E. It is often the case that a figure in the subtrahend represents a greater value than the corresponding figure of the minuend. In that case we take one of a higher denomination in the minuend, reduce it to the required denomination, adding its value to the value of the figure already expressed,' and subtract from the sum thus obtained the value of the subtrahend figure.

Suppose, for example, that we wish to find the difference between 62.7 and 35.86.

Explanation of process. As there are no hundredths expressed in the minuend, we reduce one of the 7 tenths to hundredths, leaving 6 tenths. 1 tenth = 10 hundredths, from which subtracting 6 hundredths, leaves a remainder of 4 hundredths. As 8 tenths cannot be subtracted from 6 tenths, we reduce one of the 2 units to tenths, leaving 1 unit. 1 unit = 10 tenths, which added to the 6 tenths left in the tenths' place, equal 16 tenths; 8 tenths from 16 tenths 8 tenths. As 5 units cannot be taken from the 1 unit left in the units' place, we reduce one of the 6 tens to units, leaving 5 tens. 1 ten = 10 units, which added to 1 unit equal 11 units; 5 units from 11 units 6 units. 3 tens from 5 tens leave 2 tens.

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 ?

Example 2. Find the difference between 14 lb. 6 oz. 5 dwt. 7 gr. and 8 lb. 7 oz. 18 dwt. 23 gr.

13, 17, 24, 31. Minuend changed in form.

gr.

Explanation. As we cannot subtract 23 gr. from 7 gr., we take 1 dwt. from the 5 dwt., which reduced to grains and added to the 7 gr. equals 31 from 31 gr. gr.; 23 8 gr. But as 18 dwt. cannot be taken from the 4 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 4 dwt. equal 24 dwt.; 18 dwt. from 24 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, 5 dwt. 7 gr., or 4 dwt. 31 gr., and why? 6 oz. 4 dwt., or 5 oz. 24 dwt.? 14 lb. 5 oz., or 13 lb. 17 oz.? 14 lb. 6 oz. 5 dwt. 7 gr., or 13 lb. 17 oz. 24 dwt. 31 gr. ?

How would the remainder obtained by subtracting 8 lb. 7 oz. 18 dwt. 23 gr. from 14 lb. 6 oz. 5 dwt. 7 gr., compare with that obtained by subtracting it from 13 lb. 17 oz. 24 dwt. 31 gr.? For another example take the following: How many are 83004 dollars minus 24765 dollars?

7, 12, 9, 9, 14. $8 3 0 0 4.

$2 4 7 6 5.

Minuend, changed in form.

- Minuend. = Subtrahend.

=

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; subtracting, we have

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 ten-thousands. 1 ten-thousand = ten thousands, which added to the 2 thousands in the thousands' place thousands.

4 thousands from 12 thousands 8 thousands.

=

2 ten-thousands from 7 ten-thousands 5 ten-thousands. The remainder is, therefore, 58239 dollars.

: 12

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?

Example 4. A farmer took 8 bu. 3 pk. 5 qt. of corn from a bin containing 17 bushels. How many bushels, pecks and quarts remained?

The usual reasoning process shows that 17 bu. is the minuend, and 8 bu. 3 pk. 5 qt. the subtrahend.

16, 3, 8. Minuend, changed in form.

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

= 16

16. What is the value of 7, 5s. 3d. 2qr.- 1£, 8s. 9d. 1qr.? 17. What is the value of 18£, 14s. 2d. 9£, 18s. 7d. 2qr.? 18. What is the value of 27 bu. 2 pk. 2 qt. 18 bu. 3 pk.

3 qt.?

19. What is the value of 83 yd. 2 qr. 1 na.

2 na.?

47 yd. 3 qr.

20. What is the value of 8 cwt. 2 qr. 15 lb. 7 oz. 3 dr. 2 cwt. 3 qr. 24 lb. 3 oz. 9 dr.?

21. What is the value of 37 gal. 2 qt. 1 pt. 2 gi.

3 qt. 1 pt. 3 gi.?

12 gal.

22. What is the value of 37 lb. 2 dwt. 4 lb. 7 oz. 5 dwt. 13 gr.?

23. What is the value of 24 lb, 335, 113, 43, 29,

F. It is very frequently convenient to make all the reductions and changes in the minuend before beginning to subtract, as in the following example :

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

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. 5 yd. 1 ft. 6 in., which added to the 3 yd. 1 ft.

51 yd.

=

rd.
4 in. equal 8 yd. 2 ft. 10 in.
2 rd. 4 yd. 2 ft. 5 in.

yd. 2 ft. 5 in.

=

1

The subtraction is performed in the usual method.

In performing the following examples, make the reductions

in the same manner.

1. How many are 1400.3 - 207.5?

2. How many are 16573 - 4.878?

3. How many are 5700.001

4. How many are 217.406

1847.37? 8.909?

5. How many are 18£, 13s. 4d. 1qr.

4£, 15s. 7d. 3qr.?

6. 3 R. 14 sq. rd. 7 sq. yd. 2 sq. ft. 19 sq. in 23 sq. rd. 11 sq. yd. 5 sq. ft. 238 sq. in.?

3

7. 14, 73, 5 gr. 6 lb, 33, 2, 17 gr. ?

8. 15 rd. 5 yd. 2 ft. 11 in.

9. 27 bu. 8 bu. 3 pk. 7 qt.?

16 rd. 1 ft. 4 in. ?

gr. ?

13 T. 19 cwt.

11. 15 T. 15 cwt. 2 qr. 14 lb. 12 oz. 8 dr.

qr. 27 lb. 15 oz. 15 dr.?

12. 27 sq. rd. 5 sq. ft. 17 sq. in.-26 sq. rd. 30 sq. yd. 7 sq. ft. 53 sq. in. ?

13. 6 fur. 8 rd. 1 yd. 7 ft. 1 in. — 1 fur. 8 rd. 4 yd. 2 ft. 11 in. ?

14. 18 yd. 1 na.

15. 6 m. 1 ft.

14 yd. 2 qr. 3 na. ?

- 5 m. 7 fur. 39 rd. 5 yd. 1 ft. 2 in. ?

G. Examples to be performed without writing out the changed minuend. The reductions are to be made on the same principle as before, but they are to be retained in the memory instead of being written.

1. Subtract 863.72 from 947.815.
2. Subtract 728. from 4000.

3. Subtract 5.9318 from 47.

4. Subtract 9432. from 17865.3.

5. Subtract .000765 from .420006.

6. Subtract 5 lb. 8 oz. 14 dr. from 3 qr. 1 lb. 11 oz. 7 dr.