11. What was B's tax, who paid for 4 polls, and had property to the amount of $1461 ?

12. C paid for 1 poll, and the valuation of his property was $5863. What was the amount of his tax?

13. D paid for 1 poll, and the valuation of his property was $7961. What was his tax?

14. E paid for 2 polls, and his property was valued at $14236. What was his tax?

15. F paid for 2 polls, and his real estate was valued at $21000; his personal property at $4500. What was his tax? 16. G's property was valued at $20250, and he paid for 1 poll. What was his tax?

17. H paid for 2 polls, and the valuation of his estate was $15360. What was his tax?

18. J's property was valued at $33000, and he paid for 4 polls. What was his tax?

19. K paid for 1 poll, and his property was valued at $15013. What was his tax?

20. L paid for 3 polls, and his property was valued at $4500. What was his tax?

21. M paid for 1 poll, and the valuation of his property was $30600. What was his tax?

22. The Legislature levied a tax of $5312.50 upon a certain town, having an inventory of $450000, and 1550 polls, which were assessed at $1 apiece: what was the tax on a dollar; and what was A's tax, who had $1149 real estate, $1376 personal property, and paid for 3 polls?

23. What was B's tax, who had $2175 real estate, $960 personal property, and paid for 1 poll?

24. C's personal property was $1318, his real estate $120, and he paid for 2 polls: what was his tax?

25. D paid for 2 polls, his real estate was $1538, and his personal property $1681: what was his tax?

SECTION X.

PROPERTIES OF NUMBERS.*

280. By the term properties of numbers, is meant those properties or elements which are inherent and inseparable from them. The following are some of the more prominent:

PROP. 1. The sum of any two or more even numbers, is an even number.

2. The difference of any two even numbers, is an even nuinber.

3. The sum or difference of two odd numbers, is even; but the sum of three odd numbers, is odd.

4. The sum of any even number of odd numbers, is even; but the sum of any odd number of odd numbers, is odd.

5. The sum, or difference, of an even and an odd number, is an odd number.

6. The product of an even and an odd number, or of two even numbers, is even.

7. If an even number be divisible by an odd number, the quotient is an even number.

8. The product of any number of factors, is even, if any one of them be even.

9. An odd number cannot be divided by an even number without a remainder.

10. The product of any two or more odd numbers, is an odd number.

11. If an odd number divides an even number, it will also divide the half of it.

12. If an even number is divisible by an odd number, it will also be divisible by double that number.

13. The product of any two numbers is the same, whichever of the two numbers is the multiplier. (Art. 47.)

14. The least divisor of every number, is a prime number.

QUEST.-280. What is meant by properties of numbers? What is the least divisor of every number?

• Barlow on the Theory of Numbers; also, Bonnycastle's Arithmetic.

15. Any number expressed by the decimal notation, divided by 9, will leave the same remainder as the sum of its figures or digits divided by 9. The same property belongs to the number 3, and to no other number. Thus, if 236 is divided by 9, the remainder is 2; so, if the sum of its digits, 2+3+6=11, is divided by 9, the remainder is also 2.

Note. Upon this property of the number 9, is based a convenient method of proving multiplication and division.

281. To cast the 9s out of a number, begin at the left hand, add the digits together, and as soon as the sum is 9 or over, drop the 9, and add the remainder to the next digit, and so on. For example, to cast the 9s out of 8626557, we proceed thus: 8 and 6 are 14; drop the 9, and add the 5 to. the next figure. 5 and 2 are 7 and 6 are 13; drop the 9, and add the 4 to the next figure. 4 and 5 are 9; drop the 9 as above. 5 and 7 are 12; dropping the 9, the remainder is 8.

Oss. When the sum is over 9, we may simply add its digits together, and proceed to the next figure. Thus, 8 and 6 are 14; now adding its digits, 1 and 4 are 5 und 2 are 7 and 6 are 13. Adding the digits in this sum, 1 and 3 are 4, proceed to the next figure, &c.

PROOF OF MULTIPLICATION BY CASTING OUT THE

NINES.

282. First, cast the 9s out of the multiplicand and multiplier; multiply their remainders together, and cast the 9s out of their product, and set down the excess; then cast the 98 out of the answer obtained, and if this excess be the same as that obtained from the multiplier and multiplicand, the work may be considered right.

What is the product of 565 multiplied by 356?

Operation.

Proof.

565 The excess of 9s in the multiplicand is 7.
99. " multiplier 18 5.

7x5=35; and the excess of 9s is 8.

356

66

66

3390

2825

1695

Prod. 201140

The excess of 9s in the Ans. is also 8.

QUEST.-282. How is multiplication proved by casting out the 9s?

PROOF OF DIVISION BY CASTING OUT THE NINES.

283. First, cast the 9s out of the divisor and quotient, and multiply the remainders together; to the product add the remainder, if any, after division; cast the 9s out of this sum, and set down the excess; finally, cast the 98 out of the dividend, and if the excess is the same as that obtained from the divisor and quotient, the work may be considered right.

AXIOMS.

284. In mathematics, there are certain propositions whose truth is so evident at sight, that no process of reasoning can make it plainer. These propositions are called axioms. Hence, An axiom is a self-evident proposition.

1. Quantities which are equal to the same quantity, are equal to each other.

2. If the same or equal quantities are added to equal quantities, the sums will be equal.

3. If the same or equal quantities are subtracted from equals, the remainders will be equal.

4. If the same or equal quantities are added to unequals, the Bums will be unequal.

5. If the same or equal quantities are subtracted from unequals, the remainders will be unequal.

6. If equal quantities are multiplied by the same or equal quantities, the products will be equal.

7. If equal quantities are divided by the same or equal quantities, the quotients will be equal.

8. If the same quantity is both added to and subtracted from another, the value of the latter will not be altered.

9. If a quantity is both multiplied and divided by the same or an equal quantity, its value will not be altered.

10. The whole of a quantity is greater than a part.

11. The whole of a quantity is equal to the sum of all its parts. OBS. The term quantity, signifies anything which can be multiplied, divided, or measured. Thus, numbers, yards, bushels, weight, time, &c., are cailed quantities.

QUEST.-283. How is division proved by casting out the 98? 284. What is an axiom? What is the first axiom? The Second? Third? Fourth? Fifth ? Sixth? Seventh? Eighth? Ninth? Tenth? Eleventh? Obs. What is meant by quantity?

GENERAL PRINCIPLES AND PROBLEMS.

286. When the sum of two numbers and one of the numbers are given, to find the other number.

From the sum subtract the given number, and the remainder will be the other number.

Ex. 1. The sum of two numbers is 25, and one of them is 10: what is the other number?

Solution. 25-10=15, the other number.

(Art. 40.)

PROOF. 15+10=25, the given sum. (Art. 284. Ax. 11.)

2. A and B together own 36 cows, 9 of which belong to A: how many does B own?

3. Two farmers bought 300 acres of land together, and one of them took 115 acres: how many acres had the other?

287. When the difference and the greater of two numbers are given, to find the less.

Subtract the difference from the greater, and the remainder will be the less number."

4. The greater of two numbers is 37, and the difference between them is 10: what is the less number?

Solution. 37-10-27, the less number. (Art. 40.)

PROOF. 27+10=37, the greater number. (Art. 40. Obs.)

5. A had 48 dollars in his pocket, which was 12 dollars more than B had: how many dollars had B?

6. D had 450 sheep, which was 63 more than E had: how many had E?

289. When the difference and the less of two numbers are given, to find the greater.

Add the difference to the less number, and the sum will be the greater. (Art. 40. Obs.)

QUEST.-286. When the sum of two numbers and one of them are given, how is the other found? 287. When the difference and the greater of two numbers are given, how is the less found? 289. When the difference and the less of two numbers are given, how is the greater found?