12 What number of pieces of merchandize, at 203s. per piece, are equivalent to 2404 pieces, at 124s. per piece answer 149177 pieces. 13 A lends to B 10031. for 63 months; what sum should Blend A for 3 years to requite his kindness answer 141 11s 9d. 155gr. 14 How many yards of cloth, at 81s. per yard, must be given for 26 yards, at 5 s. a yard? answer 17yds. 1qr. 3na. 14 THE DOUBLE RULE OF THREE IN VULGAR FRACTIONS. RULE. Prepare the terms, if necesary; then state, and work them agreeably to the directions given in whole numbers. Or, Invert the dividing terms, and multiply the upper figures continually for the numerator, and those below for the denominator, of the fractional answer. Note The Note to case 5, in Reduction, may be applied here. 94 EXAMPLES. 1 If of a yard of cloth that is tyd. wide cost L. what is the value of yard that is 14 yard wide, being ofthe same quality? Zyd. } 31. If yd. ૐ. Styd. X Zyd. S -708 } 7618+34=33=37.=13s 4d. answer. Or, 4×8×2×8×4 Cancelled. 3×f×8×8×4 =3L.=13s 4d. 2 If 9 students spend 103. in 18 days; what sum will 20 students spend in 30 days? 3 The labour of 3 men for 19 the same rate, what must 20 men days? answer 391 18s 4d.20 days comes to 81. at have for working 100 answer 3051 Os 3d. 39 8 4 If 5 persons drink 74 gallons of beer in a week, what quantity will serve 8 persons 22 weeks? answer 280 gallons. 5 Fourteen persons, upon examining into their expences for 20 weeks past, found they had laid out 4041. in what time, at the same rate, may 2031. be expended by 46 persons? answer 3 weeks. 6 If 13. in of a year gain 11. interest, what interest will 501. gain in of a year; and at what rate per cent. per annum? 12 answer 21 5s 1d. 24gr. at 105 per cent. 7 If 50l. in of a year gain 21 5s 1d. 23gr. in what time will 1331. gain 1. and at what rate per cent. per answer year, at 105 per cent. 8 When 12 persons use 14 pound of tea per month; how much should a family of 8 persons provide for year annum ? answer 41ib. 9 Two brothers at school compute the expence of their boarding, tuition, &c. for 2 of a year to be 5641. how much will the education of 3 sons for 5 years cost their father at that rate? answer 600l. DECIMAL FRACTIONS. DECIMAL Fraction is a part, or parts of a unit, A denoted by a point prefixed to a figure, or figures, thus, .4, .45, .456; the first figure after the point denotes 30 many tenths of a unit; the second so many hundredths of a unit, or tenths of one tenth, which are equal to, and 45 456 read as, 100, 1000: A mixt number consists of a whole number and a decimal; thus, 245, 789; which is, 245-78%. As whole numbers, counting from the right to the left, increase in a ten fold proportion; so decimals, counting towards the right, decrease in the same proportion; which is exemplified in the following. TABLE. C. of Millions C. of Thousands Thousands Hundredth Parts AX. Thousandth Parts -Tenth Part Thousandth Parts X. Millionth Parts Millionth Parts M. Millionth Parts -Units 98 6 5 4 3 2 1.1 2 3 4 9 Note. Ciphers, annexed to decimals, neither increase nor decrease their value; thus, ,25000, and ,25 are equal, but prefixed, decrease them in a tenfold proportion; thus ; .5, .05,.005, all express different decimals, and 50, 150, 1000. ADDITION OF DECIMALS. Place the numbers according to their value; viz. units under units, tenths under tenths, &c. and add as in addition of integers; putting the point in the sum total exactly under those in the example. EXAMPLES. What is the sum of 450-+31.47+376.004+1.08+456+ 76+.05 ?, anstver 1315.364 If 2476.8471 + 94.99.8941 + 867.05 +84.9 + 271.007+5.1008+1.6789 he added together, what is the sum ? answer 3811.3779 SUBTRACTION SUBTRACTION OF DECIMals. RULE. Place the number, as in addition, with the least under the greatest; and in the difference, set the point directly under those in the example. 1 From 100.17, take 84.476, what is left? answer 15.694 2 What is the difference between the sum of 841.46+ 109.62+34.691, and of 478.462×37.66+378.8 ? answer 90,849 MULTIPLICATION OF DECIMALS. RULE. Multiply as in integers, and point off as many decimal places in the product as are in both factors. Note 1. If the decimal places be wanting in the product, supply them with ciphers to the decimal point. 2. Multiplication in decimals may be contracted thus; Set the units figure of the multiplier under such place of the multiplicand as is to be the lowest retained in the product; and place all the remaining figures of the multiplier in an inverted order in multiplying, begin with the figure in the multiplicand which stands over the multiplying figure, adding the increase which may arise, by carrying one for the first five, and one more for every ten after, and place the products so, that the right hand figures stand under each other. EXAMPLES. 2 Multiply 79,347 by 23,15 facit 1836,88305 165,6995001296 ,00178600398 ,00008250704 7 Multiply 245,378263 by 72,4385, reserving 4 places of decimals in the product. facit 17774,8333 8 Multiply 674,4375 by 27,368, reserving only the integers in the product. facit 18458 9 Multiply 27,14986 by 92,41035, and retain 6 places of decimals in the product. facit 2508,928065 10 Multiply 184,8207 by 18,57493, and retain 3 places of decimals in the product. facit 2508,928 DIVISION OF DECIMALS. RULE. When the dividend has not as many decimal places as the divisor, or will not contain it, annex ciphers to supply the defect; then divide as in integers; and point off in the quotient, as many decimal places as the decimal places of the dividend exceed those in the divisor, Or, Let |