$4.28 + $3.19 + $8.625 + $5.846 + $9.738 + $5.96 -† $7.50 $3.25?

28. What is the sum of $.27+ $.63 + $1.04 + $.50 + $.375+$1.50 + $.07 + $.42 + $.625 + $.875 + $3.27 + $5.94 + $.86 + $1.83 + $.06 + $.40 + $.125 + $1.33?

29. What is the sum of $85.76 + $77.25 + $.86 + $34.50 $7.38+ $50.50 + $7.13 + $.47 + $.68+ $28.17+$29.50 +$8.07+ $5.00 +$17.84 + $.03 + $5.28?

30. What is the sum of 58694 + 67867.9432 + 45879.837628697.4+38679.58432 + 27598.542 + 36789.754 +58767.5437 + 86427.58697 + 98003.79 + 28547.3298 +28475.9767 ?

31. What is the sum of 958679.4437 + 298673.925 + 5867329678.4593 + 486.7923 + 5878.6532 + 185.3 28.673486.79635+28.7659.836 + 45173.425 ?

*In adding yards, it will usually be well to consider every eleven yards as two rods. Such a course will, to a very great extent, avoid the use of fractions. The pupil should, however, bear in mind that half of a yard 1 ft. 6 in., and that when in any number there are 5 yards, and 1 ft. 6 in. besides, the value may be better expressed as 1 rod.

+ Obtaining the answer to the 40th example in the usual method, we shall find it to be 41 m. 7 fur. 39 rd. 5 yd. 2 ft. 11 in. This is correct, but it is not in the best form, for although there are not units enough expressed of any denomination to make one of the next higher, it equals 42 m. 0.fur. Ord. 0 yd. 1 ft. 5 in. Show the truth of this statement, and show also why the answer does not at first appear in the best form.

The answer to the 41st example will take the form at first of 35 rd

56. Another, and often a shorter, Method of reducing

Compound Numbers.

It will often be more convenient to make the reductions by adding enough of one number to another to give a sum equivalent to a unit of the next higher denomination. We wil take, for illustration, the example given in 51.

Explanation. Having found the sum of the farthings column to be equal to 3 d., we add the 3 d. with the column of pence, thus:

4 yd. 2 ft. 11 in, which should be changed, for the sake of simplicity, to 35 rd. 5 yd. 1 ft. 5 in. Show the equality of the two expressions, and the method by which the reduction can be made.

* In adding square yards, it will be of service to notice that 60 sq. yd. 2 sq. rd; that 903 sq. yd. = 3 sq. rd.; that 121 sq. yd. = sq. rd; and that of a sq. yd.: sq. ft. 36 sq. in. This will avoid any difficulty in the use of fractions.

3 d. and 8 d. are 11 d., and 3 d. are 1 s. 2 d., (by adding 1 of the 3 d. with the 11 d.,) and 8 d. are 1 s. 10 d., and 10 d. are 2 s. 8 d., (by adding 2 of one 10 d. with the other 10 d.,) and 9 d. are 3 s. 5 d., (by adding 3 of the 8 d. with the 9 d.,) and 8 d. are 4 s. 1 d., (by adding 4 of the 5 d. with the 8 d.,) and 11 d. are 5 s. 0 d.

Adding the 5 s. with the shillings column, we have 5 s. and 16 s. are £1 1 s., (by adding 4 of the 5 s. with 16 s.,) and 13 s. are £1 14 s., and 18 s. are £2 12 s., (by adding 2 of the 14 s. with 18 s.,) and 18 s. are £3 10 s., (by adding 2 of the 12 s. with 18 s.,) and 19 s. are £4 9 s., (by adding 1 of the 10 s. with the 19 s.,) and 12 s. are £5 1 s., (by adding 8 of the 9 s. with 12 s.,) and 15 s. are £5 16 s. = sum of shillings column.

(b.) We have mentioned the numbers added in order to secure clearness of explanation, but in practical work the results alone should be named.

Thus, beginning with the farthings, we have,

1 qr., 2 qr., 1 d., 1 d. 3 qr., 2 d., 2 d. 3 qr., 3 d. 1 qr. Write 1 qr.

3 d., 11 d., 1 s. 2 d., 1 s. 10 d., 2 s. 8 d., 3 s. 5 d., 4 s. 1 d., 5 s. O d. Write 0 d.

5 s., £1 1 s., 1 d. 14 s., £2 12 s., £3 10 s., £4 9 s., £5 1 s., £5 16 s. Write 16 s.

The pounds are added as before.

57. Common Method of adding Compound Numbers.

By the method of adding compound numbers which is commonly given, the entire sum of each column is found before reducing to higher denominations. This method, however, will, as a general thing, be found much less expeditious than either of the others.

It is illustrated in the following solution of the example given in the last article.

Explanation. — By adding the farthings column, we find that its sum is 13 qr., which, as 4 qr. = 1 d., must equal as many pence as there are times 4 in 13, which are three times, with a remainder of 1. Therefore, 13 qr. = 3 d. 1 qr.

Writing 1 as the farthings figure of the amount, we add the 3 d. with the figures of the pence column; this gives 60 d., which, as 12 d. = 1 s., are equal to as many shillings as there are times 12 in 60, which are 5 times. Therefore, 60 d. 5 s.

£1,

Writing 0 as the pence figure of the amount, we add the 5 s. with the figures of the shillings column. This gives 116 s., which, as 20 s. = are equal to as many pounds as there are times 20 in 116, which are 5 times, with a remainder of 16. Therefore, 116 s. == £5 16 s

Writing 16 s., we add the £5 with the figures of the pounds column. This gives £48, which, being the highest denomination, we write.

As all the denominations have now been added, the sum or amount must be £48 16 s. 1 qr.

58. Examples for Practice in the Methods of 50, 51, and 56.

1. What is the sum of £4 17 s. 11 d. 2 qr. + £84 13 s. 3 d. + £7 19 s. 8 d. 3 qr. + £16 18 s. 9 d. 1 qr. + £7 15 s. 1 d. +£18 16 s. 11 d. 2 qr.?

2. What is the sum of £1386 15 s. 6d. + £3576 18 s. 10 d. £463 19 s. 4 d. + £23 5 s. 8 d. + £648 4 s. 6 d. + £100 10 s. 3 d.?

3. What is the sum of 40 lb. 7 oz. 5 dwt. 6 gr. + 9 lb. 8 oz. 19 dwt. 22 gr. + 2 lb. 11 oz. 19 dwt. 23 gr. + 7 lb. 8 dwt. 19 gr. 11 oz. 6 dwt. + 3 lb. 1 oz. 15 gr. + 8 lb. 17 dwt. + 3 lb. 23 gr. 18 dwt. 7 gr. + 9 oz. 15 gr. + 7 lb. 3 oz. 13 dwt. 15 gr.?

4. What is the sum of 18 w. 4 da. 21 h. 37 m. 5 sec. -+37 w. 5 da. 16 h. 43 m. 57 sec.+ 19 w. 3 da. 14 h. 46 m. 38 sec. 19 w. 6 da. 23 h. 56 m. 27 sec. +43 w. 5 da. 2 h. 17 m. 38 sec. + 28 w. 1 da. 1 h. 5 m. 7 sec.?

5. What is the sum of 47 gal. 3 qt. 1 pt. 2 gi. + 37 gal. 1 qt. 1 pt. 1 gi. + 85 gal. 2 qt. 2 gi. + 25 gal. 2 qt. 1 pt. 3 gi. +54 gal. 2 qt. 1 pt. 3 gi. + 18 gal. 2 qt. 1 pt. 2 gi. + 37 gal. 3 qt. 0 pt. 1 gi. + 19 gal. 3 qt. 1 pt. 2 gi. + 43 gal. O qt. 0 pt. 3 gi.?

6. What is the sum of 15 yd. 2 qr. 2 na.

1 na.+ 27 yd. 3 qr. 3 na. + 42 yd. 1 qr. +

† 18 yd. 3 qr. 87 yd. 3 na. +

3 qr. 3 na. 26 yd. 1 qr. 1 na. + 57 yd. 3 qr. 2 na. + 42 yd.

7. What is the sum of 18 lb 63 53 29 5 gr. + 7 lb 83 73 19 18 gr. 4b 113 43 29 13 gr. + 25 lb 93 19 4gr. 113 19 + 2H 53 63 09 16 gr. + 5 lb

113 43 19 14 gr.?

8. What is the sum of 2 m. 7 fur. 28 rd. 4 yd. 1 ft. 3 in. + 6 m. 5 fur. 19 rd. 2 yd. 2 ft. 11 in. + 25 m. 4 fur. 37 rd. 5 yd.