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, OPERATION. 267.496 26749.6 100 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 mul tiplied by 38.2175, in such a manner that it shall contain but four decimal places. OPERATION. 2.38645 38.2175 715935 190916 4773 239. 167 12 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. 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 86 201. 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. Ex plain the manner of multiplying two decimals together so as to retain a given number of places in the product. So, when we multiply by the last figure 5, we first say, 5 times 3 are 15, then 5 times 2 are 10 and 2 to carry, make 12, which is written down. Examples. 1. Multiply 36.74637 by 127.0463, retaining three decimal places in the product. 2. Multiply 54.7494367 by 4.714753, reserving five places of decimals in the product. 3. Multiply 475.710564 by .3416494, retaining three decimal places in the product. 4. Multiply 3754.4078 by .734576, retaining five decimal places in the product. 5. Multiply 4745.679 by 751.4549, and reserve only whole numbers in the product. DIVISION. 204. DIVISION OF DECIMALS is the operation of finding how many times one number is contained in another, when one or both are decimals 204. What is division of decimals? How is division performed? How does the number of decimal places in the dividend compare with those in the divisor and quotient? How do you determine the num ber of decimal places in the quotient? Give the rule for the division of decimals. 1. Divide the decimal .71505 by 2.043. ANALYSIS.-Division of decimals is performed in the same manner as division cf whole numbers. Since the dividend must be equal to the product of the divisor and quotient, it must contain many decimal places as both of them. Art. 200.) Therefore, OPERATION. 2.043).71505(35 6129 10215 10215 Ans. 0.35 There must be as many decimal places in the quotient as the number of decimal places in the dividend exceeds that in the divisor: Hence, Rule.-Divide as in simple numbers, and point off in the quotient, from the right hand, as many places for decimals as the number of decimal places in the dividend exceeds that in the divisor; and if there are not so many, supply the deficiency by prefixing ciphers. 7. Divide .051 by .012. 9. Divide 1.05 by 14. 75.15204, divided by 3? By 13. What is the quotient of 3? By .03? By .003? By .0003. 14. What is the quotient of 389.27688, divided by 8? By .08? By .008? By .0008? By .00008? 15. What is the quotient of 374.598, divided by 9? By .9? By .09? By .009? By .0009? By .00009? 16 What is the quotient of 1528.4086488, divided by 6? By .06? By .006? .0006? By .00006? By 000006? 17. Divide 17.543275 by 125.7. 18. Divide 1437.5435 by .7493 19. Divide .000177089 by .0374.- 25 If 25 men remove 154.125 cubic yards of earth in a day, how much does each man remove? 26. If 167 dollars 8 dimes 7 cents and 5 mills be equally_ divided among 17 men, how much will each receive? 27. Bought 45.22 yards of cloth for $97.223: how much was it a yard? 28. If 375.25 bushels of salt cost $232.655, what is the price per bushel? 29. At $0.125 per pound, how much sugar can be bought for $2.25 ? 30. How many suits of clothes can be made from 34 yards of cloth, allowing 4.25 yards for each suit? 31. If a man travel 26.18 miles a day, how long will it take him to travel 366.52 miles? 32. A miller wishes to purchase an equal quantity of wheat, corn, aud rye; he pays for the wheat, $2.225 a bushel; for the corn, $0.985 a bushel; and for the rye, $1.168 a bushel; how many bushels of each can he buy for $242.979? 33. A farmer purchased a farm containing 56 acres of woodland, for which he paid $46.347 per acre; 176 acres of meadow land at the rate of $59.465 per acre; besides which there was a swamp on the farm that covered 37 acres, for which he was charged $13.836 per acre. What was the area of the land; what its cost; and what was the average price per acre? 34. A person dying has $8345 in cash, and 6 houses, valued at $4379.837 each; he ordered his debts to be paid, amounting to $3976.480, and $120 to be expended at his funeral; the residue was to be divided among his five sons in the fol |