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have found 9, and should consequently, have been obliged to make three useless trials. We easily perceive that this can never give too great a quotient, for if it could, then might 400 be contained in the dividend a greater number of times than 397, which is impossible, Because, 14 X 397 = 319, (60 and 156,) the above ex

ample is proved thus: 6 X 397 + 319=2701. This expression is the same as to say, multiply the divisor and the integral part of the quotient together, and to the product add the remainder or numerator of the fractional part, which will give the dividend. The student may, in like manner, prove all similar divisions : also (161) by casting out the nines.

164. It is sometimes so nice a point to determine the true quotient figure, that most persons, however skilful, are liable to err; but the multiplication of the divisor by the quotient figure immediately shows when this figure is too great, because, in this case, the product will be greater than the partial dividend from which it is to be subtracted. Also, when the subtraction of the product from the partial dividend leaves a remainder equal to or greater than the divisor, as the divisor still contained in this remainder, the quotient figure is too small. 6814834 - 113582=59,999 + 113582) 6814834 (59,999+

Proof by 9.
tens 567910
units

S 1135734
1022238

1134960
tenths

See article 161. 1022238

1127220 hundredths

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{
{
{1022238
{1022238

1049820

thousandths

27582 thousandths. We here first say 1 in 6, or rather 11 in 68, six times, and place 6 in the quotient; but, having multiplied the divisor by this figure and found that the product 681492 is greater than the partial dividend 681483, and consequently the figure 6 too great, we efface it, and, having substituted the figure 5, proceed as usual.

Or thus, taking 6 for the first quotient figure, say 6 times 11 is 66, which taken from 68, leaves 2. This 2, with the next figure, makes 21. Then 6 times 3 is 18, which taken from 21, leaves 3. This, with 4, makes 34. Then 6 times 5 is 30, and 30 from 34 leaves 4. This, with 8, makes 48. Then 6 times 8 is 48, and 48 from 48, leaves 0. Lastly, as 6 times 2 is not contained in 3, we see that 6 is too great, and put 5 in its stead as before.

Proof by multiplication.

113582

59999 thousandths
1135706418
567910
6814806418 thousandths

27582 thousandths

6814834,000 Here it is plain that the product, formed by repeating 59999 thousandths 113582 times, is a number of thousandths, as well as the remainder 27582. Hence, the sum of these, which is 6814834000, is also a number of thousandths; wherefore, to obtain the units in this number, (144,) we divide by 1000, which is done (56) by cutting off the three ciphers on the right, the number on the left being the dividend, as required.

1. 728836 =907883, and 138.76=803955 2. 27065849 ; 356 3. 26.665,942 4. 3141593 - 7854= 399,999 + 5. 78539992146 ; 7854 6. 18437688851569 ;-4293913 7. 111030011 ; 199 = 557939,75 + 8. 393.8,725 = 57082,971 + 9. 12167395 ; 1219 = 9981,456 + 10. 78987116249 - 19818=3985624,9999 + 11. 0138939 = 6,1234512345, &c. 12. 5684973 X 285311670611 25937424601

= 62534703 165. The greater the dividend, or number to be divided into

16

equal parts, the greater the quotient, or value of each part. Therefore, to multiply the dividend (150) is to multiply the the quotient; and, inversely, to divide the dividend is to divide the quotient. Again, the greater the divisor, or number of equal parts, the less the quotient, or value of each. Therefore, to multiply the divisor is to divide the quotient; and inversely, to divide the divisor is to multiply the quotient ; that is to say, the operation performed upon the dividend produces upon the quotient the same effect; but that performed upon the divisor, the contrary. But (154) multiplication and division destroy the effect of each other; wherefore, to multiply or divide both dividend and divisor by the same number has no effect upon the quotient.

dividend 6
divisor 16

= 0,375 quotient. dividend mult. by 2 6 X2 12

quotient mult. by 2.

= 0,75 0,375 x 2 same divisor

16 dividend div. by 2 6 2 3

quotient div.by 2.

= 0,1875 = 0,375 = 2 same divisor

16 16 dividend unaltered 6

quotient div. by 2. divisor mult. by 2 16 X2

0,1875 0,375 • 2 same dividend 6

6

quotient mult, by 2. divisor div. by 2 16 X 2 8

= 0,75 0,375 x 2 dividend and divisor 6 x 2 12 quotient unaltered. both mult. by 2 16 X 2

0,375

32 dividend and divisor 6 + 2 3 both div. by 2

0,375 quotient the same. 16 - 2

8 166. The decimal value of the generic unit in each of the fractions , , and ly, is found by dividing a unit by the denominator, and that of the fraction by multiplying the value of the generic unit by the numerator, thus :

š=0,125 and } = 0,125 X 3=0,375
= 0,0625 and 1=0,0625 X 6=0,3750

= 0,03125 and 11 = 0,03125 X 12=0,37500 The ciphers on the right of the numbers 0,3750 and 0,37500 may be omitted, as having no effect upon the value of the number 0,375, because (165) 330 = 18% = 3088%, &c.

32

167. When any two numbers are each multiplied by the same number, the two products are called equimultiples of the two numbers. Thus, if we multiply the two terms of the fraction each by 2, the resulting terms 6 and 16 are equimultiples of 3 and 8. But we have seen (165) that the quotient i is the same as the quotient 18 or 3*, &c. Wherefore, if any equimultiples whatever be taken of the terms of a fraction, the resulting fraction is still of the same value. Hence, it is evident

that the same fraction may be represented by an infinity of different numbers, seeing that the two terms may be multiplied by each of the numbers in the natural scale 2, 3, 4, 5, 6, 7, &c., ad. inf. Thus, š====18 18 = , &c., ad. inf.

The least numbers by which a fraction can be expressed are called its lowest terms. Thus, the terms 3 and 8 are the lowest terms of the above fraction. Each of the other expressions, 16, 24, 33, &c., is called a value of 3.

We may further elucidate this by the smaller Spanish coins or parts of a dollar, namely, halves, quarters, eighths, and sixteenths, which, though they do not accord with the decimal Federal Money of the United States, are still in general use.

Thus,

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It will also be useful to the student to render himself betimes familiar with the following decimal values of these coins, thus :

1. ,50 or 50 cents
2. ,25 or 25 cts.
3. ,75 or 75 cts.
4. ,125 or 123 cts.
5. ,375 or 37} cts.
6. ,625 or 621 cts.
7. ,875 or 874 cts.
8. ,0625 or 64 cts.
9. ,1875 or 18 cts.

,3125 or 314 cts.
11.
16

,4375 or 43 cts.
12. 99
16

,5625 or 564 cts.
13. ii

,6875 or 68 cts. 14. i 3

,8125 or 814 cts. 15. ,9375 or 93 cts.

16

10. 16

16

we

168. When a fraction is expressed by equimultiples of its lowest terms, it is often necessary, in order to expedite calculation, to find those terms; for, if we would multiply a number by 122936 and divide the product by 138303, which is the same as to multiply the number by the fraction 133936 should obtain the result much more readily if we could conWeniently discover that 122936

s, in which case we should simply multiply by 8 and divide by 9 or, subtract from the number one-ninth of itself. This we shall illustrate thus : 428695731 x 122936

= 381062872
138303
First method 428695731

122936
3858261579 nine hundred times

15433046316 thirty-six times
857391462 two thousand times

5144348772 120 thousand times 138303) 52702138386216 (381062872

414909

1121123
1106424

146998
138303

Second method. 428695731

8

86953 829818

9) 3429565848

381062872

397206
276606
1206002
1106424

Third method.

428695731 one-ninth 47632859 eight-ninths 381062872

995781
968121

276606
276606

******

To facilitate the research above alluded to, as well as for other important purposes, the properties of numbers, with which we commence the succeeding book, will be found of great utility.

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