11. An army which consists of 24647 infantry and 3249 cavalry, is increased by 6473 infantry and 2464 cavalry. What is the whole number of men?

12. From January 19th, 1852, to June 17th, inclusive, how many days?

13. The smaller of two numbers is 349, the larger is 124 more. What is the sum of both numbers?

14. Required, the sum of two numbers, such, that when 48 is taken away from one, and 244 from the other, the remainder is 173.

15. A father is 49 years older than his son, who is 12 years old; the daughter is five years older than the son, and the mother is 23 years older than the daughter. What are the ages of the father and mother?

16. The land on the earth's surface is divided into five great parts; Europe contains 3720000 square miles; Asia, 17 millions; Africa, 12 millions; America, 15050000; Australia, four millions. The number of square miles of the land is required.

17. The first of five numbers is 247, and the four others increase respectively by 48, 49, 50, and 51. Required, those numbers, and their sum.

18. Six partners, A, B, C, D, E, and F, divide their profits; A gets £2458, B gets as much as A and D together, C as much as B and F, D gets £1500, E as much as C and D, and F receives £800. Find each partner's share and the whole profit.

SUBTRACTION.

52. In an orchard, there are 36 trees; if 13 be cut down, how many will be left.

Here we are led, by the meaning of the question, to take away, or subtract 13 trees from 36 trees, and the operation is called a subtraction; the resulting number is called remainder, difference,

or excess.

53. Therefore, to subtract is to find how much one number exceeds another. The greater number given is called the minuend, and the less, subtrahend.

54. The difference of 9 and 4 is 5, which is briefly expressed, The sign meaning minus or less.

9-4-5.

Write the smaller number below the larger, placing units under units, tens under tens, &c.

7569 or 7000+500+60+9

4354 or 4000+300+50+4

3215 or 3000+200+10+5

Now, 4 units

from 9 units
from 6 tens

leaves 5 units
leaves 1 tens

5 tens
3 hundreds from 5 hundreds leaves 2 hundreds
4 thousands from 7 thousands leaves 3 thousands

Therefore, the difference of 7569-4354=3215.

57. The difference of 93546 and 37874 is required.

The numbers being placed as before

935469 tens of thous. +3 thous. +5 hunds.+4 tens+6 units

+7 +1

+6

+7 +2

Beginning at the right, take 4 units from 6 units, there remain 2 units; 7 tens from 4 tens cannot be done, but if we add mentally 1 hundreds, which is 10 tens to the 4 tens, we shall now have 7 tens from 14 tens, there remain 7 tens, which are written down in the tens' place; proceeding to the next figures, it must be observed that the 8 hundreds must be increased by 1 hundreds, (since 1 hundreds has been added to the top row) then say 9 hundreds from 5 hundreds, which cannot be done, in the same manner add 1 thousands, or 10 hundreds, to the 5 hundreds, and we have now 9 hundreds from 15 hundreds, leave 6 hundreds, which are set down in the hundreds' place; for the same reason as before, we have 8 thousands from 13 thousands, leave 5 thousands, which are set down; and, lastly, 4 tens of thousands from 9 tens of thousands, leave 5 tens of thousands.

The operation, with the artifices used, might be put under this form :

Tens of Thous, Thous. Hunds. Tens. Units.

58. The inference drawn from this operation is, that the difference of two numbers is not altered by adding the same number to the minuend and the subtrahend. It is easily shown that this is also true when the same number is subtracted from both.

For instance, 18-—13=5, adding 7 to each, we have 25-20 =5; subtracting 7 from each, we have 11-6=5.

59. Let it be required to take 234639 from 700607.

Minuend 700607

Subtrahend 234639

Remainder 465968

9 units from 17 units leave 8 units; 4 tens from 10 tens leave 6 tens; 7 hundreds from 16 hundreds leave 9 hundreds;

5 thousands from ten thousands leave 5 thousands; 4 tens of thousands from 10 tens of thousands leave 6 tens of thousands ; and 3 hundreds of thousands from 7 hundreds of thousands leave 4 hundreds of thousands. Therefore, the greater number exceeds the lesser by 465968.

60. Enough has been said to show that subtraction might be defined to be an operation in which it is required to find a number called difference, which, being added to a proposed number, called subtrahend, the sum is equal to another proposed number, called minuend.

61. Therefore, to ascertain if a subtraction be correct, add the difference to the subtrahend; if the sum be equal to the minuend, it is presumed the work is right.

62. Another proof is: subtract the difference from the minuend, and if the remainder be equal to the subtrahend, the work, in all probability, is correct; for the minuend may be considered as the sum of the subtrahend and difference.

63. EXERCISES.

1. The authorised version of the Old Testament contains 592439 words, and the New 181253. How many more words are there in the Old than in the New?

2. 97-43; 549-327; 784-376; 2947-578; 1474813942.

3. 340639-247546; 461201201-359758467.

4. 2122090046-1217484566; 100076709-934567788.

5. 7900000000-4756739523.

6. Napoleon was born in 1769, and died in 1821. How long did he live?

7. The University of Cambridge was founded A.D. 915. How many years have elapsed from that date till now?

8. The expenses of an establishment are £74656, and the receipts £93758; the profits are to be ascertained.

9. Printing was invented in 1449. How many years from that time to 1853?

10. In summer, when the earth is in its aphelion, or greatest distance from the sun, that distance is about 95600000 miles; and in winter, in its perihelion, or least distance, it is about 93500000 miles. How much nearer the sun is the earth in winter than in summer?

11. An empty box weighs 76lbs., and when filled with goods, 464lbs. The weight of the goods is required.

12. Halley's comet, which appeared in 1835, was 76 years invisible. When was it seen previously?

13. If A lived 36 years longer he would be 100 years old. Find A's age.

14. The greater of two numbers is 278, and the difference 169. What is the smaller number?

15. Find the greater of two numbers, the smaller of which is 111, and the sum 327.

16. A property was bought for £9867, and sold for £8978. What is the loss?

17. If 18 years were taken from the age of the father, and added to his son's age, they would each be 30 years old. Find the age of each.

18. Two persons started from two towns, they met one had travelled 198 miles. travelled?

347 miles apart, when How far had the other

19. If I had £500 more, I could repay £1200 which I owe, and have £19 over, What sum have I ?

20. The father is born in 1796, the mother in 1801, the son in 1823, and the daughter in 1827. Find the age of each person, the differences, and the sum of their ages in 1853.