WEIGHT (AVOIRDUPOIS)
Ounce. Pound. Stone. Quarter. Hundred Ton.
weight.
I is six T48 iAs sax6
r6 i rx ~'a iii a3 a
224 14 I 2 a SBII
448 28 2 1 s's
1792 II2 8 4
35840 2240 16o 8o 20 I
(Also 7000 grains = lib avoirdupois.)
120. Change of System.—It is sometimes necessary, when a quantity is expressed in one system, to express it in another,
1 See also WEIGHTS AND MEASURES.
I
convergents are L, q , 4 , ... the number can be expressed in the form pl (I+p Q2) (1Q2) (I {pQ) . . . , and that if we go up to the factor 1 5"090+1 the product of these factors
differs from the true value of the number by less than t 1 gngn tl
The following are the ratios of some of the units; each unit is expressed approximately as a decimal of the other, and their ratio is shown as a continued product (§ 116), a few of the corresponding convergents to the continued fraction (§ 117) being added in brackets. It must be remembered that the number expressing any quantity in terms of a unit is inversely proportional to the magnitude of the unit, i.e. the number of new units is to be found by multiplying the number of old units by the ratio of the
old unit to the new unit. gg~ qq y d~ Yard
Metre 790000 =10039= 12'3 84 '83 94"(1 1 ,ii 42J •
C I entimnchet re 2 1 5.402  730093 _ 5, 62 I 1 6&gi (, 3 ., 1529 0000— 7 — 6~5 955 2 3 333 520 ).
Mile
Kilometre _ — 16093 1 62 90.1 4 0=.Q.. 184•23G9.. • (9f 23, 1SI$).
1 0 0 0 0 _— 0
Square yard =1 ss1 0000_,.9_os,1_e s_. s si_ 2s_o.
Square metre 0000—Y1900—6 305 15249 • • • (0, 61, 299)•
Acre Hectare Quart Litre
Pound
Kilogramme _4 36_ — _I_0 52 0.00 _12, 11.01 ,I3 4,22 11 (1 1, T5r,9744 ,53' 500 T~00004"6—85 29392 •
(ii.) Special Applications._
121. Commercial Arithmetic.—This term covers practically all dealings with money which involve the application of the principle of proportion. A simple class of cases is that which deals with equivalence of sums of money in different currencies; these cases really come under § 120. In other cases we are concerned with a proportion stated as a numerical percentage, or as a money percentage (i.e. a sum of money per loo), or as a rate in the £ or the shilling. The following are some examples. Percentage: Brokerage, commission, discount, dividend, interest, investment, profit and loss. Rate in the £: Discount, dividend, rates, taxes. Rate in the shilling: Discount.
Textbooks on arithmetic usually contain explanations of the chief commercial transactions in which arithmetical calculations arise; it will be sufficient in the present article to deal with interest and discount, and to give some notes on percentages and rates in the £. Insurance and Annuities are matters of general importance, which are dealt with elsewhere under their own headings.
122. Percentages and Rates in the £.—In dealing with percentages and rates it is important to notice whether the sum which is expressed as a percentage of a rate on another sum is a part of or an addition to that sum, or whether they are independent of one another. Income tax, for instance, is calculated on income, and is in the nature of a deduction from the income; but local rates are calculated in proportion to certain other payments, actual or potential, and could without absurdity exceed 20S. in the £.
It is also important to note that if the increase or decrease of an amount A by a certain percentage produces B, it will require a different percentage to decrease or increase B to A. Thus, if B is 20% less than A, A is 25% greater than B.
123. Interest is usually calculated yearly or halfyearly, at a certain rate per cent. on the principal. In legal documents the rate is sometimes expressed as a certain sum of money " per centum per annum "; here " centum " must be taken to mean " £loo.,,
Simple interest arises where unpaid interest accumulates as a debt not itself bearing interest; but, if this debt bears interest, the total, i.e. interest and interest on interest, is called compound interest. If roor is the rate per cent. per annum, the simple interest on D. for n years is £nrA, and the compound interest (supposing interest payable yearly) is £[(1+r)°1]A. If n is large, the compound interest is most easily calculated by means of logarithms.
124. Discount is of various kinds. Tradesmen allow discount for ready money, this being usually at so much in the shilling or £. Discount may be allowed twice in succession off quoted prices; in such cases the second discount is off the reduced price, and therefore it is not correct to add the two rates of discount together. Thus a discount of 20%, followed by a further discount of 25%,gives a total discount of 40 %, not 45 %°, off the original amount. When an amount will fall due at some future date, the present value of the debt is found by deducting discount at some rate per cent. for the intervening period, in the same way as interest to be added is calculated. This discount, of course, is not equal to the interest which the present value would produce at that rate of interest, but is rather greater, so that the present value as calculated in this way is less than the theoretical present value.
125. Applications to Physics are numerous, but are usually only of special interest. A case of general interest is the measurement of temperature. The graduation of a thermometer is determined by the freezingpoint and the boilingpoint of water, the interval between these being divided into a certain number of degrees, representing equal increases of temperature. On the Fahrenheit scale the points are respectively 32° and 212°; on the Centigrade scale they are o° and loo°; and on the Reaumur they are o° and 8o°. From these data a temperature as measured on one scale can be expressed on either of the other two scales.
126. Averages occur in statistics, economics, &c. An average is found by adding together several measurements of the same kind and dividing by the number of measurements. In calculating an average it should be observed that the addition of any numerical quantity (positive or negative) to each of the measurements produces the addition of the same quantity to the average, so that the calculation may often be simplified by taking some particular measurement as a new zero from which to measure.
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