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product of the larger company into what the less company expend, equal $14595; what is the expense of the larger company ? - - Ans. $1326,818-1-. 8. What number must be divided by 31, that the quotient may be the same as the quotient of 15 divided by 3; and what number must 31 be multiplied by to produce that number? Explanation.—The first number demanded must contain 31 as many times as 15 contains 3. If we divide 15 by 3 the quotient will express the number of 3's contained in 15, which is 5; and if we multiply 31 by 5, the product will contain 31, 5 times, as many times as 15 contains 3. 5 is the ratio, because it is the number by which 3 was multiplied to make the second term or first consequent, and 31, the second antecedent. must be multiplied by the ratio in order that the fourth term may be in the same proportion to 31 that 15 is to 3. The proportion demanded in the question, is, that 3 can be taken from 15 as many times as 31 can be taken from 155; or 3 : 15 :: 31 : 155. If we invert the terms, the numbers will still be in proportion, and have the same ratio; but the ratio becomes a divisor instead of a multiplier. Thus 15 : 3 :: 155 : 31; or 15 is the same multiple of 3 that 155 is of 31, for the two consequents are formed by dividing the two antecedents by 5, the ratio, which was the former multiplier. This is called inverting the terms of the proportion. It will frequently be more convenient to find the ratio by dividing the larger of the two first terms by the smaller, and multiplying or dividing the third term by it, which will

give the fourth. This method of operating will al

ways show the scholar more clearly the nature of proportion. 9. What number is the same multiple of 7 that 61 is of 97 The proportion will be, as 9 : 61 :: 7 to the anSW C P. Ans. 47#. 10. What number is the same part of 27 that 13 is of 19 Ż - Ans. 18.1%.

11. The first of three terms is 9, the square of the mean 256; what is the third proportional 2 - Ans. 284. 12. Let the product of two means be a #r, the first term ; ; required the fourth term 2 Ans. #"r. 13. Find a third proportional to 8 and 13. * - Ans. 214. 14. Find a third proportional to rāz and ##. - * . Ans. 38 or 15. If the square of a mean proportional be and the first term be to ; what is the last term 2 Ans. 16#. 16. What number is that which bears the same proportion to 79 that. 17 does to 152 Ans. 89*.

17. If 15 yards of cloth cost 16 dollars, what will 90 yards cost at the same rate 2

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Explanation.—If we divide 16 by 15, the quotient will be the price of one yard, and 90 multiplied by the price of one yard, will give the price of 90 yards. The price of one yard is the ratio, therefore 15 must be to its price 16 dollars, as 90 is to 96 dollars its price. Or if we multiply 90 yards by 16 dollars, the price of 15 yards, the product must contain the price of 9 yards 15 times; consequently dividing that product by 15 will give the true answer.

18. If a man sell 2 pounds of butter for 31 cents, what will 1000 pounds come to ? Ans. $155. 19. If a horse eat 3 tons of hay in 7 months, how much would he eat in 12 months' Ans. 54 tons. 20. If 8 square rods of ground produce 13 bushels of potatoes, how many bushels would one acre produce, at the same rate? Ans. 260. 21. If 3 pounds of sugar cost 40 cents, what must I give for 100 lbs. of the same * Ans. $18,334. 22. Wishing to know the heighth of a certain steele, l measured the shadow of the same on a horizontal plain, 27} feet. I then erected a ten feet pole on the same plain, and it cast a shadow of 23 feet Required the height of the steeple 2 * Ans. 103}. feet.

23. The earth turns upon its axis once in 23 hours and 56 minutes; and its circumference is 25000 miles. How far is a person carried in one second of time, who lives on the equator 2 Ans. 92 for rods. 24. Two men, A and B, commence a journey at the same time and from the same place. A's speed, which is 54 miles an hour, bears the same proportion to B's that 5 does to 7; how far does B travel in 9 hours ? Ans. 65%; miles. 25. A gentleman having a lot of land in the form of a parallelogram, wishes to set off 50 acres from one end which is 100 rods wide. How many rods will it require on the other side to contain 50 acres? - - Ans. 80. 26. A owes B $500, but not being able to pay the whole, B offers to compound with him for a sum which shall bear the same proportion to his debt, as # does to #; what does B receive? Ans, 357,142+

JNote. The numerator of a fraction bears the same proportion to that part of a quantity expressed by the fraction as the denominator bears to the whole quantity.

27. If a man fill ; of a cask with brandy, with wine, and 3 with water, and it lack 213 gallons of being full, how many gallons will that cask contain 2 - Ans. 100. 28. If a gamester at the first game lose # of his money, and win 3 shillings, at the second ; lose ; of his first stock at the third, and win six shillings at the fourth ; now if what he has won bears the same proportion to what he has lost, as what he has left of his first stock does to his first stock, how much money had he when he commenced playing 2* Ans. 52; shillings. 29. A and B trade until they gain 6 per cent. on their stock; then 3 of A's gain is $18. Now if A's

*This question is susceptible of an indefinite number of answers. The one given is the answer obtained without reducing the fraction,

stock is to B's as # to #, what did each gain, and what was the original stock of each Ans A's gain was $45 and his stock $750. ' ' B's gain was $37,50 and his stock $625. 30. A, B, and C united stocks in trade; A put in $160, B, $200, and C, $150; and they gain a sum equal to 3 of their whole stock: how much does each gain Ans. A gains $32, B, $40, and C$30. 31. A sells B100 lbs. of sugar at 12 cents a pound, and gains 9 per cent. ; he receives of B 100 lb. of beef at 6 cents a pounds, who gains 10 per cent. Which gains the most by the trade 2 Ans. A gains 44,53+ cents more than B. 32. What per cent. does a merchant gain who buys tea at 45 cents a pound, and sells it at 557 .As 45 cents are to what is gained on 45, so is 100 to what would be gained on 100, which will be the per cent. Ans. 22#.

33. If a man buy tea at 45 cents a pound, and sell it so as to gain 25 per cent., what does he receive per pound ! Ans. 56} cents.

34. B bought brandy for $1,10 per gallon, and

sold it so as to lose 54 cents on a dollar. What did he receive per gallon ? Ans. $1,0395.

JNote. The diameter of a circle bears nearly the same proportion to the circumference, that 7 does to 22.

35. If the diameter of a circle be 100 feet, what is the circumference 2 Ans. 3143 feet.

36. The diameter of the sun is 88.3246 miles ; what is its circumference 2 Ans. 2”775°916 miles.

37. The diameter of Jupiter and Mercury are to each other as 44585 is to 1612. Allowing the circumference of Mercury to be 101324 miles, what is the diameter of Jupiter ? Ans. 89170 miles.

JNote.—The spaces, passed over-by bodies falling freely in vacuo, are as the squares of the times. It is found by experiment that bodies falling in vacue, descend 16 FF feet in the first second ef time.

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38. If a body fall freely in a vacuo 4 seconds of time, how much farther will it descend during the fourth second than the first 2 Ans. 963 feet.

39. If a body fall 16,' feet in the first second of its descent, how far will it have descended at the end of 5 seconds ! Ans. 4023 feet.

JNote.—The squares of the periodical times of the Planets, bear

the same proportion to each other as the cubes of their mean distances from the sun.

40. If two balls revolve about a common centre, —one at the distance of 3 inches, the other at the distance of 5, and the ball nearest the centre revolve once in 3 seconds; how long will the other be completing a revolution, if both observe the same laws of motion as the planets 2 - - Ans. 6,45+seconds.

41. The Earth’s mean distance from the Sun is 94'507°428 miles. Calling the periodical time of the Earth 365 days, and that of Venus 2244, which is near the truth, what is the mean distance of Venus from the Sun ? Ans. 68°351°533+ miles.

42. If 15 men can do a piece of work in 14 days,

in what time would 18 men do the same work, working at the same rate

The greater number of men must bear the same proportion to the less, that the longer period of time does to the shorter. Ans. 11 # days.

43. If $160 worth of provisions will support 80 men 12 days, how long will the same provision support 135 men 2 Ans. 74.

-eeeeePROMISCUOUS QUESTIONS.

1. A man sold one hog, weighing 250 pounds, at 4 cents per pound; a second, weighing 300, at 44 cents; and a third, weighing 369, at 5 cents per

pound. What was the average price per pound for the whole 2

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