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other, and then finds one flock twice as numerous as the other; what number did each of them contain at first? Ans. 147.

6. From each of 16 coins, an artist filed the value of 20 cents; when the coins, being examined, were found worth only 11 dollars 68 cents; what was the original value of each? Ans. 93 cents.

7. The hold of a ship, containing 442 gallons, is emptied in 12 minutes by two buckets; the greater of which, holding twice as much as the less, is emptied twice in 3 minutes, and the less, is emptied three times in 2 minutes. Quere the number of gallons held by each bucket? Ans. 26 and 13.

8. A trader maintained himself for 3 years at an annual expense of £50; and in each of those years, augmented that part of his stock which was not expended by thereof. At the end of the third year, his original stock was doubled; what was that stock?

Ans. £740.

9. What number is that, which being added to, and subtracted from 36, the sum of the cube roots of the results shall be 6? Ans. 28.

10. Required to find a fraction, to the numerator of which if 4 be added, the value will be ; but if 7 be added to the denominator, the value will be ? Ans.

11. What two numbers are those, whose difference, sum, and product, are as the numbers 2, 3, and 5 respectively? Ans. 10 and 2.

12. A vintner having mixed a quantity of brandy and water, finds that if he had mixed 6 gallons more of each, he would have had 7 gallons of brandy for every 6 of water; but if he had mixed 6 gallons less of each, he would have had 6 gallons of brandy for every 5 of water. Quere the number of gallons of each?

Ans. 78 of brandy, and 66 of water. 13. A. and B. together, are able to perform a piece of work in 15 days; after working jointly 6 days, B. finishes

it alone in 30 days; in what time would each of them singly effect it? Ans. A. in 21 days, B. in 50 days.

14. A company of smugglers found a cave which would exactly hold their cargo, viz. 13 bales of cotton, and 33 casks of rum; but while they were unloading, a revenue cutter appeared, on which they sailed away with 9 casks and 5 bales, having filled of the cave; how many bales, or how many casks would the cave contain?

33

Ans. 24 bales, or 72 casks.

15. There are two numbers, whose sum is to their difference as 8 to 1; and the difference of whose squares is 128; what are the numbers? Ans. 18 and 14.

16. Required those two numbers, whose sum is to the less as 5 to 2; and whose difference multiplied by the difference of their squares is 135? Ans. 9 and 6.

17. A merchant laid out a certain sum upon a speculation, and found, at the end of a year, that he had gained £69. This he added to his stock, and at the end of the second year, found he had gained as much per cent. as in the first. Continuing in this manner, and each year adding to his stock the gain of the preceding, he found at the end of the fourth year, that his stock was to the sum first laid out, as 81 to 16. Quere the sum first invested? Ans. £138.

18. There are two numbers, whose sum is to the greater as 40 is to the less, and whose sum is to the less as 90 to the greater; what are the numbers?

Ans. 36 and 24.

19. The area of a rectangular parallelogram is 960 yards, and the length exceeds the breadth by 16 yards; what are the sides?

Ans. 40 and 24.

20. The area of a rectangular parallelogram is 480, and the sum of the length and breadth 52. Quere the sides? Ans. 40 and 12. 21. The sides of a right angled triangle, form an equidifferent series, whose common difference is 3; what are the sides? Ans. 15, 12 and 9.

22. There are three numbers, the difference of whose differences is 5, their sum is 20, and their continued product 130; required the numbers ?

Ans. 2, 5, and 13.

23. The sum of three numbers is 21, the sum of the squares of the greatest and least is 137; and the difference of the the differences is 3. Quere the numbers? Ans. 4, 6, and 11.

24. There is a number consisting of two digits, which divided by the sum of its digits, has a quotient greater by 2, than the first digit. But, the digits being inverted and divided by the sum of the digits increased by unity, the quotient is equal to the first digit increased by 4. Quere the number?

Ans. 24.

25. Required to find four numbers in geometrical progression, whose sum shall be 15, and the sum of the squares 85? Ans. 1, 2, 4, and 8. 26. There are five numbers in geometrical progression, whose sum is 242, and the sum of their squares 29524; what are the numbers? Ans. 2, 6, 18, 54, 162.

27. There are six numbers in geometrical progression, the sum of the extremes is 99, and the sum of the other four terms is 90. Quere the numbers?

Ans. 3, 6, 12, 24, 48, and 96.

28. The American dollar consists of 1485 parts by weight, of pure silver, and 179 of copper. Quere the specific gravity of this dollar, the specific gravity of pure silver being 11092, and that of copper 9000?

Ans. 10821.

SECTION VIII.

ON RATIOS.

61. Ratio is the relation that two quantities of the

same kind bear to each other.

Ratio is estimated by the quotient arising from the divi

a

sion of the first term by the second. Thus, if=

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is said to have the same ratio to b, that c has to d; or the quantities a, b, c, d, are termed proportionals; which is briefly expressed thus, a:b::c: d.

The first and third terms are called antecedents, the second and fourth consequents.*

62. When four quantities are proportionals, the product of the first and fourth is equal to the product of the second and third; and reciprocally,

If, by multiplying by bd, ad=bc.

ad bc

Reciprocally, if ad=bc,

a

C

or
bd bď b a

63. When four quantities are proportionals, the sum of the first and second is to the second, as the sum of the third and fourth is to the fourth.

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If the word difference be substituted for sum, the proposition will still be a true one.

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64. If four quantities be proportionals, the sum of the first and second is to their difference, as the sum of the third and fourth is to their difference.

* The terms are similarly designated when more than four are concerned.

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65. When any number of quantities are proportionals, as one antecedent is to its consequent, so is the sum of all the antecedents to the sum of all the consequents.

Let ab::cd::e: f :: g: h, &c., then (art. 62.)
ad=bc, af be, ah=bg, &c., also ab-ba.

.. ab+ad+af+ah, &c. =ba+bc+be+bg, &c. Or, ax{b+d+f+h, &c.}=b×{a+c+e+g, &c.} .. (art. 62,) a : b::a+c+e+g, &c. : b+d+f+h, &c. 66. When four quantities are proportionals, if the first and second be multiplied or divided by any quantity, and likewise the third and fourth, the resulting quantities will be proportionals.

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The demonstration is manifestly applicable when m and n are, one or both, fractional numbers.

67. When the first and third of four proportionals, are multiplied or divided by any number, and also, the second and fourth, the resulting quantities are proportionals.

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Or, ma: nb:: mc: nd, m and n being any numbers, either integral or fractional.

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