how many pairs of shoes would it take to buy 18 pairs of boots? Ans. 63 pairs. 20. If four gallons of vinegar be worth 7 gallons of cider, what quantity of vinegar would it take to buy 47 gallons of cider? Ans. 26 gallons. 21. If a man travel 377 miles in 15 days, how far would he travel in 1 day? how far in 100 days? Ans. 2513 miles. 22. If 12 men can dig a ditch 50 feet long, 4 feet wide,and 3 feet deep, in 30 days, how long would it take 1 man? how long 47 men? Ans. 734 days.

23. If 5 barrels of flour cost $25.75, what will 39 barrels

cost?

Ans. $200.85. $791.01; what would 98 Ans. $4598.90,87. $275.20; what is the Ans. $19.78+.

24. Bought 17 acres of land for acres 3 roods and 14 perches cost? 25. Bought 973 yards of cloth for price of 7 yards?

26. If a pole 7 feet long cast a shadow of 5 feet, how high is that steeple whose shadow is 97 feet? Ans. 135 feet.

27. Gave 3 cwt. of sugar, at $ 9, for 4 of an acre of land; how much sugar would it have required to purchase an acre? Ans. 5 cwt.

28. If a vessel sail 473 miles in 3 hours, how far would it sail in 1 week? Ans. 2425m. 29. James can mow a field in 7 days, by laboring 10 hours a day; how many days would it take him to perform the work by laboring 12 hours a day? Ans. 5 days. 30. If of a lot of land is worth $42.12, what is the value of of it? Ans. $29.41. 31. If a man can earn $10.27 in 4 of a week, how much would he earn in a month? Ans. $71.89.

32. If 18 men can reap 72 acres in 5 days, how long would it take 6 men to perform the labor? Ans. 15 days.

33. If 19 gallons of wine can be bought for $25, how many gallons will $71.25 buy? Ans. 54 gallons. 34. If a penny loaf weighs 8 ounces when wheat is $1 a bushel, what should it weigh when wheat is sold for $1.25 a bushel? Ans. 63 ounces.

35. If a basket which contains 14 bushels must be filled with apples 7 times to make one barrel of cider, how many barrels may be made by its being filled 126 times? Ans. 18bbl.