19. Two men divided a lot of wood which they purchased together for 27 dollars; one took 5 cords; the other 8 cords; what ought each to pay y? 20. The main-spring of a watch weighs about 1 dwt. 12 grs., Troy weight; estimating its worth at of a dollar, what would a pound Troy of steel be worth after it was manufactured into watch main-springs, allowing nothing for waste in manufacturing? 21. A hair-spring of a watch weighs of a grain, Troy; estimating its value at 3 cents, what would be the value of 1 lb. Troy, of steel, made into hair-springs, allowing nothing for waste? 22. Two men hired a horse one week for 6 dollars; one rode him 70 miles; the other, 84; how much ought each to pay? 23. A stack of hay is bought by two men for 761⁄2 dollars, to be paid for in proportion to the amount of hay each one takes; one takes 34 tons, the other the remainder, which was 2 tons; how much ought each to pay? SECTION XXI. DECIMAL FRACTIONS. [See Section XII. Part I.] Addition and Subtraction. In setting down the numbers, place those of the same order under each other, as units under units, tenths under tenths. Examples. 1. 24.5+68.3+17.14+87.96+3.125. 3. 450.61+27.134+89.4216+.984. 4. 64.25+3.125+87.25+181.7. 10. 648.62-.541-.891; 346.4-91.324. 11. 5.1-1.324; .5-.0067. 12. .81-.126; .94—.3816. Multiplication of Decimals. Sec. XII. Part. I. 13. 124.3X87; 321.67X24.3. 14. 97.125X6; 31.4×.125. Division of Decimals. Sec. XII. Part I. 21. 84.012; 965÷÷.15. 22. 1.65-15; 846÷3.4. 23. 1640.96; 425÷.055. 24. 1.001; 2÷÷0002. 25. .0012; 384÷0012. SECTION XXII. REDUCTION OF VULGAR FRACTIONS TO DECIMALS. [See Section XIII. Part I.] Examples. 1. Reduce to a decimal. 2. Reduce to a decimal. 3. Reduce to decimals 4. Reduce to decimals ; 3. ; ; T6• 5. Reduce to decimals ; t ; 18. If the fractions are reducible to decimals without a remainder, obtain the answer exactly; if they are irreducible, obtain the proximate answer to four places, and annex the fractional remainder. In order to know if a fraction is exactly expressible in decimals, see Section XIII. Part I, as directed above. 6. Reduce to decimals 32; 3; 31. 7. Reduce to decimals; 8. Reduce to decimals; ; 34. 9. Reduce to decimals; 14; 33. 10. Reduce to decimals; rta; 6šr• In ordinary transactions it is usual to carry the decimal answer to three or four places; the remainder is then so small in value that it may be dropped as of no importance. At whatever place you stop, however, the decimal obtained, and the fractional remainder, when added together, will exactly equal the original fraction. 11. In order to show this, we will take. Reducing it, we obtain at the first step 1 tenth 7)10 of 1 tenth; 1+3 adding these,+%=48=4, which is the original faction. We now carry the reduction one step further, 7)100, 14+7 we obtain 14 hundredths,+ of a hundredth. Adding these, 70%+780488=4, the original faction. 7)1000 142+. We obtain 142 thousandths+ of a thousandth. Adding these by using the common denominator 7000, +7000=1888 =, the original fraction. We will carry the reduction one step further; 12. Reduce to a decimal, of one figure, with the remainder; carried to 2 places, with the remainder; carried to 3 places, with the remainder. 13. Reduce 14. Reduce to a decimal of 7 places. 15. Reduce to a decimal of 10 places. Repeating and Circulating Decimals. When a fraction is irreducible, the decimal figure will either repeat, as 1.333+; or the decimal figures obtained by the partial reduction will, after a time, recur again, in the same order as at first. Thus, gives .090909+ and so on, without end. When the same figure is repeated continually, it is called a repeating decimal; when the same series of different figures recurs, it is called a circulating decimal. SECTION XXIII. REDUCTION OF DENOMINATE INTEGERS TO DECIMALS. 1. Reduce 5s. 11d. to the decimal of a £. First, reduce the quantity to the vulgar fraction of a £; then reduce that vulgar fraction to a decimal. 2. Reduce 3s. 24d. to the decimal of a £. SECTION XXIV. TO FIND THE INTEGRAL VALUE OF DENOMINATE DECIMALS 1. What is the value of .7 of a rod? Supposing the quantity was 7 rods, its value in feet would be found by multiplying it by 163; 16X7-115, or 115.5; but it was not 7 rods, but 7 tenths, of a rod, whose value we wish to find; the answer obtained, therefore, is 10 times too large; dividing by 10, it is 11.55,-11 feet and 55 hundredths. In order to find the value in inches of 55 hundredths of a foot, we will call it 55 feet; the answer is, 55×12=660 – 660 feet; but, as we regarded the 55 as 100 times greater in value than it is, the answer is 100 times too large; dividing it by 100, the answer is 69.60 inches, hundredths, or 6 tenths. 6 inches and 60 The above analysis shows the nature of the operation in all cases. 2. What is the value, in feet and inches, of .3 of a rod? 3. What is the value of .94 of a rod? 4. What is the value of .26 of a rod? 5. How many shillings and pence are there in .65 of a £? 6. How many shillings and pence are there in .8 of a £? 7. How many pence are there in .7 of a shilling? 8. How many pence are there in .16 of a shilling? 9. What is the value of .19 of a £? 10. What is the value of .74 of a bushel? 11. What is the value of .9 of a bushel? 12. What is the value, in rods and feet, of .7 of an acre? 13. What is the value of .9 of an acre ? 14. What is the value of .12 of an hour? 15. How many minutes and seconds in .15 of an hour? 22. How many feet in .13 of a cord? SECTION XXV. PRACTICAL EXAMPLES. 1. Add $1.50+$.375+$.0625+$.1875+$5.00. 2. Add $34.75+$6.00+$.375+$.08. 3. A man had $50, and spent $.375 of it; how much had he left? 4. A man had $10.00, and spent $.875 of it; how much had he left? 5. A watch cost $45.675; the chain and key, $4.845; what Idid the whole cost? 6. The owner then sold the watch, chain, and key, for $48.375; how much did he lose? 7. A man set out on a journey with $10.00; the first day he spent $1.125; how much had he left? |