The answer, then, is 2 tens, 6 units, 8 tenths, and 4 hundredths, or 26.84.

This may be proved in the same way that the preceding examples

were.

Questions on the above.

Which expresses the larger number, 7 tenths, or 6 tenths and 10 hundredths, and why? 2 units and 6 tenths, or 1 unit and 16 tenths? 5 tens and 1 unit, or 4 tens and 11 units? 6 tens, 2 units, and 7 tenths, or 5 tens, 11 units, 16 tenths, and 10 hundredths?

How then will the remainder, obtained by subtracting 35.86 from 62.7, compare with the remainder obtained by subtracting it from 5 tens, 11 units, 16 tenths, and 10 hundredths?

Explanation. — As we cannot take 5 dollars from 4 dollars, and as there are no tens or hundreds expressed in the minuend, we take 1 thousand from the 3 thousands, leaving 2 thousands; 1 thousand = 10 hundreds, and taking 1 of these hundreds to reduce to tens, we have 9 hundreds left. 1 hundred 10 tens, and taking 1 of these tens to reduce to units we have 9 tens left. 1 ten 10 units, which added to the 4 units in the units' place = 14 units. Now, by subtracting, we have

9 units from 14 units = 5 units.

6 tens from 9 tens 3 tens.

7 hundreds from 9 hundreds = 2 hundreds.

4 thousands cannot be taken from 2 thousands, therefore we take 1 ten-thousand from the 8 ten-thousands in the minuend, leaving 7 tenthousands. 1 ten-thousand 10 thousands, which added to the 2 thousands in the thousands' place 12 thousands.

4 thousands from 12 thousands = 8 thousands.

2 ten-thousands from 7 ten-thousands = 5 ten-thousands.

The remainder is, therefore, 58239 dollars.

Questions upon the above. How can it be shown that 3004 is equal to 2 thousands, 9 hundreds, 9 tens, and 14 units? That 83004 is equal to 7 ten-thousands, 12 thousands, 9 hundreds, 9 tens, and 14 units?

Explanation.

10 16 18 Remainder.

As we cannot subtract 23 gr. from 17 gr., we take 1 dwt. from the 15 dwt., which reduced to grains and added to the 17 gr. equals 41 gr.; 23 gr. from 41 gr. = 18 gr.

But as 18 dwt. cannot be taken from the 14 dwt. left in the minuend, we take 1 oz. from the 6 oz. and reduce it to pennyweights; 1 oz. = 20 dwt., which added to the 14 dwt. equal 34 dwt.; 18 dwt. from 34 dwt = 6 dwt.

As 7 oz. cannot be taken from the 5 oz. left in the minuend, we take 1 lb. from the 14 lb. and reduce it to ounces; 1 lb. = 12 oz., which added to the 5 oz. equal 17 oz.; 7 oz. from 17 oz. = 10 oz. 8 lb. from 13 lb. 5 lb.

The answer is, therefore, 5 lb. 10 oz. 6 dwt. 8 gr., which may be proved as before.

Questions. Which expresses the greater quantity, 15 dwt. 17 gr., or 14 dwt. 41 gr., and why? 6 oz. 14 dwt., or 5 oz. 34 dwt.? 14 lb. 5 oz., or 13 lb. 17 oz. ? 14 lb. 6 oz. 15 dwt. 17 gr., or 13 lb. 17 oz. 34 dwt.

41 gr.?

How would the remainder obtained by subtracting 8 lb. 7 oz. 18 dwt. 23 gr. from 14 lb. 6 oz. 15 dwt. 17 gr. compare with that obtained by subtracting it from 13 lb. 17 oz. 34 dwt. 41 gr.?

Second Example. A farmer took 8 bu. 3 pk. 5 qt. of corn from a bin containing 17 bushels. How many bushels, pecks, and quarts remained?

Reasoning Process.· If the bin contained 18 bu., and he took out 8 bu. 3 pk. 5 qt., there would remain the difference between 17 bu. and 8 bu. 3 pk. 5 qt. This shows that 17 bu. is the minuend, and 8 bu. 3 pk 5 qt. the subtrahend.

Explanation. As there are no pecks or quarts expressed in the minuend, we take 1 bu. from the 17 bu. and reduce it to lower denominations. 1 bu. = 3 pk. 8 qt. Therefore 17 bu. = 16 bu. 3 pk. 8 qt. The subtraction can now be performed as before.

(a.) In examples involving fractional denominations, it will usually be more convenient to make all the reductions and changes in the minuend before beginning to subtract, as in the following:

Third Example. What is the difference between 8 rd. 3 yd. 1 ft. 4 in. and 2 rd. 4 yd. 2 ft. 5 in.?

WRITTEN WORK.

7 8 8 2 10 Minuend, changed in form.

---

Remainder.

Explanation. Since there are more yards, feet, and inches expressed in the subtrahend than in the minuend, we will take 1 rd. from the 8 rd. and reduce it to lower denominations. 1 rd. = 5 yd.

5 yd. 1 ft. 6 in., Therefore

which added to the 3 yd. 1 ft. 4 in. equal 8 yd. 2 ft. 10 in. 8 rd. 3 yd. 1 ft. 4 in.- 2 rd. 4 yd. 2 ft. 5 in. 7 rd. 8 yd. 2 ft. 10 in. 2 rd. 4 yd. 2 ft. 5 in.

NOTE.

Had not the 1 rod been reduced to yards, and the yard to feet and inches, before commencing the subtraction, the answer would have taken the form of 5 rd. 3 yd. 1 ft. 11 in., from which, by reducing the yard to feet and inches, we should get 5 rd. 3 yd. 2 ft. 17 in. = 5 rd. 4 yd. 0 ft. 5 in. : = the answer obtained directly by first method.

Х

What is the value of

824.57 347.28?

X 13. £17 6 s. 8 d. 1 qr. £13 17 s. 3 d. 2 qr.?

14. 25 cwt. 1 qr. 13 lb. 10 oz. 7 dr. 13 cwt. 2 qr. 15 lb 3 oz. 11 dr.?

19.

£487 6 s. 0 d. 1 qr. £236 11 s. 8 d. 3 qr.?

20. 27° 24' 47"

21. 38 m. 4 fur.

22. 54 m. 6 fur. 3 rd. 8 in.

28 rd. 5 yd. 1 ft. 5 in.?

2 ft. 10 in.?

48 m. 3 fur. 3 rd. 2 yd.

23. 3 R. 14 sq. rd. 7 sq. yd. 2 sq. ft. 19 sq. in. - 23 sq. rd. 11 sq. yd. 5 sq. ft. 138 sq. in. ?

24. 15 rd. 5 yd. 2 ft. 11 in.

16 rd. 1 ft. 4 in. ?

68. The Changed Minuend not usually written. (a.) The changed form of the minuend has been written in the preceding examples to insure that the nature of the reductions and changes shall be understood by the pupil. It is not, however, customary to write it. The full explanation is the same whether it is written or omitted; but when it is omitted, and every step of the process is understood and mastered, abbreviated explanations like the following may be adopted :

WRITTEN WORK OF FIRST EXAMPLE UNDER 65.

First Abbreviated Explanation.

6 hundredths cannot be taken from O hundredths, but 1 tenth = 10 hundredths, and 6 hundredths from 10 hundredths leave 4 hundredths.

8 tenths cannot be taken from 6 tenths, but one unit equal 10 tenths, and 6 tenths added 16 tenths. 8 tenths from 16 tenths = 8 tenths. 5 units cannot be taken from 1 unit, but 1 ten = added 11 units. 5 units from 11 units leave 6 units.

3 tens from 5 tens leave 2 tens.

Hence the answer is 26.84.

10 units, and 1 unit

Second Abbreviated Explanation.—6 hundredths from 10 hundredths leave 4 hundredths; 8 tenths from 16 tenths leave 8 tenths; 5 units from 11 units leave 6 units; 3 tens from 5 tens leave 2 tens.

This gives 26.84 for an answer, as before.

(b.) All explanations should finally be dropped, and only results named, thus: 4 hundredths, 8 tenths, 6 units, 2 tens; giving for the answer 26.84, as before.

WRITTEN WORK OF FIRST EXAMPLE Under 66.

5 10 16 18 Remainder.

First Abbreviated Explanation.—23 gr. cannot be subtracted from 17 gr.; but 1 dwt. = 24 gr., and 17 gr. added are 41 gr. 23 gr. from 41 gr.

= 18 gr.

= 16 dwt.

18 dwt. cannot be taken from 14 dwt.; but 1 oz. 20 dwt., and 14 dwt. added = 34 dwt. 18 dwt. from 34 dwt. 7 oz. cannot be taken from 5 oz.; but 1 lb.

are 17 oz. 7 oz. from 17 oz. 10 oz.

12 oz., and 5 oz. added

Hence the remainder is 5 lb. 10 oz. 16 dwt. 18 gr.

69. Subtrahend Figure may be increased instead of diminishing Minuend.

It is obvious that the result would be the same, if, instead of considering the minuend figure of a denomination from which a reduction has been made to be one less, we should consider the corresponding subtrahend figure to be one greater. In the former case, we subtract 1 (on account of the reduced unit) before subtracting the subtrahend figure, while in the latter we add 1 to the subtrahend figure, and subtract both together.