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0.08814 Principal 500. The Logarithm of which is
2.69897 Logarithm of the amount
2.78711 The natural number to 2.78711 is 612.5, the amount required.
PROOF TO EXAMPLE 1.
7 rate per cent.
8572.45 principal 3d year. 5001 principal.
7 rate per cent. 535 amount of 1st year:
40.07.15 interest 3d year. 8535 principal 2d year.
572 45 principal.
8612 5215 amount.
EXAMPLE 2. What is the compound interest of 6000 cents, for 4 years, at 5 per cent, per annum?
Difference or Logratio,
Leaves 1293 cents for interest, or answer.
PROOF OF EXAMPLE 2. 6000 cents, principal.
6615 cts. principal for 3d year. 5 rate per cent.
5 rate per cent.
300100 interest for 1st year. 6000 principal.
330175 interest for 3d year. 6615 principali
6300|00 amount for the first year. 694575 amount for 3d year. 6300 principal for 2d year. 6945.75 cts. principal for 4th yr. 5 rate per cent.
5 rate per cent.
315/00 interest for 2d year. 6300 principal.
347/28.75 interest for 4th year. 694575 principal
Logarithm Tables may be had at book-stores; they are printed and bound like a pamphlet. They are also inserted in books on Surveying, Navigation, &c.
Mr. Chapman's patent sliding Interest Table, is an ingenious piece of of work, and very convenient in computing simple interest.
RATE-BILL. If the value of the real and personal estates, in any city or town, be rated at 34562 dollars, and a tax of 565 dollars be imposed thereon, it is required to ascertain how much that will be on a dollar?
34562 : 565.00000000 :: 1
DIRECTIONS. 1. Find how much this tax will be on a dollar, by making a sfatement in Proportion as above; by that we find it will be .01634743 of a dollar; but in order to make the result great enough to prove the work after calculating several different shares, we will alter the last figure 3 in the de cimal number .01634743 and make it a 4. This method will cause us to gain a small fraction rather than to lose.
2. Multiply this decimal number, or common multiplier, or ratio if you please, by 2, by 3, by 4, by 5, by 6, by 7, by 8, by 9, and by 10. Place these products under each other for a scale, which will enable us to calculate any number or share by inspection.
OPERATION AND EXAMPLE.
2 3 4 5 6 7 8 9
X X X X X O
.0 .0 .0 .0 .0 .0 .1 .1 .1 .1
1 6 3 4 3 2 6 9 4 9 0 4 6 5. 3
8 8 1 7 3 9 8 0 8 1 4 4 3 3 0 7 7 4 7
7 4 4
1 2 6 3 4 7
Now to elucidate this scale we will take .01634744 for the tax on one dollar ; .03269488 for the tax on two dollars; .04904232 for the tax on three dollars ; .06538976 for the tax on four dollars; .08173720 for the tax on five dollars; .09808464 for the tax on six dollars; .11443208 for the tax on seven dollars; .13077952 for the tax on eight dollars; .14712696 for the tax on nine dollars, and .16347440 for the tax on ten dollars.
3. When we want to know the tax on 20, move the deciinal point of No. 2, one figure to the right, thus, 0.3269438; on 30, move the deci. mal point of No. 3, one figure to the right, thus, 0.4904232 ; on 40, move the decimal point of No. 4, one figure to the right, thus, 0.6538976; &c. to ten.
4. For the tax on 100, move the decimal point of No. 1, two figures to the right, thus, 01.634744; or, move the point of 10 one figure to the right thus, 1.6347440; for the tax on 200 dollars, more the decimal point of No. 2, two figures to the right, thus, 03.269488; for the tax on 300 dollars, more the decimal point of No. 3, two figures to the right, thus, 04.904232; and proceed down with all the numbers in the same manner as the case may require. But, mind when coming to ten, that the moving of two figures to the right will make ten hundred, that is, 1000. This number 10 must be taken notice of particularly, or some great error may arise in calculation.
It may be remarked also, that, whoever undertakes to apportion as. sessments by this scale, ought to be, previously, acquainted with decimals: But, let us proceed.
5. If we want to know the tax on 1000 dollars, move the decimal point of No. 1. three figures to the right, thus, 016.34744;* on 2000 dollars, move the point of No. 2. three figures to the right, thus, 032.69488; on 8000, inove the decimal point of No. 8. three figures to the right, thus, 130.77952; and so for the rest of the numbers as occasion may require.
6. When the number of dollars to be taxed consists of tens, hundreds, thousands, or tens of thousands, as, 13440 the value of A's estate, hereafter mentioned, begin A's tax with ten thousand by inspecting the scale and moving the decimal point of No. 1. four figures to the right; or yoų may for the same tax move the point of No. 10. three figures to the right, then the tax on ten thousand dollars will be 0163.4744 or, 163.4744 the same in value.
This same tax may be taken from No. 10. by moving the decimal point only two figures to the right as before mentioned ; because 1 is already multiplied by 10, and ten multiplied by 100
7. Set 0163.4744 in a memorandum, thus, 0163.4744
Theu find the tax on three thousand by in- b. 049.04232 specting the scale at No. 3, and moving the c. 06.538976 decimal point three figures to the right and d. 0.6538976 place your work in the memorandum as at b.
Then look at the scale for four hundred, and e. 219.7095936 move the point of No. 4, two figures to the right, and place it in the memorandum as at c.
8. Now we want the tax on 40 dollars : look at No. 4. and move the point one figure to the right, and place it in the momorandum as at d.
Add these several sums in the memorandum for A's share of the tax, which will be $ 219.7095936 as at e, over A.
9. Now we will suppose that four men, A, B, C, and D, must raise the whole tax of 565 dollars: that the estate of A is worth 13440 dollars as before mentioned, that of B, 8651, that of C, 10452, and that of D, 2019 dollars; which several sums added together, produce a sum equal to that of their real and personal estates, viz. 34562 dollars.
10. In the next place find B's part of the tax: 8651 dollars is the amount of his estate; inspect the scale at No. 8. for 8000 dollars and move the decimal point three figures to the right, thus, 130,77952; place this number in a memoranduin as in the case of A, thus, 130.77952 ; then for 600, inspect No. 6. and move the point two f. 09.808464 figures to the right as at f; for 50 move the point g. 0.8173720 of No. 5. one figure to the right as at g; and for 1, h. .01634744 insert No. 1, as at h.
Thus we find B's share of the tax to bej. $ 141.42170344 $ 141.42170344 as atj, over B.
B. 11. C's estate is 10452 dollars. Inspect the scale at No. 1. and move the point four figures to the right, thus,
0163.4744; for 400, two figures to the right of No. 4. thus,
06.538976; for 50, one figure to the right of No. 5. thus,
0.8173720; for 2 insert No. 2. Thus,
.03269488: Thus the share of C, is
с 12. D's estate 2019 dollars :
Move the point at No. 2. three figures to the right for 2000 dollars, thus,
032.69488; for 10 dollars, copy No. 10. thus,
..16347440; for 9 dollars, copy No, 9. thus,
D's share of the tax is
Proof, $ 565.00022128 If we wish to know how much the tax would be on a cent or cents, move the decimal point two figures to the left by prefixing ciphers which is the same as dividing by 100.
EXAMPLES For the tax on one cent, prefix two ciphers to No. 1. thus, .00016347+; for two cents prefix two ciphers to No. 2: thus, .000326944; for three cents prefix two ciphers to No. 3. thus, .00049042+; and so on for the residue of the scale.
This rule is divided into two parts, Alligation Medial and Alligation Alternate. It signifies a linking together; it enables us to find the value or mean rate of a pound, an ounce, a bushel, a gallon, &c. of any mixture when several sorts are combined at different prices. And to the contrary, we find the several quantities to be mixed, when their prices or rates are given.
is to the amount of their combined value;
EXAMPLE 1. Suppose we mix 4 bushels of pease at $ 1. a bushel with 6 bushels of oats at 50 cents a bushel, 2 bushels of rye at $ 1.25 a bushel, 3 bushels of corn at 75 cents a bushel, and 8 bushels of buckwheat at 40 cents a bushel; what will a bushel of this mixture be worth?
Answer 65 cent.
OPERATION. 4 bushels
at 100 cents
of peas, of oats,
23 bushels make the Amount of their combined value, 1495 cents, sum total of the different articles.
A. Cents. A. Now say, As 23 articles, or bushels,
23 : 1495 :: 1 are to 1495 cents combined value; 23)1495(65 cents. So is 1 bushel,
138 to its particular value.
A. A. Cents. Say, As 23 articles,
1 :: 1495 are to 1 article;
23) 1495(65 cents. So are 1495 cents,
138 to 65 cents, the answer.
EXAMPLE 2. Mix 5 gallons of wine at $2. a gallon, with 8 gallons at $2.50 a gallon, 10 gallons at $3. a gallon, and 7 gallons at $4. a gallon; at what price can a grocer afford to sell this mixture by the gallon?
Answer, $2.99 and .33+ of a cent.
OPERATION. 5 gallons of Wine
at 200 cents
1000 cents, 8 do.
2000 10 do.
3000 7 do.
30 Sum total of the different articles.
8800 Amount of their combined value,