EXAMPLE. Reduce 7s. 8d. to the fraction of £3 7s. 8±d. – 185 halfpence, and £3 = 1440 halfpence, and .128472. Ex. 14. Find the value of the following expressions :— 178. We have seen that the processes of Addition, Subtraction, Multiplication, and Division of Decimal Fractions are precisely similar to those of the simple rules of integral arithmetic. This arises from the fact that the notation of decimal fractions is constructed on the same base or scale as that of whole numbers. It would be the same with concrete arithmetic, represented by the various tables of money, weights, and measures, if all those tables were constructed on the same decimal base. With a uniform decimal scale for concrete as well as for abstract quantities, the student of arithmetic would find no more difficulty in working compound than in working simple arithmetic. A system of this kind is in use in France, and a description of the French system is given in the second part of this work. It is not at all improbable that sooner or later a similar system will be introduced into England; and it is generally considered that a trial of it might be made first in connection with our coinage. Several plans have been proposed, the adoption of any one of which would necessitate the introduction of new coins, and the abolition of several or all of our current coins. The plan which will probably be adopted proposes to retain the £ as the principal unit, the two-shilling piece, or florin, as the second, and to introduce new coins for the tenth and the hundredth part of the florin, to be called, respectively, the cent and mil. With this system the money table would be— 10 mils =1 cent. 10 cents. =1 florin. It would not be necessary to write down the names of the units, for they will be indicated by the positions of the figures. Thus £3.4751 will be read £3 4fl. 7c. 5.1m. The rule for reducing a sum of money expressed in the current coinage to this proposed decimal coinage, will be the same as that for reducing a sum of money to the decimal of a £.* Ex. 15. Express as £'s, florins, cents, and mills £ S. d. £ S. d. £ 8. d. For an abbreviated method, see "Approximate and Contracted Rules," in part ii. of this work. £ fl. c. m. £ fl. c. m. (7) 7 8 6 17 Miscellaneous Questions on Decimals. (1) If 625 of an article cost £13 16s. 1ąd, what will ⚫125 cost? (2) The circumference of a circle is 3.14159 of the diameter: find the circumference of a circle whose diameter is 14.25 feet. (3) What is the difference between the circumference of two circles, the diameters of which are respectively 2.6 yds., and 11.75 feet? (4) A French metre is 39-371 English inches: express in metres 2 miles 3 fur. 8 ps. (5) The length of a seconds pendulum in London is 39-1393 inches. Express this as a fraction of a foot, of a yard, and of a furlong. (6) The imperial gallon contains 277-274 cubic inches: what are the solid contents of 75 of a pint and of a gill ? (7) The year consists of 365-24224 days. In what time would the error arising from taking it as 365 days amount to an error of one day? (8) A owns 583 of an estate, and B the rest. of B's portion is £500 less than A's. What is the worth of the whole estate? (9) 025 of a certain sum is £22 15s. 4ğd.: what is the sum ? (10) After cutting from an apple 346 and then 4756 of the whole, what fraction is left? (11) There are two fractions whose sum is 7.54, and whose difference is 1.53: find the fractions. (12) Show that a vulgar fraction can be reduced to a decimal which is either finite or repeating. (13) Show that ·4=·400=·40000, and that 7·463=7+fʊ +1880=7488. (14) A house and land cost £762-275. The house was pulled down, and the materials sold for ·025 of the first outlay. A new house cost building 12-3 times the sum for which the old materials were sold. What was the whole cost of this new house and land? (15) 18-25 tons of coal are bought at 20 shillings a ton, and 27.125 tons at 18:375 shillings a ton. The whole is sold at the average price of 945 shillings a cwt. How much was gained by the transaction? (16) Divide 3.765 into two parts, such that one is ⚫96874 less than the other. (17) Reduce 3.675 of a lb. to the decimal of '07625 of a ton. (18) By selling a book for 8s. 6d., 2142857 of the cost price was gained: what was the cost of the book? (19) If 30 men reap 374-4 acres in 11.25 days, how many acres will they reap in a day? How many acres in 5.5 days? How many will one reap in 5.5 days, and how many will 20 men reap in 5.5 days? (20) If 75 men can dig 287-82 yds. of a trench in 5.85 days, find how long it would take 100 men to dig 600-54 yds. ? Work this sum by taking steps similar to those indicated in the previous question. (21) State the rule for Division of Decimals. Find by what number 8.25 must be multiplied to produce the difference between 6.746 and 23. (22) A clock gains 075 hours in a day. It is set right at midday on Saturday; what will be the true time when it is 6 p.m. by the clock on the Saturday following? (23) A crow flies at the rate of 5 miles in 4 minutes, and a train travels at the rate of 42.75 miles in 75 hours, how many times as fast as the train does the crow fly? (24) A dishonest tradesman uses a false balance, such that ⚫857142 of a lb. at one end balances 1 lb. at the other. He weighs 154 lbs. of tea @ 3·75s. per lb. Then his assistant puts the weight in the other scale, and weighs out 150 lbs. what has the grocer gained? (25) Show that (1) When a denominator of a vulgar fraction contains no other factor than 2 or 5, the equivalent decimal will be finite. (2) When the denominator contains neither 2 nor 5, the resulting decimal will repeat, the repeating period commencing immediately. (3) When the denominator contains 2 or 5 and other factors, the resulting decimal will repeat, but the repeating period will not commence immediately. Takes as examples,, and }, and make six others. (26) Find the G.C.M. and L.C.M. of 4.2237 and 755-82. (27) Two men, A and B, start together from Hyde Park corner to walk to Richmond and back. A walks at the rate of 7.46 miles in 13 hours, and B at the rate of 11-428571 miles in 24 hours; after walking 2 hours they meet. Find the distance between Hyde Park and Richmond. K 2 Practice. 179. Practice is a method of finding by the help of fractions the value of a number of articles when the price of one is given. Thus 3, 6 180. When one number is a multiple of another the second is termed an Aliquot Part of the first. 8, 4, 12 are all aliquot parts of 24. 181. Practice may be Simple or Compound. SIMPLE PRACTICE. 182. In Simple Practice the price of one unit of a certain denomination is given, and the value of a number of these units is required. 183. RULE.-Multiply the number of articles by the number of pounds, shillings, or pence, and take aliquot parts for the rest. 184, Case I.—When the price is less than 1d. EXAMPLE. Required the cost of 754 pencils, at ad. each. 185. Students are advised, when working the exercises, to write out in full all the particulars indicated in the above example. The first column shows of which line in the sum the part is to be taken; namely, in the first case, of the line which is the cost at 1d., and in the second case, of the line which is the cost at d. |