when taken from the greater, to diminish it, so that it shall be equal to the less; for you perceive, when we take 14 from the minuend, it is reduced to 32, a number equal to the subtrahend. Consequently subtraction may be made to prove itself. NOTE. We might very properly have offered subtraction, as a proof of addition, had you been acquainted with the rule. It is a saying of the school boy, that it is a poor rule that will not work both ways. We have already proved subtraction by addition; then to make the rule good according to the test of the school boy, we will prove addition by subtraction. Let 46 and 32 be added, the amount is 78; now it is plain, if the numbers which compose this amount, be taken away, nothing will be left; thus, if from 78 you subtract 46, the first number, 32, will remain; and if 32, the other number, be taken away from 32 the remainder, nothing will remain. Now you may treasure up in your mind this fact; that if you subtract from the amount of any sum the several numbers that compose the amount, you destroy it or reduce it to nothing. 2. From 4 2 Take 2 4 In this example, we find that the 4 units of the subtrahend cannot be taken from the 2 in the minuend; therefore we add 10 to the 2, which increases it 1 8 to 12; then we say, 4 from 12 leaves 8, placing it under. Then we add 1 to the 2, the next figure of the subtrahend, which increases it to 3; we then say, 3 from 4 leaves 1, placing it directly under, which leaves 18 for the difference of the 2 given numbers. DEMONSTRATION.-The reason of this operation is plain when we recollect that the 1 ten which we add to the subtrahend is equal to the 10 units which we add to the minuend; because one in a superiour column is equal to 10 in an inferiour column; and adding equal sums to two numbers, or subtracting equal sums, the difference between the two numbers must ever remain the same; thus, 1 5+1 0=2 5—6-1 9 1 2+1 0-2 2-6=1 6 NOTE. Adding 10 to a figure in the minuend before we can subtract, and then one to the next left hand figure of the subtrahend, is by some called borrowing. 3. Again, to show the principles of this rule in a different, though, if possible, in a clearer light, we introduce the following example. From 5 4 (3) DEMONSTRATION.-When no figure in the subtrahend is greater than that directly above it in the minuend, the student finds no difficul28 difference. ty. And even when a figure in the subtrahend is greater than that directly above it in the minuend, the difficulty vanishes, when he properly understands the local value of numbers. In this example our minuend (54) consist of 5 tens and 4 units; and our subtrahend (26) consists of 2 tens and 6 units. Now to take the 6 units of the subtrahend from the 4 units of the minuend is impossible; yet it is evident that the minuend (54) is greater than the subtrahend (26;) therefore, we will resolve our minuend, 5 tens and 4 units, into 4 tens and 14 units, which can be evi"dently done. And now we can take the 6 units of the subtrahend from the 14 units of the minuend and have 8 units remain; we then take the 2 tens of the subtrahend from the 4 tens of the minuend, and have 2 tens left; then our remainder or difference is 2 tens and 8 units, or 28, which is the same. Although further examples may seem unnecessary to illustrate this rule; yet as the following example includes every variety in subtraction, it may not be thought amiss to introduce it here. We begin at the right hand as our rule directs; and since we cannot -take 2 from 1, we add 10 to the 1, which makes 11, and then say, 2 from 11 leaves. 9, placing it directly under. We then add 1 to the next left hand figure of the subtrahend (for reasons already given under the 2d example) which is a cipher (0,) and say 1 from 4 leaves 3, placing it under. Next we say take a cipher (0) from a (0) and a cipher (0) remains. We next say, take 7 from 5 we cannot, but according to our rule, we add 10 to the 5 and then say, 7 from 15 leaves 8, placing it directly under. Next, according to our rule, (which has been explained,) we add 1 to the 8, which makes 9, and then say, 9 from a cipher (0,) we cannot; but when we add 10 to the cipher (0,) we say 9 from 10 leaves 1, placing it under. We now add 1 to the 7, which makes 8, and say, 8 from 2 we cannot subtract; we then add 10 to the 2, which increases it to 12, we then say 8 from 12 leaves 4, placing it under. Lastly, we say 1 from 3 leaves 2, placing it under. We obtain this 1 in the vacant place of the subtrahend at the left hand, on account of our having added 10 to the next right hand figure of the minuend. 5. What is the difference between 478 and 320 ? 6. If 1 be taken from a 1000; what will remain? Ans. 158. Ans. 999. 7. If one be subtracted from a million; what will remain ? Ans. 999,999. 8. What is the difference between one million and ten thousand? Ans. 990,000. 9. At the census taken in 1810, the number of inhabitants in the six N. England states, was 1,471,973; and at the census taken in 1820, the number of inhabitants was 1,659,854 j what was the increase of population in New England between 1810 and 1820 ? Ans. 187,881 increase. 10. At the census in 1810, the number of inhabitants in the State of New-York was 958,066; and on taking the census in 1820, the number of inhabitants had increased to 1,372,812; then what was the increase between 1819 and 1820 ? Ans. 414,746 increase. 11. According to the two preceding sums, how much more was the increase of population in the State of New-York, than in the six New England States between 1810 and 1820 ? Ans. 226,865. 12. The number of square miles in New England is 65,047, and the number of square miles in New-York is 46,000; then how many more square miles are there in New England than in New-York? Ans. 19,047. 13. Newspapers were first printed at Paris in 1651; how many years since, up to the year 1829? Ans. 198 years. 14. Bought 2000 yards of shirting for 466 dollars, and sold 1476 yards for 369 dollars; how many yards have I left, and how many dollars do I want to make up the first cost? Ans. I have 524 yards, and want $97. 15. What number must be subtracted from 2081, that the remainder may be 1104? Ans. 977. 16. America was discovered by Christopher Columbus in 1492; how many years have since elapsed up to the year Ans. 337 years. 17. Union College was incorporated in 1794; how many years since:? 1829 ? 18. Yale College at New-Haven was incorporated in 1700; how many years since? 19. Gen. George Washington died in 1799, aged 67 years; in what year was he born? Ans. 1732. 20. How long since the Declaration of Independence which was declared in 1776-reckoning to the year 1829 ? Ans. 53 SUBTRACTION OF DECIMALS. F Subtraction of decimal or federal money, is exactly the same in the operation of the work, as subtraction of whole numbers. Because ten in an inferiour denomination, is equal to one in the next superiour, through all the denominations. Consequently the are proportion exists as in whole numbers. RULE. Place whole numbers the same as if no decimals were annexed: and decimals, so that tenths stand under tenths, hundredths under hundredths, and thousandths under thousandths, &c. Or, which is the same thing, place dollars under dollars, cents under cents, and mills under mills; so that the separatrix in the subtrahend shall stand directly under that of the minuend; and care must be taken to place the separatrix in the result directly under the separatrix in the given numbers. Here, as in addition of decimals, a dollar is a unit; and dimes, cents, and mills, according to their order, are tenths, hundredths, and thousandths of a dollar; eagles and dollars are expressed in dollars; dimes, cents and mills are expressed in cents and mills; thus, 4 eagles, 8 dollars, 6 dimes, 8 cents, and 3 mills, are expressed 48 dollars, 68 cents and 3 mills, or $48, 68 cts. 3 m. A dollar being a unit, a separatrix must always be placed directly after dollars when any of the inferiour denominations follow; thus, five dollars and sixty-eight cents must be expressed in figures $5,68 cts. NOTE.-Care must be taken to place ciphers in the place of vacant denominations, whenever they occur in either of the given sums, so that the significant figures may stand in their proper places; thus, you should write 1 be placed betllar and 5 mills, $1,00, cts. 5 m. and a separatrix may cents and mills as well as between dollars and cents, for the ease of expressing numbers. 5. A miller bought a quantity of corn. for which he paid $480; ne afterwards sold it for 580 dollars, 68 cents and 5 mills; what did he gain by the sale? Ans. $100, 68 cts. 5m. 6. A merchant borrowed $600, for which he gave his note; he paid at one time $240, at another $150; how much remains due on the note? Ans. $210. 2. Joseph bought 10 oranges for 30 cents, he sold 6 for 24 cents; how many oranges has he left, and how many cents does he want to make up the first cost? Ans. he has 4 oranges left, and wants 6 cents. 8. From one dollar, take one dime and one mill. Ans. ,89 cts. 9 m. 9. From two dollars, take two dimes. 10. From two dimes, take two cents. 11. From seven dollars, take six dimes. 12. From eight dollars and eight dimes, take one dime and one mill. Ans. $8,69 cts. 9 m. 13. From ten dollars, take one dollar, fifty cents and four mills. Ans. $8,49 cts. 6 m. 14. From forty dollars, take forty cents and five mills. Ans. $39,59 cts. 5 m. 15. If one mill be taken from a thousand dollars, what will remain ? Ans. $999,99 cts. 9 m. 16. A gentleman found four bags of money to the amount of $500; the first bag contained $95, the second $130, and the third $101; I wish to know what the fourth contained? T Ans. $174. 17. A clergyman's salary is $1500 a year, and he spends $800,50 cts.; how much of his salary does he lay up? 18. A gentleman bought a horse for unsound, he is willing to lose $12,37 the horse? Äns. $699,50 cts. $100, which proving cts.; how must he sell Ans. $87,624 cts. sold it for $175,121 19. A merchant bought a piece of broadcloth for $205, which proved to be damaged, so that he cents; how much did the merchant lose? 20. On a note of $105,37 cts. if you $42,20 cts. at another $37 84 cts. what remains due? Ans. $29,87 cts. received at one time Ans. $25, 33 cts, QUESTIONS ON SIMPLE SUBTRACTION. NOTE.-The teacher should not wait for the student to go over the 1 |