500) 8 9 9 | 52

A. 17 9452.

300

Cutting off two figures does in fact divide it by 100, it being one factor of the composite number 500. Then, if we mul

tiply (as in XXIV. 9,) the last remainder 4 by the first divisor 100, it makes 400, to which adding 52, the first remainder, makes 452; but by simply annexing the 52 to 4, produces the same effect, hence the rule.

4. Divide 783,456,078 by 2,100. 5. Divide 634,278,975 by 8,000. 6. Divide 854,267 by 500,000.

A. Remainder 678
2100
A. Remainder 8275

8000.

500000

A. Quotient 1354267 7. The annual expense for schools in the United States is about 15,000,000 of dollars, and the number of children about 3,750,000; what is the average expense for each child?

A. 4 dollars.

8. The number of teachers is about 95,000: how many scholars then to each teacher? A. 40 nearly.

CONTRACTION OF RULES.

XXVII. 1. To multiply easily by any number from 10 to 20.Multiply by the unit figure only of the multiplier, and having removed its product one place further towards the right of the multiplicand, add it to the multiplicand.

4. For the figures which are added in both operations are the same; the results must therefore correspond.

10. To multiply by 5.-Annex a cipher, and divide 2. 11. For annexing one cipher multiplies it by 10, then 2 times 5 being 10, dividing only by 2, leaves the number increased 5 times. 12. Multiply 6,545 by 5.

13. Multiply, by this rule, 7,521 by 5.

A. 32,725.

A. 37,605.

14. To divide by 5.-Reverse the last process by multiplying by 2, and cutting off one figure for a remainder.

XXVII. Q. How can you multiply by 11, 12, 13, &c., up to 20 expeditiously? 1. Why so? 4. How can you multiply by 5 easily? 10. Why? 11. How divide by 5 14. How many are 5 times 48?-times 5 in 240?

15. Divide, by this rule, 32,725 by 5. 16. Divide, by this rule, 37,605 by 5.

A. 6,545. A. 7,521.

17. To multiply by 25.-Annex two ciphers, and divide by 4.

18. For annexing two ciphers multiplies by 100, and 4 times 25 being 100, dividing only by 4, leaves the number 25 times the greater. 19. Multiply 6,532,405 by 25. 20. Multiply 4,230,216 by 25.

A. 163,310,125.

A. 105,755,400.

21. To divide by 25.-Reverse the last process by multiplying by 4, and cutting off two figures for a remainder.

22. Divide 163,310,125 by 25.

23. Divide 105,755,400 by 25.

A. 6,532,405.

A. 4,230,216.

24. To multiply by 125.—Annex three ciphers, and divide by 8. 25. For 8 times 125 being 1000, annexing three ciphers and dividing only by 8, leaves the number 125 times the greater.

26. Multiply 6,304,521 by 125.

27. Multiply 2,403,450 by 125.

A. 788,065,125.

A. 300,431,250.

28. To divide by 125.-Reverse the last process by multiplying by 8, and cutting off three figures from the right of the product.

29. Divide 788,065,125 by 125.

30. Divide 300,431,250 by 125.

A. 6,304,521.
A. 2,403,450.

31. To multiply by 33}.—Annex two ciphers, and divide by 3. 32. For 3 times 331 being 100, annexing two ciphers multiplies by 100, and dividing only by 3, leaves the number 33 times the greater. 33. Multiply 65,220 by 331.

34. Multiply 73,410 by 33}.

A. 2,174,000.
A. 2,447,000.

35. To divide by 331.-Reverse the last process by multiplying by

3. and cutting off two figures.

36. Divide 2,174,000 by 33.

37. Divide 2,447,000 by 33.

A. 65,220.

A. 73,410.

38. To multiply by 9, or 99, or 999, &c.-Annex to the multiplicand as many ciphers as there are 9s, and subtract the multiplicand from it.

39. For annexing one cipher, for instance, multiplies by 10, and deducting the multiplicand, leaves it 9 times the greater.

40. Multiply 467 by 9, and by 99, and by 9999.

4670 4 6 7 A. 4 2 0 3

46700 4 6 7 A. 4 6 2 3 3

4 6 7 0 0 0 0 4 6 7 A. 4 6 6 9 533 A. 652,767,579. A. 653,354,658.

41. Multiply 653,421 by 999. 42. Multiply 65,342 by 9,999.

Q. How is the multiplying by 25 abridged? 17. Why? 18. The dividing by 25 abridged? 21. How many are 25 times 8? 25 in 200? How is the multiplying by 125 abridged? 24. Why? 25. The dividing by 125? 28. Multiply 8 by 125. Divide 1000 by 125. How is the multiplying by 331 abridged? 31. Why? 32. The dividing by 331 abridged? 35. How many are 33 times 15?-times 331 in 500? What abreviation is there in multiplying by 9, or 99, or 999? 38. Why? 39.

ARITHMETICAL SIGNS.

XXVIII. 1. EQUALITY.' The sign = between two numbers shows that the number before it is equal in value to the number after it; as, 100 cents 1 dollar, meaning 100 cents are equal to 1 dollar.

2. This sign is two horizontal lines, drawn parallel3 to each other. 3. ADDITION.* The sign+shows that the number before it is to be XXVIII. Q. What is the sign of Equality? 2. What does it show? 1. What do 100 cents and 1 dollar with the sign of Equality between them mean? 1.

*PROOF OF ADDITION, SUBTRACTION, MULTIPLICATION, AND DIVISION, by casting out the 98, or by 9s as it is called.

* ADDITION. One 10 contains one 9 and 1 unit; 2 tens or 20, two 9s and 2 units; 3 tens or 30, three 9s and 3 units, and so on, leaving for a remainder each time as many units as there are tens.

Hence if we deduct from any number of tens, as many units as there are tens, the remainder will contain an even number of 9s.

One hundred [100] contains eleven 9s and 1 unit; two hundred [200,] twenty-two 9s and 2 units, and so on, leaving for a remainder each time as many units as there are hundreds.

And universally if from any number of tens, or hundreds, or thousands, &c., there be taken as many units, as there are tens, or hundreds, or thousands, &c., the remainder will contain even 9s.

The number 634, for instance, is made up of 600, 30, and 4. The 600 then contains even 9s, and 6 remainder; the 30, even 9s, and 3 remainder; and the 4 no 9s, and 4 remainder. Now the remainders 6, 3, and 4, are the very numbers that form 634, therefore we derive the following proposition, (4) on which is based (5) the proof of Addition, viz:

The sum of the figures that compose any number has an excess (6) of 9s equal to the excess of 9s in that number.

And from the nature of Addition, it follows, that the sum of the excesses of 9s in two or more numbers, always has an excess equal to the excess in the sum of those numbers.

2

68318 956 345 3 1984 4

The amount is 1984, and the proof as follows, viz: Adding the figures 6, 8, and 3 together, in the top line, makes 17, or one 9 and 8 over for the excess; reject the one 9 and write down on the right the excess above 9 which is the 8. Do the same with the 9, 5, and 6, rejecting the the 98 and writing down the excess, which is 2. The third line leaves in like manner an excess of 3; next adding these remainders, 3, 2, and 8, makes 13, or one 9 and 4 remainder; reject the 9 and write down the 4 underneath. The bottom line 1, 9, 8, and 4 makes 22, or two 9s and 4 remainder; rejecting the 9s, the remainder is 4, the same as the preceding remainder, and therefore 1984 is the true amount.

RULE. Add the figures in the uppermost row or line together, reject the 9s contained in their sum, and set the excess directly even with the figures in that row. Do the same with each row and set all the excesses of 9 together in a line, and find their sum; then if the excess of 9s in this sum (found as before,) be equal to the excess of 9s in the total sum, the work may be considered correct.

It may sometimes be more convenient to reject the 9s while adding: thus, taking 5632 for example, 5 and 6 are 11-one 9 and 2 over; the 2, over and 3 are 5 and 2 are 8

1 EQUALITY, [L. equalitas.] Agreement; evenness; uniformity.

2 HORIZONTAL. Relating to the horizon; level; not perpendicular.

3 PARALLEL. A line equally distant through its whole extent from another line

4 PROPOSITION. What is proposed; stateinent of facts; offer of terms,

5 BASED. Founded.

6 EXCESS. What is over; superfluity; remaining.

added to the number after it; as 6+4=10, meaning 6 and 4 added together are equal to 10.

4. This sign is a cross, formed by a horizontal line intersecting1 a perpendicular one, at right3 angles,a and is read plus, which means more; thus 6+4=10, means 6 plus 4 are 10. 5. How many are 375+125+100=

A. 600.

6. How many are 57,563 +1,500+1,000,000+42+100+101+5 +72? A. 1,059,383.

7. SUBTRACTION.* This sign-shows that the number after it is to be subtracted from the number before it; as, 6-4-2, meaning 4 from 6 leaves 2.

8. This sign is a single horizontal line, and is often called minus,6 signifying less; thus, 10-3=7, is read 10 minus 3 is 7.

9. How much is 10,000,000-1,001 ? A. 9,998,999. 10. How much is 37,500,209-4,209 ? A. 37,496,000. 11. MULTIPLICATION.† This sign shows that the number before it and the number after it are to be multiplied into each other; as, 10×6=60, meaning 10 times 6 are 60.

12. This sign is two lines crossing each other in the form of an x 13. How many are 5,320,065 × 801? A. 4,261,372,065. 14. How many are 423 × 100 × 200 ? A. 8,460,000.

Q. What is the sign of Addition? 4. How formed? 4. What does it show? 3. How is it read? 4. How much is 6 plus 4? 8 plus 12? What is the sign of Subtraction? 8. What is it often called? 8. What does it show? 7. How much is 10 minus 7? 25 minus 14? What is the sign of Multiplication? 12. What does it show? 11. When 10 and 6 have this sign between them, what do they mean?

11.

*SUBTRACTION. Since the subtrahend and difference added together should equal the minuend, the Proof is the same in principle, as that for Addition.

RULE. Reject the 9s from the subtrahend and difference, noting the excesses. Add these excesses together, and if the excess of 9s in their sum equal the excess in the minuend, the work is right.

6 3 4 5 6...6

5 2 0 0 1...8

1 1 4 5 5...7

The sum of the excesses 8 and 7 are 15, from which rejecting the 9s leaves 6; equaling the excess in the minuend.

+MULTIPLICATION. Since Multiplication is an abbreviation of Addition, it may be proved on the same principle.

RULE. Multiply the excess of 9s in the multiplicand, by the excess in the multiplier, and if the excess of 9s in this product equal the excess in the total product the work is right. 6 8 3 45...8

The product of the two excesses is 24, and the excess of its 9s is 6, which is the same as the excess in the product and therefore right.

1 INTERSECTING, [L. intersecto.] Cutting or crossing each other.

2 PERPENDICULAR. Hanging in a straight line from any point to the centre of the earth; upright; not level.

3 RIGHT, Straight; lawful; just; most direct. A square corner is a right angle; a square figure has four right angles; when the four angles, made by two lines crossing each other are equal, each is called a right angle.

4 ANGLE. A corner; the space between two lines that meet.

5 PLUS, from the Latin plus, signifying more.

6 MINUS, from the Latin minus, signifying less.

15. How many are 1×2×3×4×5×6×7×8×9×10?

A. 3,628,800.

16. DIVISION.* The sign÷shows that the number before it is to be divided by the number after it; as, 60÷5=12, meaning 60 divided by 5 is 12.

17. The sign

shows that the number above the line, is to be

divided by the number below the line.

18. The first sign is formed by one horizontal line passing between two dots, and the second by writing the divisor under the dividend with a line between.

19. Perform 1,236,000÷5. 20. Perform 3,756,000÷20. 21. Perform 4237500830

301

A. 247,200.

A. 187,800.

22. This sign is the proper method of expressing the remainder after division is performed. VI. 1, 2. 23. Perform 750,348÷125.

24. Perform 320658952

1017

A. 8,458,08534

A. 6,002,98 A. 315,298,886

123.

25. When we wish merely to indicate there is a remainder, it being not of sufficient importance to be expressed, the sign of Addition is generally adopted; thus 10 mills÷3 makes 3+.

26. Divide 5,608,354 drams by 117.

27. Divide 7,503,478 gills by 129?

A. 47,934+

A. 58,166+

28. When two or more signs occur in succession, each operation is to be performed in the order of the signs.

A. 1.

29. Perform 600+100-150×20÷11,000=1.† 30. Perform 100+100-5+29÷8+6x11+40×3+617×5-1295 ÷80=100.

A. 100.

Q. What are the two signs of Division? 18. What does each show? 16. 17. What two methods are there of indicating a remainder? 22. 25. What do several signs in succession indicate? 28.

* DIVISION. From the principles of proof recognized in Addition and Multiplication, we may proceed as follows:

RULE. Multiply the excess of 9s in the quotient, by the excess in the divisor, and reject the 9s from the product: to which add the excess of 9s in the remainder, and if the result equal the excess of 9s in the dividend, the work is right.

NOTE. This property of 9 belongs to it, only because it happens to be 1 less than 10 (the radix (2) of the system;) for did we reckon by 11s, then 10 would answer the same purpose; but since any number of 9s always contains an exact number of 3s, we may prove questions as well by casting out the 3s in the manner above, as by casting out the 98.

† Add 100 to 600, from the amount subtract 150, multiply the remainder by 20, and divide the product by 11,000, the quotient will be 1 the Answer.

1. ADOPTED, [L. adopto.] Taken as one's own; selected for use.

2 RADIX, [L. radix, a root.] A primitive word; root.