24. If 4 barrels of flour cost $343, how much can be bought for $175?

25. If 2 gallons of molasses cost 65 cents, what will 3 hogsheads cost?

26. What is the cost of 6 bushels of coal at the rate of £1 14s. 6d. a chaldron ?

27. What quantity of corn can I buy for 90 guineas, at the rate of 6 shillings a bushel?

28. A merchant failing in trade owes $3500, and his effects are sold for $2100: how much does B. receive, to whom he owes $420?

29. If 3 yards of broadcloth cost as much as 4 yards of cassimere, how much cassimere can be bought for 18 yards of broadcloth?

30. If 7 hats cost as much as 25 pair of gloves, worth 84 cents a pair, how many hats can be purchased for $216?

31. How many barrels of apples can be bought for $114.33, if 7 barrels cost $21.63?

32. If 27 pounds of butter will buy 45 pounds of sugar, how much butter will buy 36 pounds of sugar?

33. If 42 tons of coal cost $206.21, what will be the cost of 2 tons?

34. If 40 gallons run into a cistern, holding 700 gallons, in an hour, and 15 run out, in what time will it be filled?

35. A piece of land of a certain length and 12 rods in width, contains 14 acres, how much would there be in a piece of the same length 263 rods wide?

36. If 13 men can be boarded 1 week for $39,585, what will it cost to board 3 men and 6 women the same time, the women being boarded at half price?

37. What will 75 bushels of wheat cost, if 4 bushels 3. pecks cost $10.687?

38. What will be the cost, in United States money, of 324 yards 3qrs. of cloth, at 5s. 4d. New York currency, for 2 yards?

39. At $1.12 a square foot, what will it cost to pave a floor 18 feet long and 12ft. 6in. wide ?

CAUSE AND EFFECT.

231. Whatever produces effects, as men at work, animals eating, time, goods purchased or sold, money lent, and the like, may be regarded as causes.

Causes are of two kinds, simple and compound.

A simple cause has but a single element, as men at work, a portion of time, goods purchased or sold, and the like.

A compound cause is made up of two or more simple elements, such as men at work taken in connection with time, and the like.

232. The results of causes, as work done, provisions consumed, money paid, cost of goods, and the like, may be re- : garded as effects. A simple effect is one which has but a single element; a compound effect is one which arises from the multiplication of two or more elements.

233. Causes which are of the same kind, that is, which can be reduced to the same unit, may be compared with each other; and effects which are of the same kind may likewise be compared with each other. From the nature of causes and effects, we know that

1st Cause

2d Cause : : 1st Effect: 2d Effect:

and, 1st Effect : 2d Effect : : 1st Cause: 2d Cause.

234. Simple causes and simple effects give rise to simple ratios. Compound causes or compound effects give rise to compound ratios.

231. What are causes?

How many kinds of causes are there? What is a simple cause? What is a compound cause?

232. What are effects? What is a simple effect? What is a compound effect?

233. What causes are of the same kind? What causes may be compared with each other? What do we infer from the nature of causes and effects?

234. What gives rise to simple ratios?

DOUBLE RULE OF THREE.

236. The Double Rule of Three is an application of the principles of compound proportion. It embraces all that class of questions in which the causes are compound, or in which the effects are compound; and is divided into two parts:

1st. When the compound causes produce the same effects; 2d. When the compound causes produce different effects. 237. When the compound causes produce the same effects. 1. If 6 men can dig a ditch in 40 days, what time will 30 men require to dig the same?

ANALYSIS.-The first cause is compounded of 6 men, and $40 days, the time required to do the work, and is equal to what 1 man would do in 6×40 240 days.

The second cause is compounded of 30 men and the number of days necessary to do the same work, viz: 30 xx.

x=8 days.

But since the effects are the same, viz: the work done, the causes must be equal; hence, the products of the elements of the causes are equal. Therefore, in the solution of all like examples,

Write the cause containing the unknown element on the left of the vertical line for a divisor, and the other cause on the right for a dividend.

NOTE. This class of questions has generally been arranged under the head of "Rule of Three Inverse."

EXAMPLES.

1. A certain work can be done in 12 days, by working 4 hours a day: how many days would it require the same number of men to do the same work, if they worked 6 hours a day?

236. What is the double Rule of Three? What class of questions does it embrace? Into how many parts is it divided? What are they? 237. What is the rule when the effects are equal? Under what rule has this class of cases been arranged?

2. A pasture of a certain extent supplies 30 horses for 18 days: how long will the same pasture supply 20 horses?

3. If a certain quantity of food will subsist a family of 12 persons 48 days, how long will the same food subsist a family of 8 persons?

4. If 30 barrels of flour will subsist 100 men for 40 days, how long will it subsist 25 men?

5. If 90 bushels of oats will feed 40 horses for six days, how many horses would consume the same in 12 days?

6. If a man perform a journey of 221 days, when the days are 12 hours long, how many days will it take him to perform the same journey when the days are 15 hours long?

7. If a person drinks 20 bottles of wine per month when it costs 2s. per bottle, how much must he drink without increasing the expense when it costs 2s. 6d. per bottle?

8. If 9 men in 18 days will cut 150 acres of grass, how many men will cut the same in 27 days?

9. If a garrison of 536 men have provisions for 326 days, how long will those provisions last if the garrison be increased to 1304 men?

10. A pasture of a certain extent having supplied a body of horse, consisting of 3000, with forage for 18 days: how many days would the same pasture have supplied a body of 2000 horse?

11. What length must be cut off from a board that is 9 inches wide, to make a square foot, that is, as much as is contained in 12 inches in length and 12 in breadth ?

12. If a certain sum of money will buy 40 bushels of oats at 45 cents a bushel, how many bushels of barley will the same money buy at 72 cents a bushel ?

13. If 30 barrels of flour will support 100 men for 40 days, how long would it subsist 400 men?

14. The governor of a besieged place has provisions for 54 days, at the rate of 2lb. of bread per day, but is desirous of prolonging the siege to 80 days in expectation of succor : what must be the ration of bread?

238. When the Compound Causes produce different Effects.

In this class of questions, either a cause, or a single element of a cause may be required; or an effect, or a single element of an effect may be required.

1. If a family of 6 persons expend $300 in 8 months, how much will serve a family of 15 persons for 20 months?

ANALYSIS.-In this example the second effect is required; and the statement may be read thus: If 6 persons in 8 months expend 2 $300, 15 persons in 20 months will expend how many (or x) dollars?

STATEMENT.

1st Cause 2d Cause : : 1st Effect: 2d Effect.

Or, 6 x 8 : 15 × 20 :: 300 : X.

2. If 16 men, in 12 days, build 18 feet of wall, how many men must be employed to build 72 feet in 8 days?

ANALYSIS.-In this example an element of the second cause is required, viz: the number

of men. The question may be read thus: If 16 men, in 12 days, build 18 feet of wall, how many (or x) men, in 8 days, will build 72 feet of wall?

OPERATION.

1872
$1$
X 12

x=96 men.

3. If 32 men build a wall 36 feet long, 8 feet

high, and 4 feet thick, in 4 days, working 12 hours a day how long a wall, that is 6 feet high, and 3 feet thick can 48 men build in 36 days, working 9 hours a day?

238. When the compound causes produce different effects, what will always be required?