EXAMPLES FOR PRACTICE. 4. At $3 pr. lb., what cost 16 lb. of tea? A, $55.20. 5. At $2 pr. gal. what cost gal. of wine? A. $1.20. 6. At $5.27 pr. yd., what cost 7 yds. of cloth? A. $36.9954. 7. At 3 cts. pr. lb., what cost 7,853 lbs. of rice? A. $274.855. 8. At $133 a bl., what cost 464 bls. of sugar? A. $627.12. 9. Multiply .004 by .04. 10. Multiply 24.61 by .0529. 11. Multiply .0007853 by .035. 12. Multiply .051 into .091. 13. Multiply .0217 into .00431. NOTE. The truth of the rule will further appear, by writing the decimals in the form of vulgar Fractions, and multiplying by the rule § XLIV. Thus, multiply .02 by .03.02 and .030 X 180 = 10000 =.0006, where the decimal places of the product are equal to those in both factors. 14. Multiply .02 by 10. 15. Multiply .003 by 100. 6 A. .2 A. .3 It will be seen that multiplying a decimal by 10 does not alter the significant figures, but only moves the separatrix one place towards the right. This makes the multiplicand 10 times greater, because it causes every figure to stand one place higher. So, multiplying by 100 moves the separatrix two places towards the right; by 1,000, three places, &c. Hence, to multiply by a unit with cyphers annexed, REMOVE THE SEPARATRIX AS MANY PLACES TO THE RIGHT AS THERE ARE CYPHERS IN THE MULTIPLIER, ANNEXING CYPHERS IF NECESSARY. NOTE. If the multiplier be any number with cyphers annexed, it may be considered as a composite number, one of whose factors is 10, 100, 1000, &c., and we may first employ this factor, according to the above rule, and afterwards, the other, as usual. 16. Multiply .38179 by 10; by 100; by 1,000; by 10,000. 17. Multiply 1.876 by 1,000; by 10,000; by 1,000,000. 18. Multiply 17.9 by 1,00; .by 1,000; by 100,000,000. § LXIII. It often happens that, in practice, we have no occasion for as many decimal places as the rule will give us.. The lower decimals are so small that they may be rejected as of no importance. We will now propose a mode, by which We may obtain decimals, accurately as far as we please without the trouble of finding the others. 1. Multiply 13.1346 by 12.2876. COMMON METHOD. 7 88076 91 9422 1050 768 2626 92 26269 2 131346 161.392 71096 I wish to perform this multipli- CONTRACTION. 13.1346 6782.21 131.346 26.269 2.627 1.050 .092 .008 161.392 Juct, thus obtained. If I multiply four decimal figures of the multiplicand by the 1 in the tens' place of the multiplier,=10, I shall have four decimal places in the product, but the last one, (being a cypher) may be neglected, leaving three places, as before. If I multiply two figures of the multiplicand by the 2 in The tenths' place in the multiplier, I shall have three decimal places in the product. (S LXII.) If I multiply, in like manner, one decimal place by the 8 in the hundredths' place, I shall have again three decimal places in the product. So, if I go on, multiplying one place less in the multiplicand, for every place lower in the multiplier, I shall still have three decimal places in the product. I may therefore arrange all these products, thus obtained, with their right hand figures under each other, and put the separatrix, in each, three places from the right. They may then, it is obvious, be added as they stand, and the sum will be the total product, with only three decimal places, It is a convenient way of determining where to commence multiplying with each figure, to put the units' place of the multiplier under that decimal of the multiplicand, which you wish should be the last in the product; and then write all the other figures before and after it, in an inverted order, as in the contraction above. Then in employing each figure of the multiplier, commence with the figure of the multiplicand immediately above it, neglecting all the figures to the right, and place the right hand figures of the several products exactly under each other. They will then stand in proper order to be added. Thus, in the last example, units stand under thousandths; and, of course, the first figure of the product is thousandths; tenths stand under hundredths, and the last figure of the product is in like manner thousandths; hundredths under tenths, which produce thousandths, as before, &c. One thing more. In order to ensure sufficient accuracy, when we begin to multiply by any figure, we must first multiply the preceding figure of the mul tiplicand by the figure we are using as a multiplier, and carry from that product, to the first figure which we set down; always carrying for the nearest number of tens, whether above or below. Thus when multiplying by the 2 (units) in the above example, the first figure in the product is 2X3-6. But to this we carry 1, making 7, because 2X4 (the preceding figure) =8, which is nearer to 10 than it is to 0. In multiplying by 8, (tens,) the first figure obtained is 8X1=8, to which we carrry 2, making 10, because 8X3=24, which is nearer to 2 tens than to 3 tens, &c. Hence, the rule. 1. WRITE THE UNIT FIGURE OF THE MULTIPLIER UNDER THE DECI MAL PLACE WHICH IS TO BE LAST IN THE PRODUCT, AND THE OTHER FIGURES IN AN INVERTED ORDER. II. IN MULTIPLYING, REJECT ALL THE FIGURES IN THE MULTIPLICAND, TO THE RIGHT OF THE FIGURE YOU ARE MULTIPLYING BY; CARRY TO THE FIRST FIGURE OF THE PRODUCT FOR THE NEAREST NUMBER OF TENS CONTAINED IN THE PRODUCT OF THE PRECEDING FIGURE IN THE MULTIPLICAND, AND SET DOWN THE RIGHT HAND FIGURES OF THE SEVERAL, PRODUCTS UNDER EACH OTHER. III. THESE PRoducts, addED AS THEY STAND, WILL GIVE THE TOTAL PRODUCT, WITH THE REQUIRED NUMBER OF DECIMAL PLACES. NOTE. If the multiplier, when thus written, extends below the multiplicand, fill out the multiplicand, or suppose it to be filled out with cyphers annexed. 3. Multiply 27.14986 by 92.41035, retaining four decimal places in the product. A. 2,508.9280 4. Multiply 480.14 36 by 2.72416, retaining four decimal places. A. 1,308.0037 5. Multiply 73.8429753 by 4.628754, retaining five decimal places. A. 341.80097 6. Multiply 8,634.875 by 843.7527, retaining only the integers in the product. A. 7,285,699 § XLIV. 1. What is the value of .75 of a pound sterling in shillings, pence, &c. ? 0 7.5 I£ contains 20s. Therefore of a pound contains of a shilling, and of a pound contains 20 times as many hundredths of a shilling=1500 of a shilling=15s. This is performing the operation as in vulgar Fractions. It is obviously the same, whether the denominator 100 be expressed or not. £0.75 20 A. 15.00s. 2. Reduce .17525£ to lower denominations. A. 3s. 6d. 0.2688 qrs. Hence, to reduce decimals of higher, to whole numbers, and decimals of lower denominations, MULTIPLY AS IN REDUCTION DESCENDING, RESERVING THE WHOLE NUMBERS OF EACH DENOMINATION, AND CONTINUING THE REDUCTION OF THE DECIMAL ONLY. A. 9 d. A. 17 s. 0 d. 2.4 qrs. A. 55 gal. O qt. 1 pt. A. 9 oz. 3 dwt. 11 gr. A. 28 rds. 2 yds. 1 ft. 11.04 in. 12. Reduce .9075 acre. A. A. A. 51.999725 days. 207 d. 16 h. 26 m. 24 sec. A. 2 qrs. 19 lb. 5 oz. 2 hhd. 27 gal. 2 qts. 1 pt. A. 16 s. 7 d. 2.999936 qrs. In reducing decimals of a £. we may evidently reverse the rule LX, thus, DOUBLE THE TENTHS' FIGURE FOR SHILLINGS, AND IF THE HUNDREDTHS BE 5, OR MORE, DEDUCT THE 5, AND ADD ANOTHER SHILLING. CALL THE REMAINING FIGURES, IN THE HUNDREDTHS' AND THOUSANDTHS' PLACES, FARTHINGS, DECREASING THEM BY 1, IF BETWEEN 12 AND 36, AND BY 2, IF ABOVE 36. NOTE. The decimal below thousandths is neglected. But if it amount to more than half a thousandth, it is best to increase the 1,000ths. by 1. 21. Reduce .8971 £..13763 cwt. .19843 ml. .15634 yd. .71 lb. Troy. 71 lb. Avoirdupois. .71 b. Apothecaries'. .8934 week. .9193 month of 30 d. .7346 rd. .9874 hhd. of wine. .9874 hhd. ale. .7759 hhd. beer. .8557 oz. Troy. .9365 £. .59347 £. SUBTRACTION. § LXV. 1. I have a debt of $95 763, and I make a payment of $87.665. How much is remaining due? Of course, dimes must be taken from dimes, cts. from cts. &c. Hence, the numbers must be written as in Addition, and the point placed in the same manner. Hence, to subtract decimal numbers, $95.763 8.098 Ans. WRITE THE NUMBERS AS IN ADDITION, SUBTRACT AS IN WHOLE NUMBERS, AND PLACE THE SEPARATRIX AS IN ADDITION. 2. From a piece of cloth containing 473 yds. a merchant sold 23 yds. How much was left ? A. 24.015 3. On a debt of $383.00 there was paid $47.25. How much remained unpaid? A. $335.75 4. On a debt of $1933, there was paid $87. How much remained unpaid? A. $105.775 5. From 1,153 tons of iron, there were sold 684,5 tons. How much remained unsold? A. 468.8312 tons. 6. From 37 gals. of oil, were sold 28 gals. How much remained? A. 9.1372 gals. 7. A man having 75921 bu. of wheat, sold to one person 47, bu.; to another, 8715; to another, 94,; to another, 387. How much had he left? A. 143 bu. 8. A man had $16,73 of which he spent as follows: for a load of hay, $64; for a lead of grain, 877; for 3 bu. of corn, $13 pr. bu.; and for a load of wood, $2. What had he left? A. $0.18 9. A merchant sold a barrel of flour for $25; 5 gals. of molasses for $1 pr. gal:; and 6 gals. of wine for $193 pr. gal. In payment, he received a load of wood. worth $25, and 2 bu. of wheat worth $13 pr. bu.; and the rest in money. How much money did he receive? A. $2.428, nearly. 830 10. From 8734 take 3631. From 927,3 take 17931. From 169423 take 4738. From 82937 take 33. From 334927 take 23433. From 973 take 864,999 1008 11. From 72.345 take 63.1345. From 39.38463 take -.27953. From 125.125125 take 25.025025. From 380. 613401 take 1.7834. From 830.595003 take .000004. From .00001 take .00000001. DIVISION. § LXVI. 1. If four gallons of wine cost $8.24, what cost 1 gallon? We must divide by 4. The fourth part of 8 dollars is 4)8.24 2 dollars, and the fourth part of 24 cts., that is, 24 hundredths of a dollar is 6 cts. 6 hundredths=.06 2.06 2. At $2.06 pr. gal., how many gals. of wine may be bought witli $8.24? Here we have decimals both in the divisor and dividend. But the 8.24 is 824 hundredths, and the 2.06 is 206 hundredths. 2.06)8.24(4 8.24 Dividing hundredths by hundredths will evidently give whole numbers, just as dividing pints by pints, quarts by quarts, or units by units, gives whole numbers. 824-206-4, as seen above. From observation of these two examples, we see that when the divisor is a whole number, the quotient has just as many decimals as the dividend. And when the divisor has as many decimals as the dividend, the quotient is a whole number. 3. At $0.206 a yard, how many yds. of calico will $8.24 buy? Here the divisor has three decimal places. If .206)8.240(40 the dividend had as many, we should have a whole number quotient. But we may put a cypher on 824 the right of the dividend, without altering its value. 00 (§ LVIII.) The decimal places will then be equal, and the quotient 40 will be whole numbers. 4. At $30.5 pr. hhd., how many hhds. of molasses can be bought for $76.25? As the divisor has, here, one decimal place, I know that, until one decimal place in the dividend has been employed, the quotient will be whole numbers, and afterwards, decimals. A comma is placed here, after the 2, to show 30.5)76.2,5(2.5 610 1525 1525 the limit of whole numbers. The pupil will find it convenient to employ it. |