35. If 48 ozs. of tea cost 1s., what will 29 lbs. cost?

36. If cwt. cost £78, what will 6 lbs. cost?

37. If of a ton cost £, what will of a ton cost?

38.

A watch set accurately at noon indicates 10 min. to 5 at 5 o'clock the same afternoon, what is the time when the watch indicates 5 o'clock?

39. Two clocks point to 2 o'clock at the same instant, one loses 7 sec. and the other gains 8 sec. in 24 hours; when will one be half an hour before the other?

40. The supply of a certain number of persons with bread at 71. a loaf costs £27 188., what will be the cost of supplying of that number at 61d. a loaf ?

41. A garrison of 1000 men is victualled for 30 days, after 10 days it is reinforced and then the provisions were exhausted in 5 days; what was the number of the reinforcement?

42. A garrison was victualled for 30 days, and after 10 days it was reinforced by 3000 men, and then the provisions were exhausted in 10 days; what was the original number of the garrison?

43. If a pole 10 feet high cast a shadow 12 feet 8 inches long, how high is a tower whose shadow at the same time is 57 feet long?

44. If one tower, known to be 99 feet high, cast a shadow 73 feet 3 inches long, what length of shadow will another tower 100 feet high cast at the same time?

45. If 48 oz. avoir. cost 8s., what will 8 lbs. cost?

46. If

of of 23 of 40 lbs. of beef cost 1d., how many lbs. may be bought at the same rate for 6s. 7 d.?

47. Lead weighs 11 times, and platinum 21 times, as much as water; what weight of platinum will be equal in bulk to 112 lbs. of lead ?

48. A watch which gains uniformly tells the true time once in every 200 days; how much does it gain or lose per hour?

49. A leaden shot of 4 inches in diameter weighs 17 lbs., but the size of a shot 4 inches in diameter is to that of one 4 inches in diameter as 64000: 91125; what is the weight of a leaden ball 4 inches in diameter ?

50. The ratio of the vibrations of a pendulum to the number of seconds in which the vibrations are made is how many times will it vibrate in 24 hours?

COMPOUND PROPORTION.

199. Questions frequently arise in which five quantities are given to find a sixth, or seven quantities given to find an eighth, and so on. In such cases it becomes necessary to repeat the process already adverted to in the Rule of Three, or to combine two or more proportions so as to reduce them all to a simple proportion, and then, if any three terms of the reduced proportion be known, the fourth can be found as before; and the method by which the fourth term of a proportion is found when the antecedent and the consequent consists each of more than one quantity, is called Compound Proportion.

The following example is one of Compound Proportion.

EXAMPLE.

Ex. If the expenses of 7 persons for 3 months be 70 guineas, what will be the expenses of 10 persons for 12 months at the same rate?

Here we must find what the expenses of 10 persons for 3 months will be.

This is done by the Rule of Three in the following manner :

7 persons : 10 persons :: 70 guineas: 100 guineas (A),

Now, suppose 10 persons expend 100 guineas in 3 months, we must next inquire how much they will expend in 12 months, and by the same rule we get―

3 months: 12 months :: 100 guineas:

400 guineas (B),

4

1 X 100
३

=400 guineas.

Instead of repeating the operation of the Rule of Three, the terms of the proportion (A) may be multiplied by the corresponding terms of the proportion (B), and the product will still be proportional (§ 184).

Hence

7 X 3 10 X 12 :: 70 X 100 : 100 X 400,

But as the ratio of the third term to the fourth will not be altered by dividing each of the terms by 100, it is obvious that the operation for finding the fourth term (100) in A is superfluous and may be dispensed with entirely.

The statement of the terms of the operation will then be as follows:

200. For the sake of convenience we may divide each question into two parts, the supposition and the demand; the former being the part which expresses the condition of the question, and the latter the part which mentions the thing demanded or sought. In the question, for example, "If the carriage of 15 cwt. for 17 miles cost me £4 58., what would the carriage of 21 cwt. for 16 miles cost me?" form the demand. Adopting this distinction we may give the following rule for working out examples in Compound Proportion.

RULE LVII.

1o. Take from the supposition that quantity which corresponds to the quantity sought in the demand, and write it down as a third term. Then take one of the other quantities in the supposition and the corresponding quantity in the demand and consider them with reference to the third term only (leaving all the other given quantities out of consideration) when the two quantities are so considered, if from the nature of the case the fourth term would be greater than the third, then, as in the Rule of Three, put the larger of the two quantities in the second term, and the smaller in the first term; but if less, put the smaller in the second term, and the larger in the first term. From the remaining quantities select two others, and retaining the same third term, proceed in the same way to make one of these quantities a first term, and the other a second term.

2o. If there be other quantities in the supposition and demand, proceed in like manner with them.

3°. In each of these statings reduce the first and second terms to the same denomination. Let the common third term be also reduced to a single denomination if it be not already in that state. The terms may then be treated as abstract numbers.

4. Multiply all the first terms together for a final first term, and all the second terms for a final second term, and retain the former third term. Alultiply the third term by the product of the quantities in the second term, aud divide the result by the product of all the quantities in the first term.

NOTE. In dealing with the final statement obtained by the Rule, the two notes to Rule LVI, page 142, will often be found useful.

EXAMPLES.

Ex. I. If 6 men in 24 days of 8 hours each set up 576 yards of telegraph wire, how many yards will be set up by 5 men in 20 days of 10 hours each?

In such questions the first thing is to get a distinct idea of the supposition as distinguished from the demand.

Here the supposition is that 6 men set up 576 yards of wire in 24 days of 8 hours each. If, now, we arrange the demand in the same way as the supposition, it will run-5 men set up? yards in 20 days of 10 hours each, and the ? is the number of yards we have to fill up. Since the third term must always be of the same kind as the answer, in this example the third term will therefore be "576 yards."

We next consider thus:

"If 6 men can set up 576 yards of wire, how many yards can be set up by 5 men ?" This proportion would be stated thus:

As 6 men : 5 men :: 576 yards of wire :: the yards set up by 5 men. Again, "If 576 yards of wire are set up in 24 days, how many yards will be set up in 20 dayɛ ?” The proportion would be stated thus:

As 24 days: 20 days: 576 yards: the yards set up in 20 days. And, therefore, resuming the statement obtained above, we have

24 days : 20 days: 576 yards : the Answer.

Again, in the above example consider thus:--

"If 576 yards of wire are set up when the men work 8 hours a day, how many yards will be set up when the men work 10 hours a day?"

This proportion would be stated thus:—

As 8 hours: 10 hours : 576 yards: the yards set up in days of 10 hours each. And, therefore, resuming the statement obtained above, we have

Let us now consider the separate effect of altering each term of the supposition to suit the question.

(1) Employing 5 men instead of 6 men will reduce the number of yards set up in the ratio of 6 to 5; therefore, as far that change alone is concerned, the 576 yards will have to be multiplied by the fraction.

(2) The number of days is decreased from 24 to 20. This will decrease the number of yards in the ratio of 24 to 20. The effect of that change upon the answer will therefore be to multiply it by 21.

(3) The alteration of the working day from 8 to 10 hours will increase the number of yards in the ratio of 8 to 10. Its effect, therefore, will be equivalent to multiply by the fraction 10.

We have thus found the separate effects of those changes, each of which acts independently of the other, and simply introduces its own multiplier. The conjoint effect upon the number of yards will evidently be to multiply that number by

Ex. 2. If 15 men working 8 hours a day can plough 240 acres of land in 10 days, how many days will it take 20 men working to hours a day to plough 600 acres?

Here the supposi'ion is that 15 men plough 240 acres in 10 days of 8 hours. If, now, we arrange the demand in the same way as the supposition it runs -20 men plough 600 acres in? days of 10 hours, and the ? in the number of days is what we have to fill up.

Let us go on to consider the separate effect of altering each term of the supposition to suit the question.

We may take the first part of the question thus:

"If it take 15 men 10 days to plough a piece of land, how many days will it take 20 men ?”

By the rule of Simple Proportion we get the following proportion :

20 men : 15 men :: 10 days

the number of days 20 men will take.

Hence, employing 20 men instead of 15 will evidently reduce the number of days in the proportion of 15 to 20; therefore, as far as that change alone is concerned the 10 days will have to be multiplied by 13.

We next consider the second part of the question thus:

"If 240 acres are ploughed in 10 days, how many days will it take to plough 600 acres?" By the rule of Simple Proportion we get the following proportion: As 240 acs. : 600 acs. :: 10 dys. : the number of days required to plough 600 acs. Or again, consider that the number of acres is increased from 240 to 600. This will increase the number of days in the ratio of 600 to 240. The effect of that change upon the answer will therefore be to multiply it by 28.

We next consider the third part of the question thus:-

"If it take 10 days to plough a certain number of acres when the working day is 8 hours, how many days will it take to do the same working 10 hours a day?"

Here we have the following proportion :

10 hours: 8 hours :: 10 days: the days of 8 hours each.

The alteration of the working day from 8 to 10 hours will reduce the number of days in the proportion of 10 to 8. Its effect, therefore, will be equivalent to multiply by the fraction.

Therefore, resuming the statement obtained, we have

Having found the separate effects of three changes, each of which acts independently of the other, and simply introduces its own multiplier. The conjoint effect upon the number of days will evidently be to multiply that number by

Ex. 3. Some repairs to an engine have occupied 16 men 7 weeks, and have cost £243 168. 9 d. for wages. Some larger repairs are necessary, which is expected will occupy 20 men 11 weeks, how much will be required to pay them at the same rate of wages?