EXERCISES.-XII.

1. Add together 3 tons 5 cwt. 2 qrs. 10 lbs.; 6 tons 2 cwt. qr. 9 lbs.; 18 tons 13 cwt. 3 qrs. 20 lbs.; 16 tons 16 cwt. 1 qr. 10 lbs.; 5 tons 19 cwt. 3 qrs. 23 lbs.; and 23 tons 4 cwt. 2 qrs. 7 lbs.

2. Paid the railway carriage of the following goods: Sugar, 12 cwt. 2 qrs. 10 lbs.; tea, 5 cwt. 2 qrs.; currants, 1 cwt. 2 qrs. 14 lbs. ; rice, 10 cwt. 1 qr. 10 lbs.; and candles, 2 qrs. 21 lbs.; what was the total weight?

3. Received a bale of goods weighing 3 tons 3 qrs. 16 lbs., and another weighing 5 tons 16 cwt. 1 qr. 11 lbs.; how much was one heavier than the other?

4. Ordered 8 chests of tea, each weighing 1 cwt. 2 qrs. 16 lbs. ; what is the weight of the whole ?

5. Nine customers have offered to purchase all my lump sugar, which weighs 3 tons 16 cwt. 3 qrs. 17 lbs.; how much must Í charge to each?

6. I purchase tallow : 1 ton 13 cwt. 1 qr. from Bristol; 3 tons 24 lbs. from London; and 6 tons 1 qr. 16 lbs. from Liverpool. After selling one-half, how much have I left?

7. Divide £13,043, 6s. 44d. by 679.

8. Divide £2726, 6s. 84d. by 43.

9. Divide £560,127, 17s. 9 d. by 45.

10. Divide £226,125,587, 3s. 5d. by 602.

11. A piece of iron is 12 ft. 6 in. long; from it are cut 73 bolts each 1 in. long; how long a piece is left?

12. A man rows a mile in 10 min. 30 sec.; how long will he take to row 27 miles at the same rate?

13. A man rows 51 miles in 23 hrs. 5 min. 30 sec.; how long is he going a mile?

14. To call at six farms it takes a postman the following times; find how long he is delivering his letters at those places: (1) 10 min. 4 sec.; (2) 15 min. 7 sec.; (3) 2 min. 38 sec.; (4) 4 min. 9 sec.; (5) 9 min. 25 sec.; (6) 2 min. 2 sec.

15. A postman has to deliver 150 letters; it takes him on the average 35 seconds to deliver each; find time of delivery.

16. If 523 letters were delivered in 3 hrs. 29 min. 12 sec., how long a time was occupied by each on the average?

17. A druggist had delivered to him the following quantities; find the total: (1) 17 lbs. 11 oz. 7 drs. 2 sc.; (2) 19 lbs. 5 drs. 1 sc.; (3) 27 lbs. 8 oz. 6 drs.; (4) 9 lbs. 4 oz. 4 drs. 1 sc.; (5) 10 oz. 7 drs. 2 sc.; (6) 20 lbs. 10 oz. 5 drs. 2 sc.

18. Out of a pound of gamboge is first sold 1 oz. 7 drs. 2 sc., then 2 oz. 5 drs. 1 sc.; how much remains?

19. An apothecary divides 2 oz. 2 drs. 2 sc. of magnesia into 28 powders; what was the weight of each?

20. A box of strawberries holds 72 lbs., they are placed into 96 small round baskets; how much does each basket hold?

21. A quantity of fish (275) is delivered to a salesman; each fish weighs on the average 2 lbs. 3 oz. 8 drs.; find the total weight.

22. Each pound of butter is 1 oz. 5 drs. short; how much is wanting in the quantity delivered for a hundredweight?

23. A druggist places into a 12 ounce bottle the following quantities: 37 drs. 59 min.; 1 oz. 4 drs. 10 min.; 3 oz. 5 drs. 11 min.; 1 oz. 1 dr. 12 min.; how much water is required to fill up the bottle?

24. Out of a can of milk containing 4 gallons the following quantities were sold; how much is left in the can: 1 gal. 2 qts. 1 pt.; 1 qt. 2 gills; 1 pt. 3 gills; 1 qt. 1 pt. 2 gills; 5 gills?

25. A collector is sent on a journey to call for various sums amounting to £1000; he returns home, having received (1) £200, 16s. 8d.; (2) £98, 3s. 4d.; (3) £105, 12s. 6d. ; (4) £71, 16s. 4d.; (5) £1, 7s. 6d.; (6) £191, 4s. 11d.; (7) £49, 2s. 11d.; (8) £159, 4s.; how much remains unpaid?

26. A gardener delivers to a market woman the following quantities of potatoes; find the total she has received: 1 cwt. 3 qrs. 7 lbs.; 9 cwt. 2 qrs. 14 lbs. ; 21 stone; 4 cwt. 1 qr. 11 lbs. ; 14 cwt. 7 lbs.; and 5 cwt. 2 qrs.

27. A woman sold out of 5 bushels of peas the following quantities; how much has she still to sell: 3 pks. 1 gal. 2 qts.; 4 pks.; 1 gal. 1 qt. 1 pt.; 1 bus. 1 pk. 1 pt.; 2 pks. 1 gal.?

28. Out of 5 tons 16 cwt. 3 qrs. 12 lbs. 8 oz. of salt, there have been sold 4 tons 17 cwt. 2 qrs. 14 lbs.; how much is left unsold? 29. Multiply 13 tons 5 cwt. 3 qrs. 11 lbs. by 24.

30. Multiply 125 ac. 3 ro. 27 per. by 27.

31. A railway truck and its load weigh 10 tons 3 cwt., the truck weighs 3 tons 14 cwt. 2 qrs. 9 lbs.; what is the weight of the load?

32. Suppose a tradesman owes to five different manufacturers the following sums, find how much money he will require to pay his debts: £486, 17s. 11d.; £900; £27, 13s. 4d.; £208, 6s. 2d.; £1, 17s. 11d.

33. A grocer pays for sugar, £75, 4s. 8d.; for rice, £9, 3s. 7d.; for tea, £208, 9s. 11d.; for coffee, £81, 17s. 2d.; for biscuits, £2, 17s. 4d.; for butter, eggs, and bacon, £200, 9s. 6d.; and for carriage, £3, 10s. 4d.; find the total sum paid.

34. A roll of cloth contains 75 yds. 3 qrs. 2 na.; there was cut from it during the week 56 yds. 2 qrs. 3 na.; how much remains? 35. A man on a walking tour goes 15 m. 7 fur. 5 yds. the first day; on the second, 10 m. 6 fur. 4 yds.; on the third, 19 m. 200 yds.; and on the fourth and fifth, 7 m. 3 fur. 8 yds., and 12 m. 80 yds.; how far has he gone in the five days?

36. Multiply 18 tons 3 cwt. 2 qrs. 9 oz. by 23. 37. Divide 69 m. 7 fur. 39 po. 2 ft. by 492.

38. If 1 cwt. of cheese cost £2, 4s. 64d., what will 324 cwt. cost? 39. Bought 21 cwt. of tobacco for £318 10s.; what is the price of 1 lb. ?

40. If a farm of 175 acres be let for £502, 7s. 8d., how much is that per acre?

41. A ton of cheese was bought for £79, 6s. 8d.; how much per lb. was it sold for to gain d. per lb.?

42. How many parcels of coffee can be made out of 11 tons 10 cwt. 3 qrs., with 3 lbs. 8 oz. in each parcel?

43. A field contains 39 ac. 2 ro. 35 po.; find how many allotments of 3 ro. 35 po. it can be divided into.

REDUCTION.

Ir has already been shown that we, in our system of Weights and Measures, have many units. The units of any quantity are termed denominations. In Time, for instance, there are seconds, minutes, hours, etc.; in Long Measure, inches, feet, yards, poles, etc., which are the various denominations. The art of changing a quantity from one of these units to other units is termed Reduction. The object of Reduction is to express quantities in any given units, so as to be able to compare given quantities one with the other. All reductions are made by means of the simple rules of Multiplication and Division. By the use of the rules already established, we shall be enabled to change compound quantities from one or more denominations to others. There are two cases- -(1.) To express a simple or compound quantity, such as 14 days or 21 yds. 2 ft. 9 in., in terms of their lowest submultiples-seconds or inches. (2.) Or to convert simple quantities, such as farthings and pints, into higher denominations—pounds and gallons.

RULES FOR REDUCTION.-The two cases of Reduction are: (1.) Reduction Ascending. This is when quantities have to be reduced from a lower to a higher denomination, and is effected by Division. Divide the quantities by the numbers which connect the several denominations in order, beginning with the lowest unit or submultiple, placing the remainders on the right hand. When the division is completed, place the last quotient with the several remainders beneath the sum as the final result.

(2.) Reduction Descending.-This is when quantities have to

be reduced from a higher to a lower denomination, and is effected by Multiplication. Multiply the highest denomination by the submultiple or unit which connects it with the next lower denomination, adding to the product the quantity in the question corresponding to that denomination. Repeat the process for each succeeding denomination, till the one sought is reached.

Convert £27, 18s. 111d. into farthings.

27 18 11

20

The problem here is to change the compound number, £27, 18s. 11 d., into another of only one denomination, farthings. First we multiply the pounds by 20 (shillings in a pound), to convert them into shillings, at the £ s. d. same time adding the eighteen shillings in the top line. Next, we take the 558 shillings thus found, and change them into pence, by multiplying by 12 (pence in one shilling), and adding the 11d. Thirdly, the 6707 pence thus found, are treated in the same way, multiplying by 4, and adding the one farthing, to bring the whole into farthings.

Change 1734281 farthings into pounds, shillings, and pence.

558

12

6707

26829 Ans.

First, the 1734281 farthings are turned into pence by

dividing by 4, for the simple reason that there must be as many pence

4)1734281

12) 433570

2.0) 3613,0 10

£1806 10 10 Ans.

in the given number as there are fours of farthings. Second, the pence 433570 are divided by 12, because every 12 pence will make one shilling. Thirdly, the 36130 shillings are divided by 20 to bring them into pounds, because every 20 shillings make a pound. The answer is obtained in the form £1806, 10s. 101d., where the remainders are brought down, and follow the pounds in their proper order.

One or more illustrations will be given of these two pro

cesses.

Change 44° 3′ 33′′ of arc into seconds, and 417381 seconds of time into days.

44° 3′ 33′′

Seconds.

6.0)41738,1

6,0) 695,6 21

60

2643

60

6) 28 3

158613

4 dys. 19 hrs. 56 min. 21 sec.

The 44° 3' 33" are sexigesimals, so we simply multiply by the two sixties, adding in the 3' and 33′′ as required.

We may notice here that minutes and seconds of arc are always indicated by ' and ", while minutes and seconds of time are marked min. sec. The French grades, minutes, and seconds are marked thus: 288 14' 55". It is customary on architectural and mechanical drawings to mark feet and inches thus: 4' 5". No confusion can possibly arise in using these signs to denote widely distinct quantities, as their position always accurately determines what is meant.

In the right-hand problem, first, the seconds are changed into minutes by dividing by 60, since 60 seconds make one minute.

Second, the minutes (6956) are converted into hours by dividing by 60, since 60 minutes make one hour.

Third, the hours are turned into days by dividing by 24, as 24 hours make one day. When dividing by 4 a remainder 3 is left; then dividing by 6 a remainder 4 is left: that four is 4 times 4, or 16, which with the 3 gives the 19 hours in the answer. PROOF.-Every example of Reduction, where a compound quantity has been reduced to its lowest unit, and then the simple quantity returned to its highest denomination, is a proof that the work has been correctly performed. Take as illustration of proof the following example where it is clearly evident that the one process is a guarantee of the correctness of the other.

Express 1748151 farthings as pounds, and prove that your result is correct.

Pounds to Farthings.

£ 8. d. 1820 19 9

20

36419 shillings.

12

437037 pence.

1748151 farthings.

EXERCISES.-XIII.

1. Reduce the following sums of money to farthings: £21, 18s. 114d.; £200, 9s. 6d.; £481, Os. 10ąd.; £921, 10s. 24d.; £7, 13s. 11d.; £1486, 19s. 24d.