3.416 is the same as the last, with 3 before the fraction. 171-82753. The whole number 171 is kept before the fraction, and is not affected by the reduction. As there is only one non-recurring figure, the 8 only is subtractive from 82753 for the numerator. Four recurring figures give four nines in the denominator, with one cypher for the non-recurring figure.

EXERCISES.-XLVI.

1. Reduce the following circulators to fractions: 4, 04, 4·04, 4-004, 4-404, 4.404.

2. Express as vulgar fractions: 17-267, 17.267, 17·267, 17-267, and 17.0287.

3. Reduce to vulgar fractions: 0245, 17·0245, 1·70245, 17-0245, 17.0245.

4. Express as vulgar fractions: 018, 012, ·7854, ·112, 1·769230, 29.4963, and 54.

5. Reduce to vulgar fractions: 2·83, 2·7916, 5·83, 7·4583, ·40972, 17-275, 008497133.

6. Reduce to vulgar fractions: 1.612, 17·5841, 3·1416, ·7854, •000215i.

7. Reduce the following recurring decimals to common fractions : 1-454545, etc.; 45·11, etc.; 17.25, and 3.79999, etc.

8. What is the difference between 4.59 and 4.6?

EXERCISES.-XLVII.

1. Subtract 2·32 of an hour from 73 of a week.

2. From 254 of a yard subtract 4 in.

3. Take 4s. 6 5d. from 5.4175 shillings.

4. Subtract 01245 qrs. from 625 bush.

5. From 4.30819 subtract 2·3082; and from 2.79 of a £ subtract 12.8 of a shilling.

6. Subtract 1 of 2 miles from of a league.

7. What difference is there between of a guinea and of a £?

8. Add together 245 of a year (365 days), 1·724 of a month (30 days), 21.74 days, and 45 hours.

9. Find the sum of 4.125 tons, 94 cwt., and 7·1 lbs.

10. Find the value of 38 of a bush. −1·27 of a pk., and 1.35 of gal.+2.25 of a qt.

11. Give the values of 4.25 hours -3.75 of 45 min., and ‘25 of 1° 25′ 35′′ +27 of 20° 20′ 20′′.

12. Express 17-4219 of a bush., and 17-2184 of a barrel, as simple quantities.

13. What are the values of 71428 and 071428 of a hhd. of wine?

14. What is the sum of '09 of £1, 3s. 2d., and 51 of £19?

15. Find the value of 375 of a guinea + ·54 of 8s. 3d. + ·027 of £2, 158.

16. Add together 5.03 lbs. and 063 of a cwt., and give the answer in ounces.

17. Subtract 3:43 pence from 6 of £1, 5s. 9d.

18. Add together 2:45 of a crown, and 047 of a guinea, and give the answer in pence, and the decimal fraction of a penny.

19. Subtract 63 of a gal. from 147 of a pk., and give the answer in pints, and the decimal fraction of a pint.

20. Add together 1.36 of a furlong and 7 32 of a yard, and give the answer in feet, and the decimal fraction of a foot.

21. Add together 0.0234 of a pound sterling and 0·0234 of a guinea; express your answer in pence.

22. Find the sum of £775, 824s., and ·305 of a crown, and express it in decimals of a guinea.

23. Find the sum and difference of the following quantities: 258 of a pound, and 365 of a guinea; 325 of a shilling, and 39 of 10d.

PRACTICE AND RULE OF THREE.

THE following exercises require a slight knowledge of fractions; they are here given as exercises to test the pupil's work in Single and Double Rule of Three, Practice, and Fractions.

PRACTICE WITH FRACTIONS.

Required the price of 17283 articles at £3, 9s. 114d. each. £ S. d. £ S. d.

8. d.

4 0

5 0

of £1.

The unit, with which the sum is commenced, is £1, i.e., 5s. is the of a £; hence the first quantity is pounds, and whatever is over, after being brought into shillings and pence, and divided, gives the shillings and pence. It is always most convenient to use the aliquot parts of as high a denomination as we can; but this in Practice generally resolves into using the next higher denomination. For instance, pounds of money, tons, cwts., etc., form generally multipliers; shillings are taken as the aliquot parts of a £, pence of shillings, etc.; quarters as the aliquot parts of hundredweights, hundredweights of tons, etc.

In the example worked out, the 1728 we call £17283, or £1728, 13s. 4d. for convenience, as it greatly facilitates the working. When we come to take the aliquot part, 10d., which is the of 5 shillings, instead of obtaining farthings, it is more convenient to keep the results in fractions of a penny, viz., . The second is thus obtained mechanically: after dividing the pence, the 10 goes 0 and 6 over; we say 6×3= 18 and 2 are 20 for the numerator; for the denominator 10 x 3=30; thus is obtained =}. The is found thus: having to divide by 4, simply say 4×3=12, and the fraction becomes. The sum is added up in the usual way.

A coal merchant sold 20 tons 16 cwt. 3 qrs. 22 lbs. of coals at 25 shillings per ton; find the selling price of the whole.

=Cost of 20 tons 16 cwt. 3 qrs. 22 lbs. at 25s. per ton.

Aliquot parts are here taken of the hundredweights, quarters, and pounds, and not of the money. The fractions are added together in the ordinary manner; but the student will find it an easy method to say and make ,=, which, with the

below, make. Now, and the give 1, which are soon added to. This simple mode of adding fractions will often save a very large number of figures.

Find the price of 1794 125 articles at £1, 15s. 11 d. each. It is seldom or never wise to attempt Practice sums by decimals, for we can hardly avoid using the aliquot parts and 1, which give circulating decimals. Sometimes the fractions of a penny may be in decimals; but it is unwise to keep the shillings and pence as decimals of a pound. Hence in the annexed sum call the 125 the of a £, or half-a-crown.

1. Find the cost of 56 cwt. 3 qrs. 17 lbs. at £5, 9s. 6d. per cwt. 2. Find the price of 2 cwt. 3 qrs. 16 lbs. at £1, 8s. 6d. per cwt. 3. How much will the railway charges on 10 packages, each weighing 5 cwt. 3 qrs. 22 lbs., come to at 12s. 6d. per ton?

4. What is the value of 47 oz. 7 dwt. 9 grs. of silver at 5s. 10d. per ounce ?

5. Required the value of 21 yds. 2 ft. 11 in. at 2s. 74d. per yard. 6. Find the value of 7031 articles at 14s. 64d., and of 6754 at £2, 1s. 5d.

7. Find the value of 37 cwt. 3 qrs. 17 lbs. of coffee at £8, 17s. 4d. per cwt. by Practice.

8. What is the worth of 37 cwt. 2 qrs. 14 lbs. at £7, 10s. 9d., per cwt.?

9. Find the value of 2036 articles at £2, 13s. 8d. each.

10. Find by Practice the value of 22 qrs. 4 bush. 3 pks. of wheat at £2, 7s. 94d. per quarter.

11. Divide 28 tons 4 cwt. 3 qrs. into 36 equal portions, and find the value of one of them at £7, 10s. 8d. per cwt.

12. Find the cost of 36 m. 3 fur. and 22 yds. of telegraph wire at £14, 13s. 4d. per mile.

13. Find the price of 1 cwt. 3 qrs. 5 lbs. at £2, 4s. 3 d. per cwt. 14. Find the dividend on £2870, 10s. at 14s. 3d. in the pound.

15. Find the value of 5 cwt. 3 qrs. 19 lbs. at £2, 15s. per cwt? 16. What is the rent of 23 ac. 3 ro. 16 po. at 24 guineas per acre. 17. Find the value of 45 ac. 3 ro. 20 per. at £111, 11s. 4d. per

acre.

18. What is the value of 18 gals. 3 qts. 14 pts. at 17s. 10дd. per gallon?

19. Find the rent of 85 ac. of land for 3 yrs. 10 mo. 10 dys. (28 dys. to a month), at a yearly rental of £3, 10s. 6d. per acre.

20. A gentleman has an income of £31,286, 17s. 6d., but his taxes, cost of collection, steward's expenses, etc., are 4s. 84d. in the £; find his net annnal income.

21. Find the cost of 56 boxes of sugar, each containing 3 cwt. 3 qrs. 21 lbs., at 18s. 6d. per cwt.

22. Cheese costs £3, 12s. 8d. per cwt.; find the cost of 1 ton 18 cwt. 2 qrs. 21 lbs.

23. Bought 156 yds. 3 qrs. 3 nls. of velvet at 15s. 2d. per yard; find the cost of the whole.

24. Bought 231 boxes of raisins, each containing 84 lbs.; find the price of the whole at 39s. 3d. per cwt.

25. A cargo of 3241 boxes of American bacon arrived in port, each box containing 7 cwt. 3 qrs. 144 lbs.; find the value of the bacon at 23s. 9d. per cwt.

RULE OF THREE, SIMPLE.

In introducing the following examples in Simple Rule of Three there is little to add to the remarks already made on Proportion. The problems here are a little more difficult than those previously given, as the majority require an elementary knowledge of fractions.

It is a difficult matter to convey a clear conception of ratios by definition. We compare the relative size of magnitudes of the same class, without distingushing their absolute size; for instance A may have £400 and B £500, these are the absolute sums of money they possess; but A relatively to B has £4 to B's 5, or A's money is to B's in the ratio of 4: 5.

The ratio between two quantities is the mutiple or part one is of the other.

The following is a very useful rule for Proportion, especially when the three terms are of the same name, as money:

(1.) Take the terms as they occur in the question, and write them down on the left hand side of the slate, one under the other, as seen below.

(2.) Link like terms together, and the odd one is of the same kind as the answer, and is the third term.