5. The multiplier is 96; the product is 49164; if the multiplier be made one-fourth as large, what will be the product?

6. The multiplier is 16; the multiplicand is 41862; if the multiplier be made one-eighth as large, what will be the product?

7. The multiplier is 24; the multiplicand is 691344; if the multiplicand be made one-fourth as large, what will be the product?

8. The multiplier is 36; the product is 91884; if the multiplicand is made one-third as large, what will be the product?

9. The divisor is 9; the quotient is 6847; if the divisor be made one-third as large, what will be the quotient?

10. The divisor is 28; the quotient is 418374; if the divisor be made 8 times as great, what will be the quotient?

11. The quotient is 9186; what will the quotient be if the divisor is made 8 times as great? 13 times? If one-ninth as large? One seventeenth as large.

12. The divisor is 34; the quotient is 7641; the remainder is 11; what is the dividend?

13. The divisor is 78; the quotient is 3746; what is the dividend?

14. The product of two factors is 39644; the multiplier is 374; what is the multiplicand?

15. The sum of two numbers is 3974; their difference is 1389; what are the numbers?

16. The sum of two numbers is 49683; their difference is 3841; what are the numbers?

LESSON III.

107. CONTRACTED OPERATIONS.

(a.) To multiply by 10, 100, 1000, &c.

Annexing a cipher to any number changes the units to tens, and the tens to hundreds, &c.; and therefore multiplies the

tens.

Ounits

units.

number by ten; by annexing a cipher to 53 (530), the number is multiplied by ten.

On the same principle, annexing two ciphers multiplies by one hundred, three ciphers by one thousand, etc.

(6.) To multiply by 25.

Rule.-Annex two ciphers to the multiplicand and divide

by 4.

Proof..-If we annex two ciphers to the multiplicand we multiply it by 100; but as 100 is 4 times as large as 25, the product is 4 times greater than it should be; therefore one fourth of this product is the product required.

(c.) To multiply by 124.

Rule.-Annex two ciphers to the multiplicand and divide the result by 8.

NOTE.-Require the pupil to demonstrate each of these rules after the model given.

(d.) To multiply by 331.

Rule.-Annex two ciphers to the multiplicand and divide the result by 3.

(e.) To multiply by 125.

Rule.-Annex three ciphers to the multiplicand and divide the result by 8.

(f.) To divide by 10, 100, 1000, &c.

By removing a figure from the right of any number the local value of ten's figure is changed to units, and hundreds to tens, &c., therefore the number is divided by ten. By re

moving the right hand figure, 7, from 637, thus 63, the number is divided by ten and a remainder of 7 is left.

On the same principle, by removing two figures the number is divided by a hundred, three figures by a thousand, the figures cut off being the remainder.

100.

(g.) To divide by 25.

Rule.-Multiply the number by 4, and divide the product by

Proof.-As 100 is 4 times as great as 25, if we divide any number by 100, the quotient will be four times less than it should be; therefore, 4 times this quotient will be the quotient required. It is most convenient first to multiply the number by the 4, and then divide the product by 100.

100.

100.

(h.) To divide by 121.

Rule.-Multiply the number by 8, and divide the product by

(i.) To divide by 334.

Rule.-Multiply the number by 3, and divide the product by

(j) To divide by 125.

Rule.-Multiply the number by 8, and divide the product by

1000.

LESSON IV.

PRACTICAL PROBLEMS.

17. What will 387 oranges cost at 10 cents apiece?
18. What will 463 melons come to at 10 cts. apiece?
19. What will 100 horses cost at $375 apiece?

20. What will 1000 laths cost at ct. each?

21. What will 60 yd. of carpet cost at $2 a yd.?

22. How much must I pay for 40 chickens at 75 cts. apiece? 23. What will 6 pies cost at 124 cts. each?

24. What will be the cost of 100 bundles of wood at 4 cts. a bundle?

25. What must a man pay for 320 yoke of oxen at $125 a yoke?

26. What will 300 barrels of apples cost at $5 a barrel ? 27. How much will 300 lemons cost at 4 cts. each?

REVIEW.-What is the difference between the Roman and Arabic notations? (7.) (8.) What is the simple value of a figure? (9.) What is the local value? (10.) What is the use of a cipher? (11.) What is numeration? (12.) What is reading numbers? (13.) Recite the numeration table from the right to the left. (13., b.) From the right to the left by periods. What is to be done with vacant places or periods? (14., 15.) How are numbers usually separated? (15., a.) What is a rule? (16.) What is the rule for notation? (17.)

28. A man sold 200 shares of oil stock at $10 a share; how much did he receive?

29. What will 300 boxes of pins cost at $1.25 per box? 30. What will 100 books cost at 20 cts. apiece?

31. What must I pay for 7 sets of china at $10 per set? 32. What is the cost of 600 bushels of potatoes at 75 cts. per bushel?

33. What will 500 boxes of matches come to at 1 ct. per box?

34. A man bought 300 lb. of lead at 8 cts. per pound, what was its cost?

35. What will 113 yd. of braid cost at 33 cts. per yd.? 36. If 12 tons of coal will cost $120 what will one ton cost?

LESSON V.

37. What will 620 yd. of muslin cost at 334 cts. per yd.? 38. If 125 cows cost $6500, what will one cost?

39. If 331 yd. of calico cost $16.50, what will 1 yd. cost? 40. A farmer sold 310 bushels of turnips at 25 cts. per bushel; how much did he receive?

41. If I pay 125 cts. a yd. for lace, what will 136 yd. cost me ?

42. What will 68 acres of land cost at $100 per acre?

43. If 12 bushels of rye cost 960 cts., what will 1 bushel cost? 44. If 1 one qt. of molasses cost 12 cts., what will 12 qt. cost?

45. If 25 bananas cost 165 cts., what will 3 cost?

46. If 4 gold pens cost $7, what will one cost?

47. A man bought 230 yd. of cloth at 125 cts. a yd.; what was the amount of his bill?

48. If I can buy 125 oranges for 625 cts., what will 5 cost? 49. A grocer bought 124 pounds of rice for $1.50; what did it cost per pound?

50. What will 10000 books cost at 125 cts. each?

51. If 25 barrels of flour cost $175, what will one cost?

LESSON VI.

PROPERTIES OF NUMBERS, &c.

108. An Integer is a whole number as distinguished from a fraction.

109. A Prime number is one that cannot be divided evenly without a fractional quotient.

110. A Composite number is one that can be divided evenly by an integer.

111. Numbers are prime to each other, when the same number will not divide both without a remainder; as 7 and 4; 15 and 14.

112. A Prime Factor of a number is a prime number that will divide it without a remainder.

(a.) Every number is a prime number, or is composed of prime factors. 113. A Composite Factor of a number is a composite number that will divide it without a remainder.

NOTES. (a.) Unity or 1 is not regarded as a material factor, since multiplying or dividing a number by 1 does not alter its value.

(b.) No direct process has been discovered by which prime numbers may be distinguished, but the following facts will aid in determining whether a number is prime or not.

1. Every even number is divisible by 2.

-All prime numbers except 2 must be odd.

*

INFERENCE.*

2. If the sum of the digits of any number is divisible by 3, the number is divisible by 3.

3. Every number whose right hand figure is a cipher or 5 is divisible by 5.

4. Every number occupying three or four places, whose right *NOTE.-Let the pupil be required to illustrate each of the following facts, and draw the inference from it.

REVIEW.-What is the sum of two or more numbers? (18.) What is addition? (19.) What is the sign of addition? (20.) What is the sign of equality? (21.) What is the dollar sign? (22.) What is an axiom? (23.) Repeat the axioms of addition. (23., a.) What is a problem? (24.) What is a simple problem? (25.) What is a complex problem? (26.) What is an analysis? (27.) What is an analytical step? (28.) What is the order of solving problems? (29.) What is the rule for addition? (29., e.) How may addition be proved? (29.,ƒ.)