the first; and the third put in as much as the other two: what was the whole amount of capital invested? 73. A farmer raised in one field 240bush. 3pks. 2qts. of wheat; in another 97bush. 6qts.; in another 42bush. 1pk.: how much did he raise in the three fields? 74. Add together three hundred dollars, ten eagles, forty dimes, ninety-six cents, seven mills, nine dollars, forty-seven cents, five mills, four eagles, three dollars, and nine dimes. 75. What is the sum of £17 10s. 6d.; £25 4s. 10дd.; 18s. 6d. 3far.; £11 94d.; £1 18s.; 21s. 61d.? 76. The Deluge, according to Chronology, occurred 1656 years after the creation; the call of Abraham 427 after the Deluge; the departure of the Israelites, 430 after the call of Abraham; the foundation of the temple, 479 after the departure of the Israelites; the end of the captivity, 476 after the foundation of the temple; and the birth of Christ, 536 years after the end of the captivity: how many years from the creation to the present time, it being the year 1856? SUBTRACTION. 45. THE DIFFERENCE between two numbers is such a number as, added to the less, will give the greater. If the numbers are unequal, the larger is called the minuend, and the less the subtrahend. If they are equal, either is the minuend and the other the subtrahend. Their difference, whether they are equal or unequal, is called the remainder. SUBTRACTION is the operation of finding the difference between two numbers. 1. From 869 take 327; that is, from 8 hundreds 6 tens and 9 units, take 3 hundreds 2 tens and 7 units. 45. What is the difference between two numbers ? When the numbers are unequal, what is the larger number called? What is the less called? What is their difference called? What is Subtraction? Give the rule for finding the difference between two numbers. ANALYSIS.-Beginning with the right hand figure, we take units from units; then, tens from tens; then, hundreds from hundreds, and find the remainder to be 542. 2. From 624 take 395. OPERATION. 869 minuend. 327 subtrahend. 542 remainder. ANALYSIS.-Having written down the numbers, we subtract 3 from 4, and find a remainder 1. At the next step we meet a difficulty, for we cannot subtract 9 tens from tens. Take 1 hundred = 10 tens, from 6 hundreds and add it to 2 tens. Then 9 tens from 12 tens leaves 3 tens, and 3 hundreds from 5 leaves 2 hundreds, and the remainder is 231. 10 OPERATION. 624 units. ♡ tens. hun. 5 12 4 393 3 9 1 231 3 2 3 1 The remainder can be found by adding, mentally, 10 to 2 tens, and then saying, 9 from 12 leaves 2 tens; then adding 1 to 3 hundreds, and say, 4 from 6 leaves 2 hundreds. The process of adding 10 to a figure of the minuend and returning 1 to the next figure of the subtrahend, at the left, is called borrowing. 3. From 67. 14cwt. 2qr. 20lb. 12oz., take 47. 17cwt. 1qr. 217b. 10oz. ANALYSIS.-Taking 10oz. from 12oz., 2oz. remain. At the next step we find a difficulty, for 21lb. cannot be taken for 20lb. We then take 1qr. = 25lb. from 2qr. and add it to 20lb., making 45lb.; then say, 21lb. from 45lb. leaves 24lb.; we then add 1 to the next left hand figure of the subtrahend, and say 2qr. from 2qr. leaves 0; then 17cwt. from 34cwt. leaves 17cwt, and 5 from 6 leaves 1 ton. T. cut. gr. lb. 02. 20 25 Hence, to find the difference between two numbers, I. Set down the less number under the greater, so that units of the same value shall fall in the same column. II. Begin with the units of the lowest denomination and subtract each from the units above it. III. When the units of any denomination in the subtrahend exceed those of the same denomination in the minuend, suppose so many units to be added as make one unit of the next higher denomination; after which add 1 to the next denomination of the subtrahend, and subtract as before. FIRST PROOF. 46. The difference or remainder, is such a number as added to the subtrahend, will give a sum equal to the minuend (45); hence, Add the remainder to the subtrahend. If the work is right, the sum will be equal to the minuend. SECOND PROOF. Since the remainder added to the subtrahend is equal to the minuend (Art. 45), it follows that the excess of 9's in these two numbers is equal to the excess of 9's in the minuend; hence, Find the excess of 9's in the minuend, in the subtrahend and in the remainder; if the work is right, the excess of 9's in the two last numbers will be equal to the excess of 9's in the first. What is the difference between 874136 and 45302 ? 47. What is the difference between 426 and 295 ? By the common method, which is spelling, we say, 5 from 6 leaves 1; 9 from 12 leaves 3 1 to carry to 2 are 3; 3 from 4 leaves 1. OPERATION. 426 295 131 By reading the words which express the final result, we make the operations mentally, and say, one, three, one. 46. What is the difference or remainder? How do you prove Subtraction? How do you prove Subtraction by the second method? 47. Explain the process of reading in Subtraction. TIME BETWEEN DATES. 48. What time elapsed between the inauguration of Mr. Jefferson, March 4th, 12 o'clock, M., 1801, and July 4th, 3 P. M., 1855 ? ANALYSIS.-Place the earlier date under the later, writing the number of the year, reckoned from the beginning of the Christian Era, on the left. Then, write in the same line the num OPERATION. yr. mo. da. hr. 1855 7 4 15 3 4 12 1801 ber of the month, reckoned from the first of January, the number of the day, reckoned from the first of the month, . the number of the hour, reckoned from 12 at night, and write the number of minutes and seconds, if there are any, still to the right. Hence, to find the time between two dates, Write down as above, and subtract the earlier date from the latter. NOTE 1.—In finding the difference between dates, as in casting interest, the month is regarded as the twelfth part of the year and as containing 30 days. 2. The civil day begins and ends at 12 o'clock at night. 48. How do you find the difference of time between two dates? In this computation, what part of a year is a month? How many days are reckoned to the month? |