difference between 75 and 43. From an inspection of these examples, it will be seen that Subtraction is, in effect, the separating of the minuend into two parts, one of which is the subtrahend, and the other the remainder. Hence, to show the correctness of the operation, we have only to recompore the minuend by adding together the subtrahend and remainder. 96. 3. A person owed: 727 dollars, of which he paid 542 dollars; bow much remains unpaid? Here we take 2 from 7, and write the difference, 5, 727 dolis. below the line in the place of units. We now proceed 542 dolls. to the tens, but find we cannot take 4 tens from 2 tens. We may, however, separate 7 hundreds into two parts, Ains. 185 dolls. one of which shall be 6 hundred, and the other 1 hundred, or 10 tens, aird this 10 we can join with the 2, making 12 tens. From the 12 we now subtract the 4, and write the remainder, 8, at the left hand of the 5, in the ten's place. Proceeding to the hundreds, we must remember that 1 unit of the upper figure of this order has already been borrowed and disposed of; we must therefore call the 7 a 6, and then taking 5 from 6, there will remain 1, which being written down in the place of hundreds, we find that 185 dollars remain unpaid. 4. A boy having 12 chesnuts, gave away 7 of them; how many had be left? 12 Here we cannot take 7. units from 2 units; fore take the 1 ten=10 units, with the 2, making 12 units ; then 7 from 12 leaves 5 for the answer. 5 Ans. 97. 5. A man has debts due him to the amount of 406 dollars, and he owes 178 dollars; what is the balance in his favour? Here we cannot take 8 units from 6 units; we must therefore 406 borrow 10 units from the 400, denoted by the figure 4, which 178 leaves 390. Now joining the ten we borrowed with 6, we have the minuend, 406, divided into two parts, which are 390 and 16. 228 Taking 8 from 16, the remainder is 8; and then we have 390, or 39 tens in the upper line, from which to take 170, or 17 tens, Thus the place of the cipher is occupied by a 9, and the significant figure! je diminished by 1, making it 3. We then say, 7 from 9 there remains 2, which we write in the place of tens, and proceeding to the next place, say 1 from 3 there remains 2. Thus we find the balance to be 228 dollars. we must there 7 SIMPLE SUBTRACTION. 98. Simple Subtraction is the taking of one simple number from another, so as to find the difference between them. The greater of the given numbers is called the minuend, the less the subtrahend, and the difference between them the remainder. RULE. 99. Write the least number under the greater, with units under units, and tens under tens, and so on, and draw a line: helow. Beginning at the right hand, take each figure of the subtrahend from the figure standing over it in the minuend, and write the remainders in their order brlow. If the figure in the lower line be greater than the figure standing over it, suppose ten to be added to the upper figure, and the next sige nificant figure in the upper line to be diminished by 1, (96) regarding ciphers, if any come between; as Is,(97); or, wlrich gives the same result, suppose 10 to be added to the upper figure, and the next figure in the lower line to be increased by 1, with which proceed as before, and so on till the whole is finished. PROOF. 100. Add together the remainder and the subtrahend, and if the work be right, their sıımı will equal the minuend. QUESTIONS FOR PRACTICE. 6. In 1810, Montpelier con- 12. What number is that tained 1877 inhabitants, and which taken from 365, leaves in 1820, 2308 inhabitants; 159? Ans. 206. what was the increase, and in what time? 13. Supposing a man to 1820 2308 have been born in 1796, how 1810 1877 old was he in 1828? Ans. 32 years. Time 10 years 431 increase. 14. If a man have 125 head 7. Dr. Franklin died in / of cattle, how many will he 1790, and was 84 years old, have after selling 8 Oxen, cows, 9 steers and 13 heifersi in what year was he born? Ans. 1706. Ans. 84. 15. What number is that to 8. A man deposited 9000 which if dollars in a bank, of which he you add 643, it will become 1826? Ans. 1183. took out 112 dollars; how much remains in the bank? 16. How many years from Ans. 8888 dolls. the flight of Mahomet in 622, 9. If a man sell 29 out of a to the year 1828? Ans. 1206. flock of 76 sheep, how many 17. America was discoverwill there be left? Ans. 47. ed by Columbus in 1492; how 10. Sir Isaac Newton was many years since? porn in the year 1642, and 18. If you lend 3646 dollars died in 1727; how old was he and receive in payment 2998 when he died? Ans. 85 years. dollars, how much is still due.? 11. If you lend a neighbor Ans, 648 dolls. 765 dollars, and lie pay you 19. A owed B $4850, of at one time, 86 dollars, and at which he paid at one time another 125 dollars, how much $200, at another, $475, at an is still due? Ans. 554 dolls. other $40, at another $1200, and at another $150; what 21. Gunpowder was inremains due? Ans. 2779. vented in the year 1330; then 20. The sum of two num- how long was this before the bers is 64892, and the greater | invention of printing, which number is 46234: what is the was in 1441 ? smallest number? Ans. 18658. 25. From 3287625 5327467 7820004 12345678 Take 2343756 2100438 2780009 4196289 Ans. 111 years. 22. Rem. 943869 28. 48764+93—139=66. 4. DIVISION. ANALYSIS, 101. 1. Divide 24 apples equally among 6 boys, how many wil each receive? The most simple way of doing this would be, first to give each boy 1 apple, then each boy l'apple more, and so on, till the whole were distribated, and the number of l's, which each received, would denote bis share of the apples, which would in this case be 4. Or as it would take 6 apples to give each boy one, each boy's share will evidently contain as many apples as there are sixes in 24. Now this may be ascertained by subtracting 6 from 24, as many times as it can be done, and the number of subtractions will be the number of times 6 is contained in 24; thus, 246=18, 18—6=12, 6–12=6, and 6—6=0. Here we find that by performing 4 subtractions of 6, the 24 is completely exhausted, which shows that 24 contains 6 just 4 times. Now as Subtraction is the reverse of Addition, 94) it is evident that the addition of 4 sixes, (6+6+6+6=24) must recompose the number, which we have separated by the subtraction of 4 sixes. “But when the munibers to be added are all equal, Addition becomes Multiplication,(83; and 24 is therefors the product of 4 and 6, (4X6=24). A number to be divided, and which is called a dividend, is then to be regarded as the product of two factors, one of which, called the divisor, is given to find the other, called the quotient; and the inquiry how many times one number is contained in another, as 6 in 24, is the same as how many times the one will make the other, as how many times 6 will make 24, and both inust receive the same answer, viz. 4, Hence to prove Division, we multiply the divisor and quotient together, and il the work be right, the product will equal the dividend. 2. How many yards of cloth will 63 dollars buy, at 9 dollars a yard? 102, 103. DIVISION. .17 102. When the dividend does not exceed 100, nor the divisor exeeed 10, the whole operation may be performed at once in the mind: but when either of them is greater than this, it will be found most convenient to write down the nuinbers before performing the operation. 3. Divide-552 dollars equally between 2. men, how many dollars will each have? Here we cannot say at once: how many times 2 is con:2)552 tained in 552, we therefore write down the dividend, 552, and place the divisor, 2, at the left hand. We then pro400-200 ceed to separate the dividend into such parts as may read.140 -70 ily be divided by 2. These parts we find to be 400, 141, :12 and 12. Now 2 is contained in 4, 2 times, and therefore in 400,200 times; 2 in 14, 7 times, and in 140, 70 times, 552-276 and 2 in 12, 6. times; and since these partial dividends, 400+1407-12=552, the whole dividend, the partial quotients, 200+TO+6=276, the whole quotient, or whole number of times 2 is contained in 552. But in practice we separate the dividend into parts no faster than we proceed in the division. Having written down the divi dend and divisor as before, we first seek how many Divis. Divid. Quot. times 2 in 5, and find it to be completely contained 2,552 (276 in it only 2 times. We therefore write 2. for the 4 2 ..highest figure of the quotient, which, since the 5 is 500, is evidently 200; but we leave the place of 15 552 tens and units blank to receive those parts of the 14 proof. quotient which shall be found by dividing the remain ing part of the dividend. We now multiply the 12 divisor 2, by the 2 in the quotient, and write the 12 product, 4, (400) under the 5 hundred in the divi dend. We have thus found that 400 contains 2, 200 times, and by subtracting 4 from 5, we find that there are 1 hundred, 5 tens, and 2 units, remaining to be divided. We next bring down the 5 tens of the dividend, by the side of the'l hundred, making 15 tens, and find 2 in 15, 7 times. But as 15 are so many tens; the 7 musi be tens also, and must occupy the place next below hundreds in the quotient. We now mnltiply the divisor by 7, and..write the product, 14, under the 15. Thus we find that 2 is contained in 15 ţens 70 times, and subtracting 14 from 15, find that I ten remains, to which we bring down the 2 units of the dividend, making 12, which contains 2, 6 times; which 6 we write in the unit's place of the quotient, and multiplying the divisor by it, find the product to bc 12. Thus have we completely exhausted the dividend, and obtained 276 for the quotient as before, 103. 4. A prize of 3349 dollars was shared equally among 16 men, how many dollars did each man receive? We write down the numbers as before, and find 16 16)3349 209- Ans. in 32, 2 timeas we write 2 in the quotient, multiply 32 the divisor by it, and place the product, 32, under 33, the part of the dividend used, and subtracting, 149 find the remainder to be 1, which is 1 hundred. To 144 theo 1 we bring down the 4 tens, making 14 tens; but · as this is less than the divisor, there can be no tens 5 in the quotient. We therefore put a cipher in the ten's place in the quotient, and bring down units of the dividend to thio IX tene, making 149 units, which contain 16 somewhat more than 9 times. Placing 9 in the unit's place of the quotien., and multiplying the divisor by it, the product is 444, which, subtracted from 149, leaves a remainder of 5. The division of these 5 dollars may be denoted by, writing the 5 over 16, with a line between, as in the example. Each man's share then will be 209 dollars and 5 sixteenths of a dollar. (21) The division of any number by another may be denoted by writing the dividend over the divisor, with a line between, and an expression of that kind is called a Vulgar Fraction. 104. 5. A certain cornfield contains 2688 bills of corn planted in rows, which are 56 hills long, how many rows are there? Here, as 56 is not contained in 26, it is necessary to take 56)2688(48 three figures, or 263, for the first partial dividend: but 224 there may be some difficulty in finding how many times the divisor may be had in it. It will, however, soon be seen by 448 inspection, that it cannot be less than 4 times, and by making 448 trial of 4, we find that we cannot have a larger nuinber than that in the ten's place of the quotient, because the remainder, 44, is less than 56, the divisor. In multiplying the divisor by the quotient figure, if the product be greater than the part of the dividend used, the quotient figure is too great; and in subtracting this product, if the remainder exceed the divisor, the quotient figure is too small; and in each case the operation must be repeated until the right figure be found. SIMPLE DIVISION. DEFINITIONS. 105. Simple Division is the method of finding how many times one simple number is contained in another; or, of sepa. rating a simple number into a proposed number of equal parts. The number which is to be divided, is called the dividend; the number by which the dividend is to be divided, is called the divisor; and the number of times the divisor is contained in the dividend, is called the quotient. If there be any thing left after performing the operation, that excess is called the remainder, and is always less than the divisor, and of the same kind as the dividend. RULE. 106. Write the divisor at the left hand of the dividend; find how many times it is contained in as many of the left hand figures of the dividend, as will contain it once, and not more than nine times, and write the result for the highesť figure of the quotient. Multiply the divisor by the quotient figure, and set the product under the part of the dividend used, and subtract it therefrom. Bring down the next figure of the dividend to the right of the remainder, and divide this number as beforc; and so on till the whole is finished. |