28. Multiply 257 by 28.

257

28

2056

514

7196

The product 56 should be suggested without thinking "8 x 7 is 56"; write the 6. Next, think "forty (8×5) five" (carried); write the 5. Likewise, the next result thought should be 20. Complete the multiplication in this way. In adding the partial products, apply methods for rapid addition.

TO THE TEACHER. Until the correct method is learned, require all

work to be done orally.

29. Checking.

The best method of proving the result correct is to interchange the factors. Thus, in example 9 above, check the product of 896 × 754 by multiplying 754 x 896.

Multiplication may be checked also by casting out the The following example will illustrate the method.

nines.

536 5

173 2

1608

3752

536

92728 1

Cast out the nines in the sum of the digits of each factor; find the product of the remainders (5 and 2) and cast out the nine; the remainder (1) should be the same as that found in casting out the nines in the sum of the digits in the result.

RAPID METHODS IN MULTIPLICATION

30. To multiply by 10, 100, 1000, etc. Annex as many ciphers to the multiplicand as there are ciphers in the multiplier.

To multiply by 20, 30, 600, etc. Multiply first by the 2, 3, etc., and then annex the necessary ciphers.

Give results orally:

31. Complements (§ 23) may be used in multiplying when the figures of the factors are between 80 and 100. Multiply 97 by 95.

97 3 (comp.) 95 5 (comp.)

9215

Write 15, the product of the complements as illustrated; subtract either complement from the other factor (95-3 or 97-5) and prefix the remainder to the 15.

NOTE. If the product of the complements is one figure, prefix 0; if more than two figures, set down the last two figures and carry the other to the next part of the product.

Write the products of the following:

23724

11

260964

NOTE.

Write down the right-hand figure (4); write, in order,

the sum of the 1st and 2d figures (4 + 2), the sum of the 2d and 3d (2 +7), the sum of the 3d and 4th (7 + 3), the sum of the 4th and 5th (3 + 2) + 1 (carried); write 2, the left-hand figure.

To multiply a number of two figures by 11, write the sum of the two figures between them; thus, 35 x 11 = 385.

33. To multiply by a number, one part of which is a multiple of another part.

Multiply 265 by 357.

265

357

1855

9275

94605

The product of 265 × 7 is 1855; 35 is 5 times 7, hence multiply 1855 by 5 and set down the first figure under the 5 of the 35.

Multiply 285 by 728.

285

728

1995 7980 207480

It does not matter which number in the multiplier is first used, if the product is written under the number by which we multiply.. Therefore, multiply first by 7, setting down 1995 so that the first figure (5) comes under the multiplier (7); multiply that result (1995) by 4 (28 = 4 × 7), writing 7980 so that the 0 is under the 8 of 28.

NOTE. In all multiplication work be careful to keep all columns straight, especially when there are a number of partial products.

16. A factory employs 2317 men at an average wage of $2.50 per day. If there are 26 working days in the month of July, what amount will be paid out in wages?

17. The average taken in by a street-car conductor per day is $53.75. What amount will be taken in by 197 conductors in 31 days?

18. Multiply the sum of 84354 and 17690 by the difference between 53643 and 11319.

19. Following the form in § 27, prepare a multiplication table of numbers from 16 to 25.

20. An automobile manufacturer makes in one year 2678 machines of an average value of $1275. What is the value of his output?

DIVISION

RAPID METHODS IN DIVISION

NOTE. Special methods depending on fractions will be taken up after that subject has been presented.

34. To divide by 10, 100, 1000, etc.

Beginning at the right-hand figure of the dividend, or at the decimal point, point off as many places to the left as there are ciphers in the divisor.

35. To divide by 30, 400, 5000, and like numbers.

Point off the required number of places, and divide by the left-hand figure of the divisor.

36. To divide by the factors of a number.

Divide 3759 by 21.

3)3759

7)1253

179

The factors of 21 are 3 and 7; divide by short division.