11. Take 75.304 from 175.01. 12. Required the difference between 17.541 and 35.49. 13. Required the difference between 7 tenths and 7 millionths. 14. From 396 take 67 and 8 ten-thousandths. 15. From 1 take one-thousandth. 16. From 6374 take fifty-nine and one-tenth. 17. From 365.0075 take 5 millionths. 18. From 21.004 take 98 ten-thousandths. 19. From 260.3609 take 47 ten-millionths. 20. From 10.0302 take 19 millionths. 21. From 2.03 take 6 ten-thousandths. 22. From one thousand, take one-thousandth. 23. From twenty-five hundred, take twenty-five hundredths. 24. From two hundred, and twenty-seven thousandths, take ninety-seven, and one hundred twenty ten-thousandths. 25. A man owning a vessel, sold five thousand seven hundred sixty-eight ten-thousandths of her: how much had he left? 26. A farmer bought at one time 127.25 acres of land; at another, 84.125 acres; at another, 116.7 acres. He wishes to make his farm amount to 500 acres: how much more must he purchase? 27. Bought a quantity of lumber for $617.37, and sold it for $700 how much did I gain by the sale? 28. Having bought some cattle for $325.50; some sheep for $97.12; and some hogs for $60.87; I sold the whole for $510.10: what was my entire gain? 29. A dealer in coal bought 225.025 tons of coal: he sold to A, 1.05 tons; to B, 20.007 tons; to C, 40.1255 tons; and to D, 37.00056 tons: how much had he left? 30. A man owes $2346.865: and has due him, from A, $1240.06; and from B, $1867.98: how much will he have left after paying his debts? MULTIPLICATION. 200. MULTIPLICATION of decimals is the operation of taking one number as many times as there are units in another, when one or both of the factors contain decimals. Rule.-Multiply as in simple numbers, and point off in the product, from the right hand, as many figures for deci mals as there are decimal places in both factors; and if there be not so many in the product, supply the deficiency by prefixing ciphers. PROOF. The same as in whole numbers. Examples. 1. Multiply 2.125 by 375 thousandths. 5. Multiply 473.54 by 57 thousandths. 6. Multiply 137.549 by 75 and 437 thousandths. 7. Multiply 3, .7495, and 73487, together. 200. What is multiplication of decimals? After multiplying, how many decimal places will you point off in the product? When there are not so many in the product, what do you do? Give the rule for the multiplication of decimals. 8. Multiply .04375 by 47134 hundred-thousandths. 12. Multiply .375494 by 574 and 375 hundredths. 14. Multiply 300.27 by 62. 15. Multiply 93.01401 by 10.03962. 20. Multiply thirty-six thousand by thirty-six thousandths. 21. Multiply 125 thousand by 25 ten-thousandths. 22. Find the product of 50 thousand by 75 ten-millionths. 23. Find the product of 48 hundredths by 75 ten-thousandths. 24. What are the contents of a lot of land, 16.25 rods long, and 9.125 rods wide? 25. What are the contents of a board, 12.07 feet long, and 1.005 feet wide? 26. What will 27.5 yards of cloth cost, at .875 dollars per yard? 27. At $25.125 an acre, what will 127.045 acres of land cost? 28. Bought 17.875 tons of hay, at $11.75 a ton: what was the cost of the whole? 29. A gentleman purchased a farm of 420.25 acres, at $35.08 an acre; he afterwards sold 196.175 acres to one man at $37.50 an acre, and the remainder to another person, at $36.125 an acre: what did he gain? 30. A merchant bought two pieces of cloth, one containing 37.5 yards, at $2.75 a yard, and the other, containing 27.35 yards, at $3.125 a yard; he sold the whole at an average price of $2.94 a yard: did he gain or lose by the bargain, and how much? CONTRACTIONS IN MULTIPLICATION. 201. CONTRACTIONS, in the multiplication of decimals, are short methods of finding the product. CASE I. 202. To multiply by 10, 100, 1000, &c. 1. Multiply 267.496 by 100. ANALYSIS.-Removing the decimal point one place to the right, increases the value of the decimal ten times; removing it two places, one hundred times, &c. To multiply by 10, 100, &c., we' remove the decimal point as many places to the right as there are ciphers in the multiplier: hence, Rule.-Remove the decimal point as many places to the right as there are ciphers in the multiplier; annexing ciphers, if necessary. Examples. 1. Multiply 479.64 by 10; also, by 100. CASE II. 203. To multiply two decimals, and retain in the product a certain number of decimal places. 1. Let it be required to find the product of 2.38645 multiplied by 38.2175, in such a manner that it shall contain but four decimal places. OPERATION. 2.38645 38.2175 715935 190916 4773 239 ANALYSIS.-Write the unit figure of the multiplier under that place of the multiplicand which has the same number, counted from the decimal point, as the figures to be retained in the product, and write the other figures in their proper places. Now, the product of the unit figure of the multiplier, by the figure of the multiplicand directly over it, will have the unit value of the required product. The product of the next figure at the right, in the multiplicand, by the tens of the multiplier, will also give a product of the required unit value; and the same will be true for the product of any two figures equally distant from the unit figure of the multiplier and the figure of the multiplicand directly over it. 167 12 91.2042 In regard to the decimals, we observe, that the tenths multiplied by the figure at the left of the one standing over the unit figure of the multiplier, will give a product of the required unit value; and the same will be true for any two figures equally distant from the decimal point and from the figure standing over the unit place of the multiplier. We therefore begin the operation with the highest unit figure of the multiplier, and the corresponding figure of the multiplicand, and then multiply in succession by the figures at the right. We must remember that the whole of the multiplicand should be multiplied by every figure of the multiplier. Hence, to compensate for the parts omitted, we begin with one figure to the right of that which gives the true unit, and carry one when the product is greater than 5 and less than 15; 2, when it falls between 15 and 25; 3, when it falls between 25 and 35; and so on for the higher numbers. For example, when we multiply by the 8, instead of saying 8 times 4 are 32, and writing down the 2, we say first, 8 times 5 are 40, and then carry 4 to the product 32, which gives 36. 201. What is contraction in the multiplication of decimals ?—202. How do you multiply by 10, 100, &c? If there are not as many decimal places in the product as there are ciphers, what do you do?-203. Explain the manner of multiplying two decimals together so as to retain a given number of places in the product. |