123. 1. What is the difference between 43.25 rods and 22.5 rods? 43.25 Ans. 20.75 rods. We write down the numbers as for Addition, with the largest uppermost. As there are no hundredths in the subtrahend, we bring down the 5 hundredths. Proceeding to the 10ths, we are unable to take 0.5 from 0.2; we therefore borrow a unit from the 3 units, which being 10 tenths, we join 10 to the 2, making. 12 tenths; from which we take 5 tenths, and write the remainder, 7 tenths, in the place of tenths below the line. The rest of the operation must be obvious. 2. From 24 hours take 18.75 hours, what remains? 24. Ans. 5.25 h. Here, as we cannot take 5 from nothing, we borrow 0.10 from the 4 units, or 400 hundredths; then taking 5 (0.05) from 0.10, the remainder is 0.05. The 400 hundredths has now become 390 hundredths, or 39 tenths, or 3.9; then 0.7 from 0.9 leaves 0.2, and so on. RULE. 124. Write down the numbers as in Addition of Decimals, observing to place the largest number uppermost. Beginning at the right, subtract as in Simple Subtraction, (99) and place the decimal point in the remainder directly under those in the given numbers. NOTE 1.-When the numbers are all properly written, and the results correctly pointed, the decimal points will all fail in one vertical column, or directly under one another, both in Subtraction and Addition. NOTE 2-In numbers given for Addition or Subtraction, the decimal places may all be made equal by annexing ciphers to a part of them,(116) without altering their value, and then all the decimals will express similar parts of a unit, or be of the same denomination. ANALYSIS. 125. 1. If 14.25 lb. of butter be divided into 3 equal shares, how many pounds will there be in each? 3) 14.25 (4.75 12 22 21 15 15 Here we wish to divide 14.25 into two factors, one of which shall be 3, and the other such a number as, multiplied by 3, (101) will produce 14.25. We first seek how many times 3 in 14, and find it 4 times, and 2 units over. The 2 units being 20 tenths, we join them to the 2 tenths, making 22 tenths, and, dividing these by 3, the quotient is 0.7, and 0.1 over: but 0.1 being 0.10, (116) we join the 1 to the 5 hundredths, making 0.15, and dividing by 3, the quotient is 5 hundredths. The whole quotient then is 4.75 lb. To prove that this is the true quotient, we multiply it by the divisor, 3, (4.75×3=14.25), and reproduce the dividend. Since any dividend may be regarded as the product of the divisor and quotient taken as factors (101), and since the product must have as many decimal places as are contained in both the factors (121), it follows, that the number of decimal places in the divisor and quotient, counted together, must be just equal to the number of decimal places in the dividend. 126. 2. If 18 bushels of wheat be divided equally among 4 men, how much will each receive? Here we find that 18 bushels will give each man 4 4) 18 (4.5 bu. bushels, and that there will be 2 bushels left. We now 16 20 20 add a cipher to the 2, which multiplying it by 10, (91) reduces it to tenths, and dividing 20 tenths by 4, the quotient is 0.5; each man will, therefore, receive 4.5 bushels. Hence by annexing ciphers to the remainder of a division, the operation may be continued, and in pointing the result, the ciphers annexed are to be regarded as decimals belonging to the dividend. 127. 3. What is the quotient of 0.0084 by 0.42 ? 84 Omitting the ciphers, we find 42 in 84 just 2 0.42 )0.0084 (0.02 Ans. times; but since there are 4 places of decimals in the dividend, and only 2 in the divisor, there must be 2 places also in the quotient: we therefore place a cipher at the left of the 2 in the quotient, between it and the separatrix, to make up the deficiency. We see by this example that if a quantity be divided by a decimal, the quotient will be larger than the dividend. RULE. 128. Write down the divisor and dividend, and divide as in whole numbers. Point off as many places for decimals from the right hand of the quotient, as the decimal places in the dividend exceed those in the divisor. NOTE 1.-If there are not so many figures in the quotient as the number of decimal places required, supply the deficiency by prefixing ciphers. 2. Should the decimal places in the divisor exceed those in the dividend, make them equal by annexing ciphers to the latter. 3. Whenever there is a remander after division, by annexing ciphers to it, one or more additional figures may be obtained in the quotient. (126) VULGAR FRACTIONS CHANGED TO DECI MALS. ANALYSIS. 129. If we divide an apple equally between 2 boys, the part which each will receive will be an apple, or the quotient of 1 divided by 2; if we divide 1 apple between 3 boys, each will receive, or the quotient of A divided by 3. In like manner, if 3 apples be divided between 4 boys. each boy will receive of an apple, or the quotient of 3 divided by 4, and generally a Vulgar, or Common Fraction, denotes the division of the numerator by the denominator. (22,103) The fraction, for example, denotes that 1 is divided by 2, but since 1 does not contain 2, the quotient is less than 1, and must therefore be expressed in parts of unity. Now if we add a cipher to the dividend, 1, it becomes 10 tenths, (126); and 10 tenths divided by 2, the quotient is 0.5. (125) Hence the decimal 0.5 is equiv alent to 1. Again, in the fraction 1. if we add a cipher to the 1, it becomes 10 tenths, as before, and 10 tenths divided by 3, the quotient is 0.3, and 0.1 remains. Joining a cipher to 0.1, it becomes 0.10, and dividing again by 3, the quotient is 0.03, and thus may we go on as far as we please, getting by each additional cipher a 3 in the quotient, which is 10 times less than the preceding, as 0.333-, which is the decimal expression for And again in the fraction 2, adding a cipher to 3, and dividing by 4, the quotient is 0.7, and 0.2 remain; adding a cipher to 0.2, and dividing again by 4, the quotient is 0.05;-0.75 then is the decimal expression for 3: And generally, 130. To change Vulgar Fractions to Decimals. RULE-Annex ciphers continually to the numerator, and divide by the denominator, so long as there shall be a remainder, or until the decimal be obtained to a sufficient degree of exactness. The quotient will be the decimal required; and it must consist of as many decimal places, as the number of ciphers annexed. If the quotient does not contain so many figures, make up the deficiency by prefixing ciphers.(127) 131. Having become familiar with the method of changing Vulgar Fractions to Decimals, whenever fractions occur, the pupil has only to substitute for them their equivalent decimal values, and proceed as if they had been given in decimals. illustrate this remark, take the following QUESTIONS FOR PRACTICE. 1. There are 3 pieces of cloth, one contains 4 yards, one 3 and the other 54 yds. how many yards in the whole? 44-4.5 32-3.75 5=5.25 Ans. 13.50-133. 2. There are 4 boxes, each of which contains 5% lb. of sugar; how many pounds in the whole? 58-5.375. Ans. 21.5 lb. 3. A person having 17 tons of hay, sold 67 tons; how much had he left? Ans. 10.925 tons. To By these examples it appears that a number is diminished by multiplication and increased by division, when the 4. What is the product multiplier and divisor are of 24 by +? fractions or decimals. 132. Federal Money is the established currency of the United States. Its denominations are all in a decimal or ten-fold proportion, as exhibited in table 1. page 31. The dollar is considered the unit money, and all the lower denominations |