11. In England the unit of work is the foot-pound, and in the metric system it is the kilogram-metre. Reduce 62 metric units of work to English units, taking 1 gram 15.4323 grains, and 1 metre

=

39.3708 inches.

=

12. The pressure of the atmosphere is 143 lbs. upon the square inch. Find the pressure in kilograms upon the square

centimetre.

we have ratio 6 8 ratio 15 : 20;

and we say that the numbers 6, 8, 15, 20 form a proportion. We generally express the fact thus

6:8 :: 15: 20.

It is easy to find by trial that the product of the extreme terms is equal to the product of the means.

Thus, we have 6 × 20 = 8 x 15.

We may prove this property of the terms of a proportion to hold generally as follows:

-

Suppose we have given the proportion 12: 21 :: 20:35. It follows, from our definition above, that = , and multiplying each of these fractions by the product of their denominators, viz., by 21 x 35, we have

1 × 21 ×

Now (Art. 8), 1† × 21

=

12 = 12, and 3g × 35

and we hence have 12 × 35 = 20 × 21.

Now, we have not in our reasoning taken into account the actual value of the terms of the given proportion; and it is therefore evident that a similar result will follow from every proportion, and we may hence conclude generally :—

In every proportion the product of the extremes is equal to the product of the means.

39. Having given any three terms of a proportion, to find the remaining one.

Since the product of the extremes is equal to the product of the means, the following rule is evident :

RULE.—If the required term be a mean, divide the product of the extremes by the other mean; but if the required term be an extreme, divide the product of the means by the other extreme.

Ex. 1.-28, 24, 30 are respectively the 1st, 3rd, and 4th terms of a proportion, required the 2nd term.

28 required term: 24: 30

We have

Ex. 2.—10, 45, 16 are respectively the 1st, 2nd, and 3rd terms, required the 4th term.

Ex. 3.-2 hours, 45 minutes, 8 men are respectively the 1st, 2nd, and 3rd terms, required the 4th term.

We must express (Art. 6) the 1st and 2nd terms in the same denomination, and the proportion will stand thus—

Now, the ratio of the first two terms is the same as the ratio of the abstract numbers 120 and 45; and the 4th term must be of the same denomination as the 3rd term, otherwise the 3rd and 4th terms could not form a ratio.

40. In Arithmetic we divide Proportion into Simple and Compound. Simple Proportion is the equality of two simple ratios, and therefore contains four simple terms; and the usual problem is to find the fourth term, having given the first three terms.

When we know the exact order of the given terms, the fourth term is, of course (Art. 38), found thus—

RULE.-Multiply the 2nd and 3rd terms together, and divide by the 1st.

The formal arrangement of the three given terms in their proper order is called the statement; and the only difficulty, therefore, in working a sum in Simple Proportion, or Single Rule of Three, as it is called, consists in stating it.

We shall work a few examples to illustrate the mode of doing this.

Ex. 1.—If 12 men earn £18, what will 15 men earn under the same circumstances?

We have here two kinds of terms, men and earnings, and whatever ratio any given number of men bears to any second given number of men, it is evident that it must be equal to the ratio of the earnings of the first lot of men to the earnings of the second lot, and we may therefore write—

or

12: 15 :: £18: 2nd earnings.

As the first two terms are of the same denomination, their ratio is not altered by treating them as abstract quantities, and the denomination of the 4th term must be the same as that of the 3rd.

Ex. 2.—If 18 men do a piece of work in 25 days, in what time will 20 men do it?

The two kinds of terms we have here to consider are men and time. In doing work we know that the time will diminish exactly as the number of men increases, and hence the ratio of the second lot of men to the first lot will be equal to the ratio of the given time to the time required. We therefore have

We have reasoned out the above examples thus to show that the working of problems in Rule of Three depends upon the principle of the equality of ratios. Practically, however, we proceed as follows:

Ex. 1.-If 12 men earn £18, what will 15 men earn under the same circumstances?

We are required to find earnings, and we therefore put down for the 3rd term the given earnings, thus—

The question is with regard to 15 men instead of 12 men, and we know their earnings must be greater. We therefore place the greater of these terms in the 2nd place and the other in the 1st, and the statement becomes—

Ex. 2.—If 18 men do a piece of work in 25 days, in what time will 20 men do it?

We are required to find time, and we place therefore the given time, viz., 25 days, in the 3rd place.

18 men.

Again, the question is with regard to 20 men instead of Now, we know that 20 men require less time than 18 men to do a piece of work, and we hence place the less of these terms in the 2nd place. The statement then becomes— Men. Men. Days.

20: 18 :: 25 : required time.

1. If 12 articles cost £15, what will 624 cost?

2. What is the price of 35 loaves, when 29 loaves cost 15s. 81d.?

3. If I get 140 metres of cloth for 541 fr. 70 c.,

what must

4. If 4 cubic metres of water run into a cistern in 18 minutes, in what time will it be full, supposing it to be 4 metres long, 6 metres, 25 centim. deep, and 35 decim. wide?

5. If the carriage of a parcel for the first 50 miles be 1s. 3d., and if the rate be reduced by one-third for distances beyond, how far can the parcel be carried for 1s. 7d.?

6. If a half-kilogram of sugar cost 1 fr. 10 c., what will be the cost of 3 kilog. 625 grams.?

7. There are two pieces of the same kind of cloth, measuring 43 yards and 57 yards respectively, and the second costs £1. 9s. 2d. more than the first. What is the cost of the

first?

8. A garrison of 720 men have provisions for 35 days, and after 7 days 120 more men arrive. How long will the provisions last?

9. After paying 4d. in the pound income-tax a person has £299. 18s. 4d. left. What was the amount of his original income?

10. Two clocks, one of which gains 3 minutes and the other loses 5 minutes per day, are put right at noon on Monday. What is the time by the second clock when the first indicates 4 P.M. on the following Thursday?

11. When will the hands of a clock be exactly 30 minute divisions apart between 2 and 3 o'clock?

12. If I lend a friend £120 for 9 months, how long ought he to lend me £270}

Compound Proportion.

41. Compound Proportion is an equality between ratios, one of which at least is a ratio compounded of two or more simple ratios.

Arithmetical questions depending on Compound Proportion are generally said to belong to the Double Rule of Three; and the proportion consists of an equality between a ratio, on the one hand, compounded of two or more simple ratios; and, on the other hand, a simple ratio, whose consequent is required.

The following examples will illustrate the method of working questions in this rule :—