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3. Divide 17,0146 by 3.53. Quot. 4.82.
228. To contract the operation.
Rule. Take as many of the left hand figures of the divisor as the quotient is intended to consist of; divide by these only, at the first step of the operation, and at each succeeding step cut off one figure (or more figures if necessary") from the divisor, instead of bringing down from the dividend, using only the figures not cut off; in multiplying these by the quotient figure, you must observe to carry from the product of the preceding figure cut off, in the same manner you did in contracted multiplication, viz. I from 5 to 15, 2 from 15 to 25, &c. but carry as usual after you begin to set down.
To determine the place of the decimal mark in the quotient, observe under what figure of the dividend the units place of the product of the divisor by the quotient figure stands, and that will be the value of the first figure in the quotient; if it stand under units, the first figure only will be a whole number; if under tens, two figures will be whole numbers; if under primes, the first figure will be primes; if under seconds, a cipher must be prefixed to the quotient, &c."
y Observing to put a cipher in the quotient whenever a figure is cut off, and the remaining figures will not go in the number to be divided.
* The place of the decimal mark in the quotient may be known two ways, viz. either from the decimals, or from the whole numbers; the former has been explained, and the latter may be shewn as follows. Let 10.9 divide 1234.56; now it is plain that if we had only the integers to operate with, (namely, 1234 to divide by 10,) the quotient would be 123, that is, the highest denomination 1 is of the same denomination with the figure 2 of the dividend under which the units figure 0 of the divisor stands, namely, (in this instance,) hundreds ; wherefore, in the above example, seeing the units place (0) in actual division stands under the hundreds place (2), the highest place in the quotient will evidently be hundreds. Again, let 99.9 divide 1.1000; now 99 will not divide 1 (unit), and therefore will not produce units in the quotient; it will not divide II tenths, and therefore will not produce tenths; but it will divide 110 hundredths, and therefore will produce hundredths, or, the first or
8. Divide 89.12543 by 12,34567, reserving only 5 figures in the quotient.
First, I cut of 2 figures from the divisor, leaving 5 on the left hand to divide by ; this divisor goes 7 times in the 5 left hand figures of the dividend; I therefore begin, 7 times 6 are 42; carry 4; then 7 times 5 are 35 and 4 are 39; put down 9, carry 3, and proceed to multiply and set down in the usual way. Next I subtract, and 2706 remains; then I cut off one more figure, viz. 5, from the divisor, leaving 1234 to divide by ; this goes twice in 2706; then twice 5 are 10; carry l; twice 4 are 8 and 1 are 9; put down 9, and proceed as in common division. Next I cut off the 4, then the 3, and so on cutting off 1 figure at every step, and carrying from the figure cut off, 1 from 5 to 15, &c. as I did in contracted multiplication. The units place of the product of the divisor multiplied by 7, falling under the 9 (or units place), shews that 7 must be considered as standing in the place of units.
highest figure of the quotient will be hundredths, that is, of the same name with the place under which the units place of the divisor stands; and the same in other instances.
9. Divide 357.6543218 by 27.1234567; let there be 7 figures in the quotient.
10. Divide 23.41005 by 7.9863. Quot. 2.93.12.
229. In division, the products of the divisor into the several quotient figures need not be set down; each figure of any product as it arises may be subtracted from the figure under which it should stand, (if set down,) and the remainder set underneath, bringing down successively the figures of the dividend in order, or cutting off those of the divisor; observing to carry for the multiplication and subtraction both in one, whenever the carrying for both occurs".
* This is usually called the Italian method, and differs from the common method only as this is a mental and that a visible operation. Let no one suppose himself master of division until he can readily work examples in this rule, both by the common and contracted way, as is shewn in ex. 16.
15. Divide 123.456789 by .432.
OPERATION. .432)123.456789(285.779604 &c. quotient. 3705 2496 Earplanation. 3367 I first find that 432 goes twice in 1234 ; I then 3438 say, twice 2 are 4; 4 from 4 and 0; put down 0, 41.49 and carry l ; twice 3 are 6; 6 from 13 and 7; put down 7, and carry 1 for the borrowing; twice 2610 4 are 8 and l are 9 ; 9 from 12 and 3 remain.
1800 To this second line bring down 5; then 432 in
Rem, 72 3705 will go 8 times; then 8 times 2 are 16; 6 from 15 and 9; carry 2, (viz. 1 for multiplying and 1 for subtracting); 8 times 3 are 24 and 2 are 26; 6 from 10, and 4; carry 3, (viz. 2 for multiplying and 1 for subtracting); 8 times 4 are 32 and 3 are 35; subtract this from 37 and 2 remain; to the remainder 249 bring down 6, and proceed as before.
16. Divide 791.0312345 by 35481.7.
Common method. Contracted method. 35481.7)791.0312345(.022294 35481.7)791.0312345(022294 813972 81397 1043,383 - 10434 333.7494 3.338 1441415 145 22147 rem. 3 rem.
17. Divide 17.01.46 by 4.82. Quot. 3.53. 18. Divide 4.5172834 by 12.34. Quot. .366068 &c. 19. Divide 0064 by .51863. Quot. 01234 &c. 20. Divide 2508.928065051 by 27.1498. Quot. 92.41035.
230. When the divisor is a whole number, consisting of an unit, with ciphers subjoined. RULE. Remove the decimal mark as many places to the left hand as there are ciphers in the divisor".
21. Divide 123.45 by 10. Quot. 12.345.
* The truth of this appears from decimal notation; it may likewise be proved by actually dividing, and marking off for decimals in the quotient, according to the common method, Art. 227.
REDUCTION OF DECIMALS.
231. Reduction of decimals teaches to change decimal fractions from one form to another, without altering their value.
232. To reduce a decimal to a vulgar fraction.
RULE I. Under the given decimal write 1, with as many ciphers subjoined as the decimal has places for a denominator. II. Reduce this fraction to its lowest terms for the answers. ExAMPLEs.
1. Reduce .24 to a vulgar fraction. Thus, 100 the denominator.
24 G - - - Then = g; the vulgar fraction required.
2. Reduce .5 to a vulgar fraction. Ans. #.
5 3. Reduce .625 to a vulgar fraction. Ans. TS’
- IOI 4. Reduce .02525 to a vulgar fraction. Ans. Ho. 4000
233. To reduce a vulgar fraction to a decimal. Rule I. Subjoin as many ciphers as are necessary to the mumerator, and place the decimal mark between the numerator
and those ciphers. II. Divide this number by the denominator, and the quotient,
with the decimal mark prefixed, will be the answer.
7 5. Reduce 8 to a decimal. OPERATION. Earplanation. 8)7.000 I add 3 ciphers as decimals to the numerator 7, and divide by the denominator 8, marking off de
875 the answer. cimals by the rule, Art. 227.
* This rule will be easily understood, as it is a natural consequence of the
decimal inode of notation.