(b) Rule of Three inverse.

RULE.-Put the quantity which is of the same kind as the answer in the third term; put the corresponding value of the other quantity in the second, and the remaining value in the first.

Proceed then in the same manner as above.

Ex. (3) If a garrison be supplied with provisions which will last 1450 men for 36 days, how long would they last 870 men?

Here the varying quantities are number of men and time, and applying the test of (69 b), we see that the time will vary inversely as the number of men. Therefore

Ex. (4) At an average rate of

of a mile per minute a train travels a certain distance in 24 hrs., how long would it take at the rate of 5 of a mile per minute?

The varying quantities are rate and time of travelling, and a little consideration will show that the time will vary inversely as the rate. Therefore

(c) It will often be easier, for stating the question, to apply the following rule, which includes the two former :

RULE.-Having determined whether the quantities in the question vary as one another in either of the above ways, put that which is of the same kind as the answer in

the third term; and consider whether, from the nature of the question, the answer will be greater or less than this. If greater, put the greater of the other quantities in the second term; if less, put the less in the second; and the remaining one in the first.

Quantities sometimes occur in a question, which will not affect the result and must therefore be left out of consideration.

Ex. (5) If 3 mėn can mow a field of 13 acres in 6 days, how long would 7 men take to do it?

The size of the field, being the same in both cases, will not affect the result. The time required will vary (inversely) as the number of men; and 7 men will clearly require a shorter time than 3 men; therefore applying (c), we have 7: 3 :: 6 days.

3×6 18

and

24 days. Ans.

Ex. (6) A bankrupt's debts amount to £5145 10s. 8d., and his assets to £1929 118. 6d.; how much can he pay in the pound? and how much will a creditor lose on a debt of £350?

In the first part of the question, there are two quantities to be considered, debt and money that can be paid in discharge of debt, and amounts of the latter must vary directly as the corresponding amounts of the former. Two amounts of debt are given, viz. £5 145 108. 8d., and £1; and to the first of these corresponds the amount of money, £1929 118. 6d.; we have to find the amount of money corresponding to £1 of debt. Therefore

The first two terms are reduced into fourpenny pieces, and the third into sixpences.

The second part may be obtained either by another proportion, or directly from the first by multiplication. Adopting the latter method,

Money paid on £350 of debt = 78. 6d. × 350

= 131 58.

and therefore the loss on £350 = £350 - £131 58.

= £218 158. Ans.

Exercise 61.

a. (1) If 5 cwt. I qr. 7 lbs. cost £12 5s., what is the value of 1 cwt. 3 qrs. 8 lbs.?

(2) If 3 men can reap 10 ac. 2 rds. in a certain time, how much could 8 men reap in the same time?

(3) What would be the cost of 3 cwt. at the rate of 167. for 5 lbs.?

(4) The rent of 9 ac. 2 rds. being £46 11s., how much of the same land may be rented for £113 18s. 6d. ?

(5) How far, at the rate of 8 knots an hour, could a ship sail in the same time that another ship sailed 154 miles at the rate of II knots an hour?

ton cost £6, what would 75 cwt. cost at the

(7) If 3s. 6d. is paid for 3 oz. 10 dwts. 3 grs., what must be paid for 2lbs. 1 oz. 19 dwts. 18 grs.?

(8) If it cost £22 108. to cover a floor with carpet at 4s. 6d. per yard, what would be the expense at 5s. 6d.?

2

(9) 4 sq. yds. 2 ft. 16 in. cost of 14 guineas, how much

(10) If a drug cost £4 10s. per lb. of the old Apothecaries' weight, what is its value per lb. Avoirdupois?

b. (11) If 24 men can complete some work in 14 days, how long would 21 men take to do it?

(12) If a journey occupies 5 hours, when the average rate of travelling is 28 miles an hour, what was the rate of travelling when it required 14 hours?

(13) A well is dug in 13 days of 9 hours each, how many days would it have taken of 10 hours each?

(14) A man whose steps are 2 ft. 5 in. long takes 2480 in walking a certain distance; how many would be taken in the same distance by another who steps 2 ft. 7 in. each time?

(15) 6 horses require a certain quantity of corn in 15 days, how long would it last 4 horses?

(16) How long would a ship sailing 7 knots an hour take for a voyage which required 30 days for another vessel whose rate was 8 knots?

(17) A court was paved with 950 stones each 15 sq. ft., and is repaved with 836 stones of a uniform size. Find the size of each.

(18) At the rate of

5

13

of a mile per minute a train takes

34 hours to reach a station, how long would it take at the

(19) How many yards of carpet 2 ft. 4 in. wide, will be required for a room which has taken 42 yards of 2 ft. 3 in. wide?

(20) How much land worth £53 28. 6d. per acre will be equivalent to 14 ac. 3 rds. 20 po. worth £93 158. per acre?

(21) If 24 tons of ore yielded 1 ton 5 cwt. of metal, how much metal should 42 tons 15 cwt. yield?

(22) If the fare for 92 miles is 15s. 6d., what should be the distance on the same line for which £2 18. 4d. is charged?

(23) How much planking 11 inches broad, will be required for a floor which took 180 yds. 2 ft. 6 in. of deals 9 inches broad?

(24) If 5 tons 12 cwt. 2 qrs. cost £15 38. 4d., what would 12 tons 17 cwt. 16 lbs. cost?

(25) A farmer sells some corn for £15, the price being 478. 6d. per quarter, what would he have received for it at 50s. 8d. per quarter?

(26) If on an income of £971 128. a gentleman pays £20 48. 10d. tax, what would another pay on an income of £820 88.?

(27) How long will it take to fill a cistern of 165 gallons by a pipe which fills one of 120 gallons in 7' 16"?

(28) A block of granite weighing a ton is of the same size as a block of Yorkshire stone weighing 14 cwt. I qr. 4 lbs., what weight of granite will be equal in size to a ton weight of Yorkshire stone?

(29) The same being given as in the last question, what would be the solidity of a block of granite weighing the same as 3 cubic feet of Yorkshire stone?

(30) A ship has sailed 1800 miles in a fortnight, how long at the same rate will it take for a voyage of 5000 miles?

(31) The wheels of a carriage are 6 ft. 9 in., and 9 ft. 6 in. in circumference; how many times will the larger turn while the smaller turns 3762 times?

(32) If 12 dwts. 18 grs. cost £1 os. 6 d., how much might be bought for £56 128. 2 d.?

(33) A bankrupt owes to two creditors £285 and £323; the former receives out of the estate £65 18s. 11⁄2d., what should the latter receive?

(34) What is the value per cwt. when 2 lbs. 11 oz. Troy are worth 58. 34d.?

of a ship are worth £433 10s., what is the

of it?

(36) What will be the weight of 18 cub. ft. 432 in. of stone, of which 10 cub. ft. 864 in. weigh 12 cwt. 2 qrs. 7 lbs. ?

(37) 280 lbs. of flour will make 360 lbs. of bread, how many 4 lb. loaves can be made from a cwt. of flour?