Page images
PDF
EPUB

APPENDIX.

SOME OF THE PROPERTIES OF 9.

511. Any number may be separated into two parts, one of which is divisible by 9, and the other of which is equal to the sum of its digits.

ILLUSTRATION. Let 5864 be the number considered.

[blocks in formation]

5864

(5 × 999+8 × 99 +6 × 9) + (5 + 8 + 6 + 4).

..5864 is separated into two parts, the first (5 × 999 +8 × 99 +6 X 9) being divisible by 9, and the second (5+8+6+4) being the sum of its digits. The same can be shown of number.

any

[ocr errors]

The following principles are derived directly from Art. 511: 512. Any number is divisible by 9, if the sum of its digits is divisible by 9.

513. If any number is divided by 9, the remainder is equal to the remainder when the sum of its digits is divided by 9.

514. PROOF OF MULTIPLICATION BY CASTING OUT THE 9's. (See Art. 50.)

MULTIPLICATION.

326

42

652

1304

13692

PROOF.

3+2+6=9+2
4+2 6

=

12. 1+23, remainder. 1+3+6+2 = 12 = 9 + 3, remainder. These remainders being equal, the work is probably correct.

(325)

[blocks in formation]

ted, we obtain four terms for a product, the first three of which are divisible by 9, and the last is the product of the two excesses. The entire product divided by 9 must, therefore, leave the same remainder as the product of the excesses in the multiplicand and multiplies divided by 9.

515. PROOF OF DIVISION BY CASTING OUT THE 9's.

DIVISION.

75) 3929 (52 375

179

150

29

(See Art. 62.)

PROOF.

7+5=12=9+3
5+2= 7

21 2+1=3, remainder. 3929—29 = 3900,3, remainder.

These remainders being equal, the work is probably correct.

DEMONSTRATION OF PROOF.

The dividend, minus the remainder, equals the product of the divisor and integral part of the quotient; therefore, if we divide the dividend, minus the remainder, by 9, the remainder thus obtained must be the same as that which results from dividing the product of the excess of 9's in the divisor and the integral part of the quotient by 9.

CONTRACTIONS IN MULTIPLICATION AND DIVISION.

Arithmetical operations may sometimes be shortened materially by the use of contractions in Multiplication and Division. A few have been suggested in Articles 52, 53, and 64. Some additional contractions are here given, which pupils are cautioned against using until they are so familiar with the common methods as to make no mistakes.

516. TO MULTIPLY BY 9, 99, 999, &c.

9 being one less than 10, 99 one less than 100, and 999 one less than 1000, &c.,

To multiply by any number whose terms are all 9's: Annex as many zeros to the multiplicand as there are 9's in the multiplier, and from that product subtract the multiplicand; thus, 27 × 99

2700-27=2673.

EXAMPLES.

1. 36 X 99?

4. 36841 X 9999?

2. 264 × 999 = ?

3.58 × 9999 = ?

7. 241 × 998=

8. 356 X 9995 = ?

5. 7 X 9999999 = ?

6. 245 × 999999 = ?

-

(241 × 998241 × 1000 — 241 × 2.)

Ans. 240518. | 9. 54932 × 999997 = ?

517. TO MULTIPLY BY A. COMPOSITE NUMBER, i. e., BY A NUMBER THAT IS ITSELF THE PRODUCT. OF TWO OR MORE NUMBERS.

Separate the multiplier into convenient factors, multiply the multiplicand by one of the factors, and that product by another factor, and so on, till all the factors are employed; the last product is the true answer; thus, 41 × 25 = 41 × 5 × 5.

EXAMPLES.

1. Multiply 368 by 72; by 36.
2. Multiply 4079 by 81; by 48.
3. Multiply 2145 by 108; by 144.

4. Multiply 50411 by 55; by 150.

518. To MULTIPLY BY ALIQUOT PARTS OF 10, 100, 1000, &c.

Multiply by 10, 100, 1000, &c., as the case may require, and then find the required part; thus, to multiply by 5, multiply by 10, and divide the product by 2; to multiply by 25, multiply by

100, and divide by 4; by 125, multiply by 1000, and divide by 8; by 333, multiply by 100, and divide by 3; by 163, multiply by 100, and divide by 6; by 121, multiply by 100, and divide by 8.

EXAMPLES.

1. 8743008 × 5 = what? 2. 8003478 X 25 what? 3. 786342 × 12 = what?

4. 875402 × 3} = what?
5. 1090806163=what?
6. 543297 X 125 what?

519. To DIVIDE BY A COMPOSITE NUmber.

ILL. EX. Divide 390 by 15.

OPERATION.

3) 390
5) 130

To divide by a composite number: Separate the divisor into convenient factors, divide by one factor, and the quotient thus obtained by 26, Ans. another factor, and so on till all the factors are employed. The last quotient is the answer required.

[blocks in formation]

To find the true remainder: Commence with the remainder resulting from the second division, and multiply each partial remainder by all the preceding divisors except the one which gave that remainder, and add the sum of the products to the remainder resulting from the first division.

[blocks in formation]

521. TO DIVIDE BY ALIQUOT PARTS OF 10, 100, 1000, &c.

To divide by 5, divide by 10, and multiply the quotient by 2; to divide by 25, divide by 100, and multiply by 4; by 125, divide by 1000, and multiply by 8; by 33, divide by 100, and multiply by 3; by 16%, divide by 100, and multiply by 6; by 166, divide by 1000, and multiply by 6, &c.

1. 987625=?

2. 34543125 =?

3. 87096163=?

EXAMPLES.

4. 432872121 = ?

5. 687904250 =?

6. 1107481663 =?

MODES OF ESTIMATING THE TIME BETWEEN TWO DATES.

522. The mode adopted in this book for estimating the time between two dates, when interest is computed in months and days, is in common use among business men, and is the one most consistent with the ordinary method of computing interest at 30 days to the month.

523. Another mode of estimating the time between two dates, is to find the number of ENTIRE calendar months, and count the remaining days.

524. A third mode of finding the time between two dates, consists in counting the exact days from one date to the other.

ILLUSTRATION.

By the first of these modes, if a note for months, which matures* Feb. 10th, 1865, were discounted at a bank the 11th of December previous, the time would be estimated at 1 day less than two months; viz., 1 month and 29 days.

By the second mode, it would be estimated at 1 month and 30 days, i. e., 2 months.

* A note is said to mature when it becomes due.

« PreviousContinue »