8. Divide 89.19543 by 12.34567, reserving only 5 figures in First, I cut off 2 figures from the divisor, leaving 5 on the left hand to di vide 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 1; 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 I 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. 27.12345|67)357.6543 218(13.18616 quotient. 271 2346 864197 813704 50493 27123 23370 21698 1672 1627 45 27 18 16 2 rem. 10. Divide 23.41005 by 7.9863. Quot. 2.9312. 11. Divide .019876843 by .012345678. Quot. 1.6100244. 12. Divide 721.17562 by 2.257432. Quot. 319.467. 13. Divide 165.6994 by 52.7438. 14. Divide 357.6543 by 13.18616. Quot. 3.14159. Quot. 27.12347. 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". a 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 3367 3438 4149 2610 Rem. 1800 Explanation. I first find that 432 goes twice in 1234; I then say, twice 2 are 4; 4 from 4 and 0; put down 0, and carry 1; twice 3 are 6; 6 from 13 and 7; put down 7, and carry 1 for the borrowing; twice 4 are 8 and 1 are 9; 9 from 12 and 3 remain. To this second line bring down 5; then 432 in 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. Quot. 92.41035. 19. Divide .0064 by .51863. Quot. .01234 &c. 20. Divide 2508.928065051 by 27.1498. 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. Quot. .987654. 22. Divide 9876.54 by 10000. 23. Divide 1 by one million. 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 1. 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 answer, 5 3. Reduce .625 to a vulgar fraction. Ans. 101 4. Reduce .02525 to a vulgar fraction. Ans. 4000 233. To reduce a vulgar fraction to a decimal. RULE I. Subjoin as many ciphers as are necessary to the numerator, 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. This rule will be easily understood, as it is a natural consequence of the decimal mode of notation. 4. To reduce money, weights, and measures, to decimals. Reduce 5bu. 3pks. to the decimal of a chaldron. 11 Thus, 5bu. 2pks. = =.152777 &c. chald. Ans. 72 The foundation of this rule is sufficiently obvious without explanation. |