7.

3 lb. 11 oz. 15 dwt. 18 gr.

1 lb. 8 oz. 13 dwt. 4

gr. ? 6 T. 17 cwt. 2 qr. 26 lb. 13 oz. 11 dr. 2 T. 6 cwt. 15 lb. 8 oz. 3 dr.?

8.

9. 16 83 53 20 18 gr. 5 tb 43 23 196 gr.?

10. 1757 gal. 3 qt. 1 pt. 3 gi. - 1323 gal. 1 qt. 2 gi.? 11. 18 rd. 4 yd. 2 ft. 11 in.

- 6 rd. 2 yd. 1 ft. 5 in.?

65. Simple Subtraction. Method when Reductions are

necessary.

When, as is often the case, a figure in the subtrahend represents a greater value than the corresponding figure of the minuend, we take one of a higher denomination in the minuend, reduce it to the required denomination, add its value to the value of the figure already expressed, and subtract the value of the subtrahend figure from the sum thus obtained. First Example. What is the difference between 62.7

Minuend, changed in form.
Minuend.

Subtrahend.

Difference.

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

8 tens from 5 tens leave 2 tens.

6 units

The answer, then, is 2 tens, 6 units, 8 tenths, and 4 nundredths, 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?

9 Remainder.

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 Now, by subtracting, we have

= 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 teus, and 14 units?

Explanation. 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 denomina. tions. 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 2 10 Minuend, changed in form.

Explanation.

Minuend.

Subtrahend.

Remainder.

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.,

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

8 rd. 3 yd. 1 ft. 4 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.

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

14.

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

qr.

13 cwt. 2 qr. 15 lb

15. 8 lb. 4 oz. 17 dwt. 13 gr. - 2 lb. 8 oz. 19 dwt. 20 gr.?

19.

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

20. 27° 24′ 47′′ 19° 37′ 51′′?

21. 38 m. 4 fur. 23 rd. 4 yd. 0 ft. 3 in. 32 m. 5 fur 28 rd. 5 yd. 1 ft. 5 in.?

22. 54 m. 6 fur. 3 rd. 8 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. 11 sq. yd. 5 sq. ft. 138 sq. in. ?

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

16 rd. 1 ft. 4 in. ?

23 sq. rd.

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, ab breviated explanations like the following may be adopted :

WRITTEN WORK OF FIRST EXAMPLE UNDER 65.