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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
18 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—
Men. Men. f.
12 : 15 : 18 : required earnings. .. as before Ans.
= £22. 10s. 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. 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. ... as beforeAns. :
days days = 22-5 days.
5 X 3
1. If 12 articles cost £15, what will 624 cost? 2. What is the price of 35 loaves, when 29 loaves cost 15s. 8}d. ?
3. If I get 140 metres of cloth for 541 fr. 70 c., what must I pay for 89 metres, 3 decim.?
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 metras, 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 ls. 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. Ad. 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 £2703
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 work. ing questions in this rule :
Ex. 1.-If 12 horses eat 20 bushels of corn in 8 days, in what time will 24 horses eat 16 bushels ?
24 horses : 12 horses 20 bushels : 16 bushels
: : 8 days : required time, EXPLANATION.-We are required to find time ; and so, as in simple proportion, we put in the 3rd place the given time, viz., 8 days.
Leaving, for the present, the quantity eaten out of consideration, we know that 24 horses require less time to consume a given quantity of food than 12 horses do; we therefore place the less of these two terms in the 2nd place, and the other in the 1st place. (The statement
up to this point is 24 horses : 12 horses :: 8 days : required time, and we might obtain 4 days as an answer, irrespective of the quantity eaten. We might now place this answer in the 3rd term of another simple proportion, and take the quantity eaten into consideration, irrespective of the number of horses, thus getting an answer depending both upon the number of horses and the quantity eaten. It is more convenient, however, to proceed thus :)
Again, taking into consideration the quantity eaten, and leaving out of consideration the other given pair of terms, we see that less time is required to eat 16 bushels than to eat 20 bushels. We, therefore, put the less term in the 2nd place, and the other in the 1st.
Now, treating the terms of the ratios which occupy the 1st and 2nd places as abstract quantities, and compounding them, we have:
24 x 20 : 12 x 16 :: 8 days : required time. .. required time
days Ex. 2.—How much bread can I get for 9d. when wheat is at 18s. a bushel, if the fourpenny loaf weigh 3 lbs. when wheat is at 20s. a bushel ? Proceeding as in Example 1, we have 4d. : 9d.
:: 3 lbs. : weight required. 18s. 20s. Or, 4 x 18 : 9 x 20 :: 3 lbs. : weight required. ,, weight required
1 2 X 16 X 8
24 X 20
lbs, = 7.5 lbs.
4 X 18
1. If 15 men can build a wall 81 feet long in 18 days, how many men can build 135 feet of the same kind of wall in 30 days?
2. In 4 days, 18 workmen can dig a ditch 162 yards long, 7 feet wide, and 12 feet deep. What must be the depth of a ditch which 45 workmen can dig in 7 days, supposing it to be 387 yards long and 5 feet wide ?
3. A traveller, going 15 hours a day, walks 1500 kilometres in 20 days. How far will he go in 30 days, walking 12 hours a day with the same velocity? Express your answer in English miles.
4. Two men are partners; one puts in a capital of £800, and receives as 6 months' profit £120. What is the capital of the other, who receives £3375 as 9 months' profit?
5. Two tourists having spent £1. 16s. 8d. in 2 days, meet three others with whom they continue their tour, and they spend while together £21. ls. 8d., at the same rate per day. Required how long they were in company.
6. If 16 men and 10 boys do a piece of work in 10 days, in how many days would 8 men and 18 boys do a piece 7 times as great, supposing the work of 5 boys equal that of 2 men?
7. Supposing the rate of carriage to be diminished one-third after the first 50 miles, find the cost of carrying 16 cwt. for 40 miles, when 12 cwt. can be carried 100 miles for 4s. 2d.
8. A cistern is 8 metres, 4 decim. long, 1 metre, 8 centim. wide, and 275 centim. deep. Find the depth of another cistern of equal capacity whose length is 7 metres, 2 decim., and width 11 decim.
9. Persons whose incomes are less than £300 per annum are taxed upon * £80 less than their income. Supposing 3 persons, having equal incomes, to pay £7 in the aggregate, at 4d. in the pound, find the total tax upon 14 persons, each having incomes 3 times as great.
10. If 12 horses eat 10 acres of grass in 16 weeks, and 18 horses eat 10 acres in 8 weeks, how many horses would eat 40 acres in 6 weeks, the grass being supposed to grow uniformly?
11. A boat, propelled by 8 oars, which take 28 strokes per minute, goes at the rate of 9} miles per hour. Find the rate of a boat propelled by 6 oars, which take 36 strokes per minute, the work done by each stroke of the latter being one-sixth less than that by each stroke of the former.
12. If 4 men and 10 women can do a piece of work in 8 days, which 12 women and 20 children can do in 4 days, in what time will 6 men, 18 women and 5 children do a work three times as great ?
APPLICATION TO ORDINARY QUESTIONS OF COMMERCE
42. Interest is the money paid for the use of money.
The Principal is the money lent, and the Amount is the sum of the interest and principal.
The Rate of interest is the money paid for a given sum for a given time. £100 is in practice the given sum, and one year the given time. Thus, if £4 be paid for the use of £100 for one year, the rate of interest is £4 per cent. per annum, or as we generally say,
per cent. Simple Interest is interest calculated on the original principal only.
Compound Interest is the interest which arises from adding the interest for each year to the principal of that year, and calculating interest for the next year upon the amount so obtained.,
Simple Interest. 43. RULE.—Multiply the principal by the rate per cent. and by the number of years ; then divide the product by 100, and the quotient will be the simple interest,